id sid tid token lemma pos iajs-2928 1 1 ihjpas ihjpas PROPN iajs-2928 1 2 . . PUNCT iajs-2928 2 1 36(1)2023 36(1)2023 NUM iajs-2928 2 2 355 355 NUM iajs-2928 2 3 this this DET iajs-2928 2 4 work work NOUN iajs-2928 2 5 is be AUX iajs-2928 2 6 licensed license VERB iajs-2928 2 7 under under ADP iajs-2928 2 8 a a DET iajs-2928 2 9 creative creative ADJ iajs-2928 2 10 commons common NOUN iajs-2928 2 11 attribution attribution NOUN iajs-2928 2 12 4.0 4.0 NUM iajs-2928 2 13 international international ADJ iajs-2928 2 14 license license NOUN iajs-2928 2 15 quasi quasi NOUN iajs-2928 2 16 semi semi ADJ iajs-2928 2 17 and and CCONJ iajs-2928 2 18 pseudo pseudo NOUN iajs-2928 2 19 semi semi ADJ iajs-2928 2 20 ( ( PUNCT iajs-2928 2 21 𝑝 𝑝 NOUN iajs-2928 2 22 , , PUNCT iajs-2928 2 23 𝐸)-convexity 𝐸)-convexity NOUN iajs-2928 2 24 in in ADP iajs-2928 2 25 non non ADJ iajs-2928 2 26 - - ADJ iajs-2928 2 27 linear linear ADJ iajs-2928 2 28 optimization optimization NOUN iajs-2928 2 29 programming programming NOUN iajs-2928 2 30 abstract abstract NOUN iajs-2928 2 31 the the DET iajs-2928 2 32 class class NOUN iajs-2928 2 33 of of ADP iajs-2928 2 34 quasi quasi NOUN iajs-2928 2 35 semi semi ADJ iajs-2928 2 36 ( ( PUNCT iajs-2928 2 37 𝑝 𝑝 PROPN iajs-2928 2 38 , , PUNCT iajs-2928 2 39 𝐸)-convex 𝐸)-convex NOUN iajs-2928 2 40 functions function NOUN iajs-2928 2 41 and and CCONJ iajs-2928 2 42 pseudo pseudo NOUN iajs-2928 2 43 semi semi ADJ iajs-2928 2 44 ( ( PUNCT iajs-2928 2 45 𝑝 𝑝 NOUN iajs-2928 2 46 , , PUNCT iajs-2928 2 47 𝐸)-convex 𝐸)-convex NUM iajs-2928 2 48 functions function NOUN iajs-2928 2 49 are be AUX iajs-2928 2 50 presented present VERB iajs-2928 2 51 in in ADP iajs-2928 2 52 this this DET iajs-2928 2 53 paper paper NOUN iajs-2928 2 54 by by ADP iajs-2928 2 55 combining combine VERB iajs-2928 2 56 the the DET iajs-2928 2 57 class class NOUN iajs-2928 2 58 of of ADP iajs-2928 2 59 𝑝-convex 𝑝-convex NOUN iajs-2928 2 60 functions function NOUN iajs-2928 2 61 with with ADP iajs-2928 2 62 the the DET iajs-2928 2 63 class class NOUN iajs-2928 2 64 of of ADP iajs-2928 2 65 quasi quasi NOUN iajs-2928 2 66 semi semi ADJ iajs-2928 2 67 𝐸-convex 𝐸-convex PROPN iajs-2928 2 68 functions function NOUN iajs-2928 2 69 and and CCONJ iajs-2928 2 70 pseudo pseudo NOUN iajs-2928 2 71 semi semi ADV iajs-2928 2 72 𝐸-convex 𝐸-convex PROPN iajs-2928 2 73 functions function NOUN iajs-2928 2 74 , , PUNCT iajs-2928 2 75 respectively respectively ADV iajs-2928 2 76 . . PUNCT iajs-2928 3 1 various various ADJ iajs-2928 3 2 non non ADJ iajs-2928 3 3 - - ADJ iajs-2928 3 4 trivial trivial ADJ iajs-2928 3 5 examples example NOUN iajs-2928 3 6 are be AUX iajs-2928 3 7 introduced introduce VERB iajs-2928 3 8 to to PART iajs-2928 3 9 illustrate illustrate VERB iajs-2928 3 10 the the DET iajs-2928 3 11 new new ADJ iajs-2928 3 12 functions function NOUN iajs-2928 3 13 and and CCONJ iajs-2928 3 14 show show VERB iajs-2928 3 15 their their PRON iajs-2928 3 16 relationships relationship NOUN iajs-2928 3 17 with with ADP iajs-2928 3 18 ( ( PUNCT iajs-2928 3 19 𝑝 𝑝 PROPN iajs-2928 3 20 , , PUNCT iajs-2928 3 21 𝐸)convex 𝐸)convex NOUN iajs-2928 3 22 functions function NOUN iajs-2928 3 23 recently recently ADV iajs-2928 3 24 introduced introduce VERB iajs-2928 3 25 in in ADP iajs-2928 3 26 the the DET iajs-2928 3 27 literature literature NOUN iajs-2928 3 28 . . PUNCT iajs-2928 4 1 different different ADJ iajs-2928 4 2 general general ADJ iajs-2928 4 3 properties property NOUN iajs-2928 4 4 and and CCONJ iajs-2928 4 5 characteristics characteristic NOUN iajs-2928 4 6 of of ADP iajs-2928 4 7 this this DET iajs-2928 4 8 class class NOUN iajs-2928 4 9 of of ADP iajs-2928 4 10 functions function NOUN iajs-2928 4 11 are be AUX iajs-2928 4 12 established establish VERB iajs-2928 4 13 . . PUNCT iajs-2928 5 1 in in ADP iajs-2928 5 2 addition addition NOUN iajs-2928 5 3 , , PUNCT iajs-2928 5 4 some some DET iajs-2928 5 5 optimality optimality NOUN iajs-2928 5 6 properties property NOUN iajs-2928 5 7 of of ADP iajs-2928 5 8 generalized generalized ADJ iajs-2928 5 9 non non ADJ iajs-2928 5 10 - - ADJ iajs-2928 5 11 linear linear ADJ iajs-2928 5 12 optimization optimization NOUN iajs-2928 5 13 problems problem NOUN iajs-2928 5 14 are be AUX iajs-2928 5 15 discussed discuss VERB iajs-2928 5 16 . . PUNCT iajs-2928 6 1 in in ADP iajs-2928 6 2 this this DET iajs-2928 6 3 generalized generalize VERB iajs-2928 6 4 optimization optimization NOUN iajs-2928 6 5 problems problem NOUN iajs-2928 6 6 , , PUNCT iajs-2928 6 7 we we PRON iajs-2928 6 8 used use VERB iajs-2928 6 9 , , PUNCT iajs-2928 6 10 as as ADP iajs-2928 6 11 the the DET iajs-2928 6 12 objective objective ADJ iajs-2928 6 13 function function NOUN iajs-2928 6 14 , , PUNCT iajs-2928 6 15 quasi quasi NOUN iajs-2928 6 16 semi semi ADJ iajs-2928 6 17 ( ( PUNCT iajs-2928 6 18 𝑝 𝑝 NOUN iajs-2928 6 19 , , PUNCT iajs-2928 6 20 𝐸)-convex 𝐸)-convex PROPN iajs-2928 6 21 ( ( PUNCT iajs-2928 6 22 respectively respectively ADV iajs-2928 6 23 , , PUNCT iajs-2928 6 24 strictly strictly ADV iajs-2928 6 25 quasi quasi ADJ iajs-2928 6 26 semi semi ADJ iajs-2928 6 27 ( ( PUNCT iajs-2928 6 28 𝑝 𝑝 PROPN iajs-2928 6 29 , , PUNCT iajs-2928 6 30 𝐸)-convex 𝐸)-convex NOUN iajs-2928 6 31 functions function NOUN iajs-2928 6 32 and and CCONJ iajs-2928 6 33 pseudo pseudo NOUN iajs-2928 6 34 semi semi ADJ iajs-2928 6 35 ( ( PUNCT iajs-2928 6 36 𝑝 𝑝 NOUN iajs-2928 6 37 , , PUNCT iajs-2928 6 38 𝐸)-convex 𝐸)-convex NOUN iajs-2928 6 39 functions function NOUN iajs-2928 6 40 ) ) PUNCT iajs-2928 6 41 , , PUNCT iajs-2928 6 42 and and CCONJ iajs-2928 6 43 the the DET iajs-2928 6 44 constraint constraint NOUN iajs-2928 6 45 set set NOUN iajs-2928 6 46 is be AUX iajs-2928 6 47 ( ( PUNCT iajs-2928 6 48 𝑝 𝑝 PROPN iajs-2928 6 49 , , PUNCT iajs-2928 6 50 𝐸)-convex 𝐸)-convex PRON iajs-2928 6 51 set set VERB iajs-2928 6 52 . . PUNCT iajs-2928 7 1 ams am NOUN iajs-2928 7 2 subject subject ADJ iajs-2928 7 3 classification classification NOUN iajs-2928 7 4 : : PUNCT iajs-2928 7 5 46n10 46n10 NUM iajs-2928 7 6 , , PUNCT iajs-2928 7 7 47n10 47n10 NUM iajs-2928 7 8 , , PUNCT iajs-2928 7 9 90c48 90c48 NUM iajs-2928 7 10 , , PUNCT iajs-2928 7 11 90c90 90c90 NUM iajs-2928 7 12 , , PUNCT iajs-2928 7 13 49k27 49k27 NUM iajs-2928 7 14 keywords keyword NOUN iajs-2928 7 15 : : PUNCT iajs-2928 7 16 ( ( PUNCT iajs-2928 7 17 𝑝 𝑝 NOUN iajs-2928 7 18 , , PUNCT iajs-2928 7 19 𝐸)-convex 𝐸)-convex PRON iajs-2928 7 20 set set VERB iajs-2928 7 21 , , PUNCT iajs-2928 7 22 ( ( PUNCT iajs-2928 7 23 𝑝 𝑝 NOUN iajs-2928 7 24 , , PUNCT iajs-2928 7 25 𝐸)-convex 𝐸)-convex PRON iajs-2928 7 26 function function VERB iajs-2928 7 27 , , PUNCT iajs-2928 7 28 quasi quasi NOUN iajs-2928 7 29 semi semi ADJ iajs-2928 7 30 ( ( PUNCT iajs-2928 7 31 𝑝 𝑝 NOUN iajs-2928 7 32 , , PUNCT iajs-2928 7 33 𝐸)-convex 𝐸)-convex NUM iajs-2928 7 34 function function NOUN iajs-2928 7 35 , , PUNCT iajs-2928 7 36 pseudo pseudo NOUN iajs-2928 7 37 semi semi ADJ iajs-2928 7 38 ( ( PUNCT iajs-2928 7 39 𝑝 𝑝 NOUN iajs-2928 7 40 , , PUNCT iajs-2928 7 41 𝐸)-convex 𝐸)-convex PRON iajs-2928 7 42 function function VERB iajs-2928 7 43 1 1 NUM iajs-2928 7 44 . . PUNCT iajs-2928 8 1 introduction introduction NOUN iajs-2928 8 2 and and CCONJ iajs-2928 8 3 preliminaries preliminary NOUN iajs-2928 8 4 generalized generalize VERB iajs-2928 8 5 convexity convexity NOUN iajs-2928 8 6 has have AUX iajs-2928 8 7 drawn draw VERB iajs-2928 8 8 the the DET iajs-2928 8 9 attention attention NOUN iajs-2928 8 10 of of ADP iajs-2928 8 11 many many ADJ iajs-2928 8 12 researchers researcher NOUN iajs-2928 8 13 in in ADP iajs-2928 8 14 recent recent ADJ iajs-2928 8 15 years year NOUN iajs-2928 8 16 due due ADP iajs-2928 8 17 to to ADP iajs-2928 8 18 its its PRON iajs-2928 8 19 vast vast ADJ iajs-2928 8 20 applications application NOUN iajs-2928 8 21 in in ADP iajs-2928 8 22 different different ADJ iajs-2928 8 23 areas area NOUN iajs-2928 8 24 , , PUNCT iajs-2928 8 25 especially especially ADV iajs-2928 8 26 in in ADP iajs-2928 8 27 optimization optimization NOUN iajs-2928 8 28 and and CCONJ iajs-2928 8 29 applied apply VERB iajs-2928 8 30 sciences science NOUN iajs-2928 8 31 ( ( PUNCT iajs-2928 8 32 see see VERB iajs-2928 8 33 e.g. e.g. ADV iajs-2928 8 34 , , PUNCT iajs-2928 8 35 [ [ X iajs-2928 8 36 1][16 1][16 X iajs-2928 8 37 ] ] X iajs-2928 8 38 ) ) PUNCT iajs-2928 8 39 . . PUNCT iajs-2928 9 1 one one NUM iajs-2928 9 2 of of ADP iajs-2928 9 3 the the DET iajs-2928 9 4 well well ADV iajs-2928 9 5 - - PUNCT iajs-2928 9 6 known know VERB iajs-2928 9 7 generalizations generalization NOUN iajs-2928 9 8 of of ADP iajs-2928 9 9 convex convex NOUN iajs-2928 9 10 sets set NOUN iajs-2928 9 11 and and CCONJ iajs-2928 9 12 convex convex NOUN iajs-2928 9 13 functions function NOUN iajs-2928 9 14 is be AUX iajs-2928 9 15 the the DET iajs-2928 9 16 class class NOUN iajs-2928 9 17 of of ADP iajs-2928 9 18 so so ADV iajs-2928 9 19 - - PUNCT iajs-2928 9 20 called call VERB iajs-2928 9 21 𝐸-convex 𝐸-convex PROPN iajs-2928 9 22 sets set NOUN iajs-2928 9 23 and and CCONJ iajs-2928 9 24 𝐸-convex 𝐸-convex PROPN iajs-2928 9 25 functions function NOUN iajs-2928 9 26 introduced introduce VERB iajs-2928 9 27 by by ADP iajs-2928 9 28 [ [ X iajs-2928 9 29 1 1 NUM iajs-2928 9 30 ] ] PUNCT iajs-2928 9 31 where where SCONJ iajs-2928 9 32 mapping mapping NOUN iajs-2928 9 33 𝐸 𝐸 NOUN iajs-2928 9 34 : : PUNCT iajs-2928 9 35 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 9 36 β†’ β†’ PUNCT iajs-2928 9 37 𝑅𝑛 𝑅𝑛 NOUN iajs-2928 9 38 is be AUX iajs-2928 9 39 employed employ VERB iajs-2928 9 40 in in ADP iajs-2928 9 41 this this DET iajs-2928 9 42 type type NOUN iajs-2928 9 43 of of ADP iajs-2928 9 44 the the DET iajs-2928 9 45 generalized generalized ADJ iajs-2928 9 46 convexity convexity NOUN iajs-2928 9 47 . . PUNCT iajs-2928 10 1 due due ADP iajs-2928 10 2 to to ADP iajs-2928 10 3 some some DET iajs-2928 10 4 erroneous erroneous ADJ iajs-2928 10 5 appeared appear VERB iajs-2928 10 6 in in ADP iajs-2928 10 7 youness youness NOUN iajs-2928 10 8 's 's PART iajs-2928 10 9 first first ADJ iajs-2928 10 10 paper paper NOUN iajs-2928 10 11 , , PUNCT iajs-2928 10 12 a a DET iajs-2928 10 13 new new ADJ iajs-2928 10 14 class class NOUN iajs-2928 10 15 of of ADP iajs-2928 10 16 𝐸-functions 𝐸-functions PROPN iajs-2928 10 17 called call VERB iajs-2928 10 18 semi semi ADV iajs-2928 10 19 𝐸-convex 𝐸-convex PROPN iajs-2928 10 20 functions function NOUN iajs-2928 10 21 , , PUNCT iajs-2928 10 22 is be AUX iajs-2928 10 23 introduced introduce VERB iajs-2928 10 24 by by ADP iajs-2928 10 25 [ [ X iajs-2928 10 26 2 2 NUM iajs-2928 10 27 ] ] PUNCT iajs-2928 10 28 , , PUNCT iajs-2928 10 29 and and CCONJ iajs-2928 10 30 its its PRON iajs-2928 10 31 properties property NOUN iajs-2928 10 32 is be AUX iajs-2928 10 33 studied study VERB iajs-2928 10 34 [ [ PUNCT iajs-2928 10 35 3 3 NUM iajs-2928 10 36 ] ] PUNCT iajs-2928 10 37 . . PUNCT iajs-2928 11 1 this this DET iajs-2928 11 2 class class NOUN iajs-2928 11 3 also also ADV iajs-2928 11 4 includes include VERB iajs-2928 11 5 quasi quasi ADJ iajs-2928 11 6 - - ADJ iajs-2928 11 7 semi semi ADJ iajs-2928 11 8 𝐸-convex 𝐸-convex PROPN iajs-2928 11 9 and and CCONJ iajs-2928 11 10 pseudo pseudo NOUN iajs-2928 11 11 semi semi ADV iajs-2928 11 12 𝐸doi.org/10.30526/36.1 𝐸doi.org/10.30526/36.1 NOUN iajs-2928 11 13 . . PUNCT iajs-2928 12 1 2928 2928 NUM iajs-2928 12 2 article article NOUN iajs-2928 12 3 history history NOUN iajs-2928 12 4 : : PUNCT iajs-2928 12 5 received receive VERB iajs-2928 12 6 18 18 NUM iajs-2928 12 7 july july PROPN iajs-2928 12 8 2022 2022 NUM iajs-2928 12 9 , , PUNCT iajs-2928 12 10 accepted accept VERB iajs-2928 12 11 31 31 NUM iajs-2928 12 12 august august PROPN iajs-2928 12 13 2022 2022 NUM iajs-2928 12 14 , , PUNCT iajs-2928 12 15 published publish VERB iajs-2928 12 16 in in ADP iajs-2928 12 17 january january PROPN iajs-2928 12 18 2023 2023 NUM iajs-2928 12 19 . . PUNCT iajs-2928 13 1 ibn ibn PROPN iajs-2928 13 2 al al PROPN iajs-2928 13 3 - - PUNCT iajs-2928 13 4 haitham haitham PROPN iajs-2928 13 5 journal journal PROPN iajs-2928 13 6 for for ADP iajs-2928 13 7 pure pure ADJ iajs-2928 13 8 and and CCONJ iajs-2928 13 9 applied applied ADJ iajs-2928 13 10 sciences sciences PROPN iajs-2928 13 11 journal journal PROPN iajs-2928 13 12 homepage homepage NOUN iajs-2928 13 13 : : PUNCT iajs-2928 13 14 jih.uobaghdad.edu.iq jih.uobaghdad.edu.iq NOUN iajs-2928 13 15 revan revan PROPN iajs-2928 13 16 imad imad PROPN iajs-2928 13 17 hazim hazim PROPN iajs-2928 13 18 department department PROPN iajs-2928 13 19 of of ADP iajs-2928 13 20 mathematics mathematics PROPN iajs-2928 13 21 , , PUNCT iajs-2928 13 22 college college NOUN iajs-2928 13 23 of of ADP iajs-2928 13 24 education education NOUN iajs-2928 13 25 for for ADP iajs-2928 13 26 pure pure ADJ iajs-2928 13 27 sciences science NOUN iajs-2928 13 28 , , PUNCT iajs-2928 13 29 ibn ibn PROPN iajs-2928 13 30 al al PROPN iajs-2928 13 31 – – PUNCT iajs-2928 13 32 haitham/ haitham/ NUM iajs-2928 13 33 university university NOUN iajs-2928 13 34 of of ADP iajs-2928 13 35 baghdadiraq baghdadiraq PROPN iajs-2928 13 36 . . PUNCT iajs-2928 14 1 van.emad1203a@ihcoedu.uobaghdad.edu.iqri van.emad1203a@ihcoedu.uobaghdad.edu.iqri PROPN iajs-2928 14 2 saba saba PROPN iajs-2928 14 3 naser naser PROPN iajs-2928 14 4 majeed majeed PROPN iajs-2928 14 5 department department PROPN iajs-2928 14 6 of of ADP iajs-2928 14 7 mathematics mathematics PROPN iajs-2928 14 8 , , PUNCT iajs-2928 14 9 college college NOUN iajs-2928 14 10 of of ADP iajs-2928 14 11 education education NOUN iajs-2928 14 12 for for ADP iajs-2928 14 13 pure pure ADJ iajs-2928 14 14 sciences science NOUN iajs-2928 14 15 , , PUNCT iajs-2928 14 16 ibn ibn PROPN iajs-2928 14 17 al al PROPN iajs-2928 14 18 – – PUNCT iajs-2928 14 19 haitham/ haitham/ NUM iajs-2928 14 20 university university NOUN iajs-2928 14 21 of of ADP iajs-2928 14 22 baghdadiraq baghdadiraq PROPN iajs-2928 14 23 . . PUNCT iajs-2928 15 1 saba.n.m@ihcoedu.uobaghdad.edu.iq saba.n.m@ihcoedu.uobaghdad.edu.iq PROPN iajs-2928 15 2 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-2928 15 3 mailto:rivan.emad1203a@ihcoedu.uobaghdad.edu.iq mailto:rivan.emad1203a@ihcoedu.uobaghdad.edu.iq NOUN iajs-2928 15 4 mailto mailto PROPN iajs-2928 15 5 : : PUNCT iajs-2928 15 6 saba.n.m@ihcoedu.uobaghdad.edu.iq2 saba.n.m@ihcoedu.uobaghdad.edu.iq2 ADJ iajs-2928 15 7 mailto mailto PROPN iajs-2928 15 8 : : PUNCT iajs-2928 15 9 saba.n.m@ihcoedu.uobaghdad.edu.iq2 saba.n.m@ihcoedu.uobaghdad.edu.iq2 ADJ iajs-2928 15 10 ihjpas ihjpas PROPN iajs-2928 15 11 . . PUNCT iajs-2928 16 1 36(1)2023 36(1)2023 NUM iajs-2928 16 2 356 356 NUM iajs-2928 16 3 convex convex NOUN iajs-2928 16 4 functions function NOUN iajs-2928 16 5 . . PUNCT iajs-2928 17 1 youness youness NOUN iajs-2928 17 2 motivated motivated ADJ iajs-2928 17 3 other other ADJ iajs-2928 17 4 researchers researcher NOUN iajs-2928 17 5 to to PART iajs-2928 17 6 extend extend VERB iajs-2928 17 7 some some DET iajs-2928 17 8 concepts concept NOUN iajs-2928 17 9 from from ADP iajs-2928 17 10 convex convex ADJ iajs-2928 17 11 analysis analysis NOUN iajs-2928 17 12 into into ADP iajs-2928 17 13 𝐸-convexity 𝐸-convexity PROPN iajs-2928 17 14 and and CCONJ iajs-2928 17 15 apply apply VERB iajs-2928 17 16 this this DET iajs-2928 17 17 concept concept NOUN iajs-2928 17 18 to to ADP iajs-2928 17 19 optimization optimization NOUN iajs-2928 17 20 problems problem NOUN iajs-2928 17 21 ( ( PUNCT iajs-2928 17 22 see see VERB iajs-2928 17 23 , , PUNCT iajs-2928 17 24 [ [ X iajs-2928 17 25 4 4 NUM iajs-2928 17 26 ] ] PUNCT iajs-2928 17 27 , , PUNCT iajs-2928 18 1 [ [ X iajs-2928 18 2 5 5 NUM iajs-2928 18 3 ] ] PUNCT iajs-2928 18 4 , , PUNCT iajs-2928 18 5 [ [ X iajs-2928 18 6 6 6 NUM iajs-2928 18 7 ] ] PUNCT iajs-2928 18 8 , , PUNCT iajs-2928 18 9 and and CCONJ iajs-2928 18 10 references reference NOUN iajs-2928 18 11 therein therein ADV iajs-2928 18 12 ) ) PUNCT iajs-2928 18 13 . . PUNCT iajs-2928 19 1 another another DET iajs-2928 19 2 important important ADJ iajs-2928 19 3 recent recent ADJ iajs-2928 19 4 generalization generalization NOUN iajs-2928 19 5 of of ADP iajs-2928 19 6 convex convex NOUN iajs-2928 19 7 sets set NOUN iajs-2928 19 8 and and CCONJ iajs-2928 19 9 functions function NOUN iajs-2928 19 10 is be AUX iajs-2928 19 11 𝑝convex 𝑝convex NOUN iajs-2928 19 12 sets set NOUN iajs-2928 19 13 [ [ X iajs-2928 19 14 7 7 NUM iajs-2928 19 15 ] ] PUNCT iajs-2928 19 16 and and CCONJ iajs-2928 19 17 𝑝-convex 𝑝-convex NOUN iajs-2928 19 18 functions function NOUN iajs-2928 19 19 [ [ X iajs-2928 19 20 8 8 NUM iajs-2928 19 21 ] ] PUNCT iajs-2928 19 22 , , PUNCT iajs-2928 19 23 affecting affect VERB iajs-2928 19 24 the the DET iajs-2928 19 25 actual actual ADJ iajs-2928 19 26 number number NOUN iajs-2928 19 27 𝑝 𝑝 ADP iajs-2928 19 28 ∈ ∈ PROPN iajs-2928 19 29 ( ( PUNCT iajs-2928 19 30 0,1 0,1 NOUN iajs-2928 19 31 ] ] PUNCT iajs-2928 19 32 . . PUNCT iajs-2928 20 1 very very ADV iajs-2928 20 2 recently recently ADV iajs-2928 20 3 , , PUNCT iajs-2928 20 4 [ [ X iajs-2928 20 5 9 9 NUM iajs-2928 20 6 ] ] PUNCT iajs-2928 20 7 presented present VERB iajs-2928 20 8 the the DET iajs-2928 20 9 class class NOUN iajs-2928 20 10 of of ADP iajs-2928 20 11 ( ( PUNCT iajs-2928 20 12 𝑝 𝑝 PROPN iajs-2928 20 13 , , PUNCT iajs-2928 20 14 𝐸 𝐸 ADJ iajs-2928 20 15 ) ) PUNCT iajs-2928 20 16 -convex -convex NOUN iajs-2928 20 17 sets set NOUN iajs-2928 20 18 and and CCONJ iajs-2928 20 19 ( ( PUNCT iajs-2928 20 20 𝑝 𝑝 NOUN iajs-2928 20 21 , , PUNCT iajs-2928 20 22 𝐸)-convex 𝐸)-convex NOUN iajs-2928 20 23 functions function NOUN iajs-2928 20 24 by by ADP iajs-2928 20 25 combining combine VERB iajs-2928 20 26 𝐸-convex 𝐸-convex PROPN iajs-2928 20 27 sets set NOUN iajs-2928 20 28 ( ( PUNCT iajs-2928 20 29 respectively respectively ADV iajs-2928 20 30 , , PUNCT iajs-2928 20 31 𝐸-convex 𝐸-convex PROPN iajs-2928 20 32 functions function NOUN iajs-2928 20 33 ) ) PUNCT iajs-2928 20 34 with with ADP iajs-2928 20 35 𝑝-convex 𝑝-convex NOUN iajs-2928 20 36 sets set NOUN iajs-2928 20 37 ( ( PUNCT iajs-2928 20 38 respectively respectively ADV iajs-2928 20 39 , , PUNCT iajs-2928 20 40 𝑝-convex 𝑝-convex NOUN iajs-2928 20 41 functions function NOUN iajs-2928 20 42 ) ) PUNCT iajs-2928 20 43 . . PUNCT iajs-2928 21 1 inspired inspire VERB iajs-2928 21 2 by by ADP iajs-2928 21 3 the the DET iajs-2928 21 4 above above ADJ iajs-2928 21 5 research research NOUN iajs-2928 21 6 works work NOUN iajs-2928 21 7 and and CCONJ iajs-2928 21 8 due due ADP iajs-2928 21 9 to to ADP iajs-2928 21 10 the the DET iajs-2928 21 11 importance importance NOUN iajs-2928 21 12 of of ADP iajs-2928 21 13 studying study VERB iajs-2928 21 14 non non ADJ iajs-2928 21 15 - - ADJ iajs-2928 21 16 convex convex ADJ iajs-2928 21 17 functions function NOUN iajs-2928 21 18 close close ADJ iajs-2928 21 19 to to ADP iajs-2928 21 20 the the DET iajs-2928 21 21 convex convex NOUN iajs-2928 21 22 in in ADP iajs-2928 21 23 some some DET iajs-2928 21 24 sense sense NOUN iajs-2928 21 25 , , PUNCT iajs-2928 21 26 the the DET iajs-2928 21 27 class class NOUN iajs-2928 21 28 of of ADP iajs-2928 21 29 quasi quasi NOUN iajs-2928 21 30 semi semi ADJ iajs-2928 21 31 ( ( PUNCT iajs-2928 21 32 𝑝 𝑝 PROPN iajs-2928 21 33 , , PUNCT iajs-2928 21 34 𝐸)-convex 𝐸)-convex NOUN iajs-2928 21 35 functions function NOUN iajs-2928 21 36 and and CCONJ iajs-2928 21 37 pseudo pseudo NOUN iajs-2928 21 38 semi semi ADJ iajs-2928 21 39 ( ( PUNCT iajs-2928 21 40 𝑝 𝑝 NOUN iajs-2928 21 41 , , PUNCT iajs-2928 21 42 𝐸)-convex 𝐸)-convex NOUN iajs-2928 21 43 functions function NOUN iajs-2928 21 44 is be AUX iajs-2928 21 45 introduced introduce VERB iajs-2928 21 46 by by ADP iajs-2928 21 47 combining combine VERB iajs-2928 21 48 𝑝-convex 𝑝-convex NOUN iajs-2928 21 49 functions function NOUN iajs-2928 21 50 with with ADP iajs-2928 21 51 quasi quasi NOUN iajs-2928 21 52 semi semi ADV iajs-2928 21 53 𝐸-convex 𝐸-convex PROPN iajs-2928 21 54 and and CCONJ iajs-2928 21 55 pseudo pseudo NOUN iajs-2928 21 56 semi semi ADV iajs-2928 21 57 𝐸-convex 𝐸-convex PROPN iajs-2928 21 58 functions function NOUN iajs-2928 21 59 , , PUNCT iajs-2928 21 60 respectively respectively ADV iajs-2928 21 61 . . PUNCT iajs-2928 22 1 these these DET iajs-2928 22 2 non non ADJ iajs-2928 22 3 - - ADJ iajs-2928 22 4 convex convex ADJ iajs-2928 22 5 functions function NOUN iajs-2928 22 6 enrich enrich VERB iajs-2928 22 7 the the DET iajs-2928 22 8 study study NOUN iajs-2928 22 9 of of ADP iajs-2928 22 10 many many ADJ iajs-2928 22 11 real real ADJ iajs-2928 22 12 - - PUNCT iajs-2928 22 13 life life NOUN iajs-2928 22 14 problems problem NOUN iajs-2928 22 15 which which PRON iajs-2928 22 16 are be AUX iajs-2928 22 17 non non ADJ iajs-2928 22 18 - - ADJ iajs-2928 22 19 convex convex ADJ iajs-2928 22 20 in in ADP iajs-2928 22 21 nature nature NOUN iajs-2928 22 22 by by ADP iajs-2928 22 23 modeling model VERB iajs-2928 22 24 them they PRON iajs-2928 22 25 as as ADP iajs-2928 22 26 optimization optimization NOUN iajs-2928 22 27 problems problem NOUN iajs-2928 22 28 that that PRON iajs-2928 22 29 are be AUX iajs-2928 22 30 close close ADJ iajs-2928 22 31 to to PART iajs-2928 22 32 convex convex VERB iajs-2928 22 33 problems problem NOUN iajs-2928 22 34 . . PUNCT iajs-2928 23 1 the the DET iajs-2928 23 2 paper paper NOUN iajs-2928 23 3 is be AUX iajs-2928 23 4 presented present VERB iajs-2928 23 5 as as SCONJ iajs-2928 23 6 follows follow VERB iajs-2928 23 7 . . PUNCT iajs-2928 24 1 the the DET iajs-2928 24 2 rest rest NOUN iajs-2928 24 3 of of ADP iajs-2928 24 4 this this DET iajs-2928 24 5 section section NOUN iajs-2928 24 6 contains contain VERB iajs-2928 24 7 preliminary preliminary ADJ iajs-2928 24 8 material material NOUN iajs-2928 24 9 that that PRON iajs-2928 24 10 makes make VERB iajs-2928 24 11 this this DET iajs-2928 24 12 work work NOUN iajs-2928 24 13 self self NOUN iajs-2928 24 14 - - PUNCT iajs-2928 24 15 contained contain VERB iajs-2928 24 16 . . PUNCT iajs-2928 25 1 in in ADP iajs-2928 25 2 section section NOUN iajs-2928 25 3 2 2 NUM iajs-2928 25 4 , , PUNCT iajs-2928 25 5 the the DET iajs-2928 25 6 definitions definition NOUN iajs-2928 25 7 of of ADP iajs-2928 25 8 quasi quasi NOUN iajs-2928 25 9 semi semi ADJ iajs-2928 25 10 ( ( PUNCT iajs-2928 25 11 𝑝 𝑝 NOUN iajs-2928 25 12 , , PUNCT iajs-2928 25 13 𝐸)-convex 𝐸)-convex NUM iajs-2928 25 14 and and CCONJ iajs-2928 25 15 pseudo pseudo NOUN iajs-2928 25 16 semi semi ADJ iajs-2928 25 17 ( ( PUNCT iajs-2928 25 18 𝑝 𝑝 NOUN iajs-2928 25 19 , , PUNCT iajs-2928 25 20 𝐸)-convex 𝐸)-convex NOUN iajs-2928 25 21 functions function NOUN iajs-2928 25 22 are be AUX iajs-2928 25 23 presented present VERB iajs-2928 25 24 , , PUNCT iajs-2928 25 25 and and CCONJ iajs-2928 25 26 various various ADJ iajs-2928 25 27 examples example NOUN iajs-2928 25 28 and and CCONJ iajs-2928 25 29 relations relation NOUN iajs-2928 25 30 related relate VERB iajs-2928 25 31 to to ADP iajs-2928 25 32 the the DET iajs-2928 25 33 new new ADJ iajs-2928 25 34 functions function NOUN iajs-2928 25 35 with with ADP iajs-2928 25 36 ( ( PUNCT iajs-2928 25 37 𝑝 𝑝 NOUN iajs-2928 25 38 , , PUNCT iajs-2928 25 39 𝐸)-convex 𝐸)-convex NUM iajs-2928 25 40 functions function NOUN iajs-2928 25 41 are be AUX iajs-2928 25 42 provided provide VERB iajs-2928 25 43 . . PUNCT iajs-2928 26 1 in in ADP iajs-2928 26 2 section section NOUN iajs-2928 26 3 3 3 NUM iajs-2928 26 4 , , PUNCT iajs-2928 26 5 we we PRON iajs-2928 26 6 provide provide VERB iajs-2928 26 7 different different ADJ iajs-2928 26 8 properties property NOUN iajs-2928 26 9 of of ADP iajs-2928 26 10 quasi quasi NOUN iajs-2928 26 11 semi semi ADJ iajs-2928 26 12 ( ( PUNCT iajs-2928 26 13 𝑝 𝑝 NOUN iajs-2928 26 14 , , PUNCT iajs-2928 26 15 𝐸)-convex 𝐸)-convex NUM iajs-2928 26 16 and and CCONJ iajs-2928 26 17 pseudo pseudo NOUN iajs-2928 26 18 semi semi ADJ iajs-2928 26 19 ( ( PUNCT iajs-2928 26 20 𝑝 𝑝 NOUN iajs-2928 26 21 , , PUNCT iajs-2928 26 22 𝐸)-convex 𝐸)-convex NOUN iajs-2928 26 23 functions function NOUN iajs-2928 26 24 . . PUNCT iajs-2928 27 1 section section NOUN iajs-2928 27 2 4 4 NUM iajs-2928 27 3 is be AUX iajs-2928 27 4 specified specify VERB iajs-2928 27 5 to to PART iajs-2928 27 6 study study VERB iajs-2928 27 7 some some DET iajs-2928 27 8 optimality optimality NOUN iajs-2928 27 9 properties property NOUN iajs-2928 27 10 of of ADP iajs-2928 27 11 non non ADJ iajs-2928 27 12 - - ADJ iajs-2928 27 13 linear linear ADJ iajs-2928 27 14 optimization optimization NOUN iajs-2928 27 15 problems problem NOUN iajs-2928 27 16 in in ADP iajs-2928 27 17 which which PRON iajs-2928 27 18 the the DET iajs-2928 27 19 objective objective ADJ iajs-2928 27 20 function function NOUN iajs-2928 27 21 is be AUX iajs-2928 27 22 quasi quasi NOUN iajs-2928 27 23 semi semi ADJ iajs-2928 27 24 ( ( PUNCT iajs-2928 27 25 𝑝 𝑝 NOUN iajs-2928 27 26 , , PUNCT iajs-2928 27 27 𝐸)-convex 𝐸)-convex NUM iajs-2928 27 28 or or CCONJ iajs-2928 27 29 pseudo pseudo NOUN iajs-2928 27 30 semi semi ADJ iajs-2928 27 31 ( ( PUNCT iajs-2928 27 32 𝑝 𝑝 NOUN iajs-2928 27 33 , , PUNCT iajs-2928 27 34 𝐸)-convex 𝐸)-convex NOUN iajs-2928 27 35 functions function NOUN iajs-2928 27 36 and and CCONJ iajs-2928 27 37 the the DET iajs-2928 27 38 constraint constraint NOUN iajs-2928 27 39 set set NOUN iajs-2928 27 40 is be AUX iajs-2928 27 41 ( ( PUNCT iajs-2928 27 42 𝑝 𝑝 PROPN iajs-2928 27 43 , , PUNCT iajs-2928 27 44 𝐸)-convex 𝐸)-convex PRON iajs-2928 27 45 set set VERB iajs-2928 27 46 . . PUNCT iajs-2928 28 1 in in ADP iajs-2928 28 2 all all DET iajs-2928 28 3 the the DET iajs-2928 28 4 definitions definition NOUN iajs-2928 28 5 and and CCONJ iajs-2928 28 6 results result NOUN iajs-2928 28 7 throughout throughout ADP iajs-2928 28 8 this this DET iajs-2928 28 9 paper paper NOUN iajs-2928 28 10 , , PUNCT iajs-2928 28 11 let let VERB iajs-2928 28 12 𝑝 𝑝 PRON iajs-2928 28 13 ∈ ∈ PROPN iajs-2928 28 14 ( ( PUNCT iajs-2928 28 15 0,1 0,1 NOUN iajs-2928 28 16 ] ] PUNCT iajs-2928 28 17 and and CCONJ iajs-2928 28 18 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 28 19 is be AUX iajs-2928 28 20 the the DET iajs-2928 28 21 𝑛-dimensional 𝑛-dimensional PROPN iajs-2928 28 22 euclidean euclidean ADJ iajs-2928 28 23 space space NOUN iajs-2928 28 24 . . PUNCT iajs-2928 29 1 assume assume VERB iajs-2928 29 2 that that SCONJ iajs-2928 29 3 𝐴 𝐴 PROPN iajs-2928 29 4 is be AUX iajs-2928 29 5 a a DET iajs-2928 29 6 non non ADJ iajs-2928 29 7 - - ADJ iajs-2928 29 8 empty empty ADJ iajs-2928 29 9 subset subset NOUN iajs-2928 29 10 of of ADP iajs-2928 29 11 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 29 12 , , PUNCT iajs-2928 29 13 𝑓 𝑓 X iajs-2928 29 14 : : PUNCT iajs-2928 29 15 𝐴 𝐴 PROPN iajs-2928 29 16 βŠ† βŠ† NUM iajs-2928 29 17 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 29 18 ⟢ ⟢ NOUN iajs-2928 29 19 𝑅 𝑅 PROPN iajs-2928 29 20 be be AUX iajs-2928 29 21 a a DET iajs-2928 29 22 function function NOUN iajs-2928 29 23 , , PUNCT iajs-2928 29 24 and and CCONJ iajs-2928 29 25 𝐸 𝐸 NOUN iajs-2928 29 26 : : PUNCT iajs-2928 29 27 𝑅𝑛 𝑅𝑛 NOUN iajs-2928 29 28 β†’ β†’ PUNCT iajs-2928 29 29 𝑅𝑛 𝑅𝑛 NOUN iajs-2928 29 30 is be AUX iajs-2928 29 31 a a DET iajs-2928 29 32 given give VERB iajs-2928 29 33 mapping mapping NOUN iajs-2928 29 34 . . PUNCT iajs-2928 30 1 let let VERB iajs-2928 30 2 us we PRON iajs-2928 30 3 now now ADV iajs-2928 30 4 recall recall VERB iajs-2928 30 5 the the DET iajs-2928 30 6 concepts concept NOUN iajs-2928 30 7 related relate VERB iajs-2928 30 8 to to ADP iajs-2928 30 9 𝐸-convex 𝐸-convex PROPN iajs-2928 30 10 set set NOUN iajs-2928 30 11 ( ( PUNCT iajs-2928 30 12 respectively respectively ADV iajs-2928 30 13 , , PUNCT iajs-2928 30 14 𝐸-convex 𝐸-convex PROPN iajs-2928 30 15 function function NOUN iajs-2928 30 16 ) ) PUNCT iajs-2928 30 17 and and CCONJ iajs-2928 30 18 𝑝-convex 𝑝-convex NOUN iajs-2928 30 19 set set NOUN iajs-2928 30 20 and and CCONJ iajs-2928 30 21 function function NOUN iajs-2928 30 22 . . PUNCT iajs-2928 31 1 definition definition NOUN iajs-2928 31 2 1.1 1.1 NUM iajs-2928 31 3 . . PUNCT iajs-2928 32 1 [ [ X iajs-2928 32 2 1 1 NUM iajs-2928 32 3 ] ] PUNCT iajs-2928 32 4 , , PUNCT iajs-2928 32 5 [ [ X iajs-2928 32 6 7 7 X iajs-2928 32 7 ] ] PUNCT iajs-2928 32 8 for for ADP iajs-2928 32 9 any any DET iajs-2928 32 10 π‘₯ π‘₯ PROPN iajs-2928 32 11 , , PUNCT iajs-2928 32 12 𝑦 𝑦 PROPN iajs-2928 32 13 ∈ ∈ PROPN iajs-2928 32 14 𝐴 𝐴 PROPN iajs-2928 32 15 , , PUNCT iajs-2928 32 16 π‘Ÿ π‘Ÿ X iajs-2928 32 17 , , PUNCT iajs-2928 32 18 𝑠 𝑠 PROPN iajs-2928 32 19 ∈ ∈ PROPN iajs-2928 33 1 [ [ X iajs-2928 33 2 0,1 0,1 NUM iajs-2928 33 3 ] ] PUNCT iajs-2928 33 4 , , PUNCT iajs-2928 33 5 and and CCONJ iajs-2928 33 6 𝑝 𝑝 X iajs-2928 33 7 ∈ ∈ PROPN iajs-2928 33 8 ( ( PUNCT iajs-2928 33 9 0,1 0,1 NOUN iajs-2928 33 10 ] ] PUNCT iajs-2928 34 1 such such ADJ iajs-2928 34 2 that that SCONJ iajs-2928 34 3 π‘Ÿπ‘ π‘Ÿπ‘ ADP iajs-2928 34 4 + + NOUN iajs-2928 34 5 𝑠𝑝 𝑠𝑝 NOUN iajs-2928 34 6 = = SYM iajs-2928 34 7 1 1 NUM iajs-2928 34 8 . . PUNCT iajs-2928 35 1 the the DET iajs-2928 35 2 set set NOUN iajs-2928 35 3 𝐴 𝐴 PROPN iajs-2928 35 4 is be AUX iajs-2928 35 5 named name VERB iajs-2928 35 6 as as ADP iajs-2928 35 7 1 1 NUM iajs-2928 35 8 . . PUNCT iajs-2928 36 1 𝐸-convex 𝐸-convex PROPN iajs-2928 36 2 if if SCONJ iajs-2928 36 3 π‘ŸπΈ(π‘₯ π‘ŸπΈ(π‘₯ NOUN iajs-2928 36 4 ) ) PUNCT iajs-2928 36 5 + + CCONJ iajs-2928 36 6 ( ( PUNCT iajs-2928 36 7 1 1 NUM iajs-2928 36 8 βˆ’ βˆ’ PROPN iajs-2928 36 9 π‘Ÿ)𝐸(𝑦 π‘Ÿ)𝐸(𝑦 NOUN iajs-2928 36 10 ) ) PUNCT iajs-2928 36 11 ∈ ∈ PROPN iajs-2928 36 12 𝐴. 𝐴. NOUN iajs-2928 36 13 2 2 X iajs-2928 36 14 . . X iajs-2928 36 15 𝑝-convex 𝑝-convex PROPN iajs-2928 37 1 if if SCONJ iajs-2928 37 2 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PRON iajs-2928 37 3 + + NUM iajs-2928 37 4 𝑠𝑦 𝑠𝑦 ADP iajs-2928 37 5 ∈ ∈ NOUN iajs-2928 37 6 𝐴. 𝐴. PROPN iajs-2928 37 7 definition definition NOUN iajs-2928 37 8 1.2 1.2 NUM iajs-2928 38 1 . . PUNCT iajs-2928 39 1 [ [ X iajs-2928 39 2 1 1 NUM iajs-2928 39 3 ] ] PUNCT iajs-2928 39 4 , , PUNCT iajs-2928 39 5 [ [ X iajs-2928 39 6 2 2 NUM iajs-2928 39 7 ] ] PUNCT iajs-2928 39 8 , , PUNCT iajs-2928 39 9 [ [ X iajs-2928 39 10 8 8 NUM iajs-2928 39 11 ] ] PUNCT iajs-2928 39 12 for for ADP iajs-2928 39 13 any any DET iajs-2928 39 14 π‘₯ π‘₯ PROPN iajs-2928 39 15 , , PUNCT iajs-2928 39 16 𝑦 𝑦 PROPN iajs-2928 39 17 ∈ ∈ PROPN iajs-2928 39 18 𝐴 𝐴 PROPN iajs-2928 39 19 , , PUNCT iajs-2928 39 20 π‘Ÿ π‘Ÿ X iajs-2928 39 21 , , PUNCT iajs-2928 39 22 𝑠 𝑠 PROPN iajs-2928 39 23 ∈ ∈ PROPN iajs-2928 40 1 [ [ X iajs-2928 40 2 0,1 0,1 NUM iajs-2928 40 3 ] ] PUNCT iajs-2928 40 4 , , PUNCT iajs-2928 40 5 and and CCONJ iajs-2928 40 6 𝑝 𝑝 X iajs-2928 40 7 ∈ ∈ PROPN iajs-2928 40 8 ( ( PUNCT iajs-2928 40 9 0,1 0,1 NOUN iajs-2928 40 10 ] ] PUNCT iajs-2928 41 1 such such ADJ iajs-2928 41 2 that that SCONJ iajs-2928 41 3 π‘Ÿπ‘ π‘Ÿπ‘ ADP iajs-2928 41 4 + + NOUN iajs-2928 41 5 𝑠𝑝 𝑠𝑝 NOUN iajs-2928 41 6 = = SYM iajs-2928 41 7 1 1 X iajs-2928 41 8 . . PUNCT iajs-2928 42 1 the the DET iajs-2928 42 2 function function NOUN iajs-2928 42 3 𝑓 𝑓 PROPN iajs-2928 42 4 is be AUX iajs-2928 42 5 named name VERB iajs-2928 42 6 as as ADP iajs-2928 42 7 1 1 NUM iajs-2928 42 8 . . PUNCT iajs-2928 43 1 𝐸-π‘π‘œπ‘›π‘£π‘’π‘₯ 𝐸-π‘π‘œπ‘›π‘£π‘’π‘₯ PROPN iajs-2928 43 2 if if SCONJ iajs-2928 43 3 𝐴 𝐴 PROPN iajs-2928 43 4 is be AUX iajs-2928 43 5 𝐸-convex 𝐸-convex PROPN iajs-2928 43 6 set set NOUN iajs-2928 43 7 and and CCONJ iajs-2928 43 8 𝑓(π‘ŸπΈ(π‘₯ 𝑓(π‘ŸπΈ(π‘₯ NOUN iajs-2928 43 9 ) ) PUNCT iajs-2928 44 1 + + CCONJ iajs-2928 44 2 ( ( PUNCT iajs-2928 44 3 1 1 NUM iajs-2928 44 4 βˆ’ βˆ’ PROPN iajs-2928 44 5 π‘Ÿ)𝐸(𝑦 π‘Ÿ)𝐸(𝑦 NOUN iajs-2928 44 6 ) ) PUNCT iajs-2928 44 7 ) ) PUNCT iajs-2928 44 8 ≀ ≀ NUM iajs-2928 44 9 π‘Ÿπ‘“(𝐸(π‘₯ π‘Ÿπ‘“(𝐸(π‘₯ NOUN iajs-2928 44 10 ) ) PUNCT iajs-2928 44 11 ) ) PUNCT iajs-2928 45 1 + + CCONJ iajs-2928 45 2 ( ( PUNCT iajs-2928 45 3 1 1 NUM iajs-2928 45 4 βˆ’ βˆ’ NOUN iajs-2928 45 5 π‘Ÿ)𝑓(𝐸(𝑦 π‘Ÿ)𝑓(𝐸(𝑦 NUM iajs-2928 45 6 ) ) PUNCT iajs-2928 45 7 ) ) PUNCT iajs-2928 45 8 . . PUNCT iajs-2928 46 1 2 2 X iajs-2928 46 2 . . X iajs-2928 46 3 quasi quasi NOUN iajs-2928 46 4 semi semi PROPN iajs-2928 46 5 𝐸-π‘π‘œπ‘›π‘£π‘’π‘₯ 𝐸-π‘π‘œπ‘›π‘£π‘’π‘₯ PROPN iajs-2928 46 6 if if SCONJ iajs-2928 46 7 𝐴 𝐴 PROPN iajs-2928 46 8 is be AUX iajs-2928 46 9 𝐸-convex 𝐸-convex PROPN iajs-2928 46 10 set set NOUN iajs-2928 46 11 and and CCONJ iajs-2928 46 12 𝑓(π‘ŸπΈ(π‘₯ 𝑓(π‘ŸπΈ(π‘₯ NOUN iajs-2928 46 13 ) ) PUNCT iajs-2928 46 14 + + CCONJ iajs-2928 46 15 ( ( PUNCT iajs-2928 46 16 1 1 NUM iajs-2928 46 17 βˆ’ βˆ’ PROPN iajs-2928 46 18 π‘Ÿ)𝐸(𝑦 π‘Ÿ)𝐸(𝑦 NOUN iajs-2928 46 19 ) ) PUNCT iajs-2928 46 20 ) ) PUNCT iajs-2928 47 1 ≀ ≀ NUM iajs-2928 47 2 max max PROPN iajs-2928 47 3 { { PUNCT iajs-2928 47 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 47 5 ) ) PUNCT iajs-2928 47 6 , , PUNCT iajs-2928 47 7 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 47 8 ) ) PUNCT iajs-2928 47 9 } } PUNCT iajs-2928 47 10 . . PUNCT iajs-2928 48 1 3 3 X iajs-2928 48 2 . . X iajs-2928 48 3 pseudo pseudo NOUN iajs-2928 48 4 semi semi ADV iajs-2928 48 5 𝐸-convex 𝐸-convex PROPN iajs-2928 48 6 on on ADP iajs-2928 48 7 𝐸-convex 𝐸-convex PROPN iajs-2928 48 8 set set NOUN iajs-2928 48 9 𝐴 𝐴 NOUN iajs-2928 48 10 if if SCONJ iajs-2928 48 11 there there PRON iajs-2928 48 12 exists exist VERB iajs-2928 48 13 a a DET iajs-2928 48 14 strictly strictly ADV iajs-2928 48 15 positive positive ADJ iajs-2928 48 16 function function NOUN iajs-2928 48 17 𝑏 𝑏 NOUN iajs-2928 48 18 : : PUNCT iajs-2928 48 19 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 48 20 Γ— Γ— NOUN iajs-2928 48 21 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 48 22 ⟢ ⟢ NOUN iajs-2928 48 23 𝑅 𝑅 PROPN iajs-2928 48 24 such such ADJ iajs-2928 48 25 that that SCONJ iajs-2928 48 26 if if SCONJ iajs-2928 48 27 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 48 28 ) ) PUNCT iajs-2928 48 29 < < X iajs-2928 48 30 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 48 31 ) ) PUNCT iajs-2928 48 32 then then ADV iajs-2928 48 33 𝑓(π‘ŸπΈ(π‘₯ 𝑓(π‘ŸπΈ(π‘₯ NOUN iajs-2928 48 34 ) ) PUNCT iajs-2928 49 1 + + CCONJ iajs-2928 49 2 ( ( PUNCT iajs-2928 49 3 1 1 NUM iajs-2928 49 4 βˆ’ βˆ’ PROPN iajs-2928 49 5 π‘Ÿ)𝐸(𝑦 π‘Ÿ)𝐸(𝑦 NOUN iajs-2928 49 6 ) ) PUNCT iajs-2928 49 7 ) ) PUNCT iajs-2928 49 8 ≀ ≀ NOUN iajs-2928 49 9 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 49 10 ) ) PUNCT iajs-2928 50 1 + + NUM iajs-2928 50 2 π‘Ÿ(π‘Ÿ π‘Ÿ(π‘Ÿ PROPN iajs-2928 50 3 βˆ’ βˆ’ PROPN iajs-2928 50 4 1)𝑏(π‘₯ 1)𝑏(π‘₯ NUM iajs-2928 50 5 , , PUNCT iajs-2928 50 6 𝑦 𝑦 NOUN iajs-2928 50 7 ) ) PUNCT iajs-2928 50 8 , , PUNCT iajs-2928 50 9 for for ADP iajs-2928 50 10 0 0 NUM iajs-2928 50 11 < < X iajs-2928 50 12 π‘Ÿ π‘Ÿ X iajs-2928 50 13 < < X iajs-2928 50 14 1 1 NUM iajs-2928 50 15 . . NOUN iajs-2928 50 16 4 4 NUM iajs-2928 50 17 . . X iajs-2928 50 18 𝑝-convex 𝑝-convex PROPN iajs-2928 50 19 if if SCONJ iajs-2928 50 20 𝐴 𝐴 PROPN iajs-2928 50 21 is be AUX iajs-2928 50 22 𝑝-π‘π‘œπ‘›π‘£π‘’π‘₯ 𝑝-π‘π‘œπ‘›π‘£π‘’π‘₯ NOUN iajs-2928 50 23 set set NOUN iajs-2928 50 24 and and CCONJ iajs-2928 50 25 𝑓 𝑓 PRON iajs-2928 50 26 ( ( PUNCT iajs-2928 50 27 π‘Ÿπ‘₯ π‘Ÿπ‘₯ NOUN iajs-2928 50 28 + + NUM iajs-2928 50 29 s𝑦 s𝑦 NOUN iajs-2928 50 30 ) ) PUNCT iajs-2928 50 31 ≀ ≀ NOUN iajs-2928 50 32 π‘Ÿπ‘“(π‘₯ π‘Ÿπ‘“(π‘₯ NOUN iajs-2928 50 33 ) ) PUNCT iajs-2928 51 1 + + NUM iajs-2928 51 2 𝑠𝑓(𝑦 𝑠𝑓(𝑦 NUM iajs-2928 51 3 ) ) PUNCT iajs-2928 51 4 . . PUNCT iajs-2928 52 1 very very ADV iajs-2928 52 2 recently recently ADV iajs-2928 52 3 , , PUNCT iajs-2928 52 4 hazim hazim NOUN iajs-2928 52 5 and and CCONJ iajs-2928 52 6 majeed majeed PROPN iajs-2928 52 7 [ [ X iajs-2928 52 8 9 9 NUM iajs-2928 52 9 ] ] PUNCT iajs-2928 52 10 have have AUX iajs-2928 52 11 extended extend VERB iajs-2928 52 12 the the DET iajs-2928 52 13 concepts concept NOUN iajs-2928 52 14 of of ADP iajs-2928 52 15 𝐸-convexity 𝐸-convexity PROPN iajs-2928 52 16 and and CCONJ iajs-2928 52 17 𝑝convexity 𝑝convexity NOUN iajs-2928 52 18 defined define VERB iajs-2928 52 19 above above ADP iajs-2928 52 20 to to ADP iajs-2928 52 21 ( ( PUNCT iajs-2928 52 22 𝑝 𝑝 NOUN iajs-2928 52 23 , , PUNCT iajs-2928 52 24 𝐸)-convexity 𝐸)-convexity PROPN iajs-2928 52 25 as as SCONJ iajs-2928 52 26 follows follow VERB iajs-2928 52 27 . . PUNCT iajs-2928 53 1 definition definition NOUN iajs-2928 53 2 1.3 1.3 NUM iajs-2928 53 3 . . PUNCT iajs-2928 54 1 [ [ X iajs-2928 54 2 9 9 NUM iajs-2928 54 3 ] ] PUNCT iajs-2928 54 4 the the DET iajs-2928 54 5 set set NOUN iajs-2928 54 6 𝐴 𝐴 PROPN iajs-2928 54 7 is be AUX iajs-2928 54 8 called call VERB iajs-2928 54 9 ( ( PUNCT iajs-2928 54 10 𝑝 𝑝 PROPN iajs-2928 54 11 , , PUNCT iajs-2928 54 12 𝐸)-convex 𝐸)-convex PRON iajs-2928 54 13 set set VERB iajs-2928 54 14 if if SCONJ iajs-2928 54 15 for for ADP iajs-2928 54 16 all all PRON iajs-2928 54 17 π‘₯ π‘₯ PROPN iajs-2928 54 18 , , PUNCT iajs-2928 54 19 𝑦 𝑦 NOUN iajs-2928 54 20 ∈ ∈ PROPN iajs-2928 54 21 𝐴 𝐴 PROPN iajs-2928 54 22 and and CCONJ iajs-2928 54 23 for for ADP iajs-2928 54 24 all all DET iajs-2928 54 25 π‘Ÿ π‘Ÿ NOUN iajs-2928 54 26 , , PUNCT iajs-2928 54 27 𝑠 𝑠 PROPN iajs-2928 54 28 ∈ ∈ PROPN iajs-2928 55 1 [ [ X iajs-2928 55 2 0,1 0,1 NUM iajs-2928 55 3 ] ] PUNCT iajs-2928 55 4 , , PUNCT iajs-2928 55 5 𝑝 𝑝 PROPN iajs-2928 55 6 ∈ ∈ PROPN iajs-2928 55 7 ( ( PUNCT iajs-2928 55 8 0,1 0,1 NOUN iajs-2928 55 9 ] ] PUNCT iajs-2928 55 10 such such ADJ iajs-2928 55 11 that that SCONJ iajs-2928 55 12 π‘Ÿπ‘ π‘Ÿπ‘ ADP iajs-2928 56 1 + + NOUN iajs-2928 56 2 𝑠𝑝 𝑠𝑝 NOUN iajs-2928 56 3 = = SYM iajs-2928 56 4 1 1 NUM iajs-2928 56 5 we we PRON iajs-2928 56 6 have have VERB iajs-2928 56 7 π‘ŸπΈ(π‘₯ π‘ŸπΈ(π‘₯ NUM iajs-2928 56 8 ) ) PUNCT iajs-2928 56 9 + + PUNCT iajs-2928 56 10 𝑠𝐸(𝑦 𝑠𝐸(𝑦 NOUN iajs-2928 56 11 ) ) PUNCT iajs-2928 56 12 ∈ ∈ NOUN iajs-2928 56 13 𝐴. 𝐴. PROPN iajs-2928 56 14 ihjpas ihjpa NOUN iajs-2928 56 15 . . PUNCT iajs-2928 57 1 36(1)2023 36(1)2023 NUM iajs-2928 57 2 357 357 NUM iajs-2928 57 3 definition definition NOUN iajs-2928 57 4 1.4 1.4 NUM iajs-2928 57 5 . . PUNCT iajs-2928 58 1 [ [ X iajs-2928 58 2 9 9 NUM iajs-2928 58 3 ] ] PUNCT iajs-2928 58 4 for for ADP iajs-2928 58 5 any any DET iajs-2928 58 6 π‘₯ π‘₯ PROPN iajs-2928 58 7 , , PUNCT iajs-2928 58 8 𝑦 𝑦 PROPN iajs-2928 58 9 ∈ ∈ PROPN iajs-2928 58 10 𝐴 𝐴 PROPN iajs-2928 58 11 , , PUNCT iajs-2928 58 12 π‘Ÿ π‘Ÿ X iajs-2928 58 13 , , PUNCT iajs-2928 58 14 𝑠 𝑠 PROPN iajs-2928 58 15 ∈ ∈ PROPN iajs-2928 59 1 [ [ X iajs-2928 59 2 0,1 0,1 NUM iajs-2928 59 3 ] ] PUNCT iajs-2928 59 4 , , PUNCT iajs-2928 59 5 and and CCONJ iajs-2928 59 6 𝑝 𝑝 X iajs-2928 59 7 ∈ ∈ PROPN iajs-2928 59 8 ( ( PUNCT iajs-2928 59 9 0,1 0,1 NOUN iajs-2928 59 10 ] ] PUNCT iajs-2928 60 1 such such ADJ iajs-2928 60 2 that that SCONJ iajs-2928 60 3 π‘Ÿπ‘ π‘Ÿπ‘ ADP iajs-2928 60 4 + + NOUN iajs-2928 60 5 𝑠𝑝 𝑠𝑝 NOUN iajs-2928 60 6 = = SYM iajs-2928 60 7 1 1 X iajs-2928 60 8 . . PUNCT iajs-2928 61 1 the the DET iajs-2928 61 2 function function NOUN iajs-2928 61 3 𝑓 𝑓 NOUN iajs-2928 61 4 is be AUX iajs-2928 61 5 named name VERB iajs-2928 61 6 as as ADP iajs-2928 61 7 ( ( PUNCT iajs-2928 61 8 𝑝 𝑝 NOUN iajs-2928 61 9 , , PUNCT iajs-2928 61 10 𝐸)-π‘π‘œπ‘›π‘£π‘’π‘₯ 𝐸)-π‘π‘œπ‘›π‘£π‘’π‘₯ PROPN iajs-2928 61 11 if if SCONJ iajs-2928 61 12 𝐴 𝐴 PROPN iajs-2928 61 13 is be AUX iajs-2928 61 14 ( ( PUNCT iajs-2928 61 15 𝑝 𝑝 NOUN iajs-2928 61 16 , , PUNCT iajs-2928 61 17 𝐸)-convex 𝐸)-convex X iajs-2928 61 18 and and CCONJ iajs-2928 61 19 𝑓(π‘ŸπΈ(π‘₯ 𝑓(π‘ŸπΈ(π‘₯ NOUN iajs-2928 61 20 ) ) PUNCT iajs-2928 61 21 + + PUNCT iajs-2928 61 22 𝑠𝐸(𝑦 𝑠𝐸(𝑦 NOUN iajs-2928 61 23 ) ) PUNCT iajs-2928 61 24 ) ) PUNCT iajs-2928 61 25 ≀ ≀ NUM iajs-2928 61 26 π‘Ÿπ‘“(𝐸(π‘₯ π‘Ÿπ‘“(𝐸(π‘₯ NOUN iajs-2928 61 27 ) ) PUNCT iajs-2928 61 28 ) ) PUNCT iajs-2928 62 1 + + CCONJ iajs-2928 62 2 𝑠𝑓(𝐸(𝑦 𝑠𝑓(𝐸(𝑦 NOUN iajs-2928 62 3 ) ) PUNCT iajs-2928 62 4 ) ) PUNCT iajs-2928 62 5 . . PUNCT iajs-2928 63 1 remark remark VERB iajs-2928 63 2 1.5 1.5 NUM iajs-2928 63 3 . . PUNCT iajs-2928 64 1 from from ADP iajs-2928 64 2 the the DET iajs-2928 64 3 definition definition NOUN iajs-2928 64 4 of of ADP iajs-2928 64 5 ( ( PUNCT iajs-2928 64 6 𝑝 𝑝 NOUN iajs-2928 64 7 , , PUNCT iajs-2928 64 8 𝐸)-convexity 𝐸)-convexity NUM iajs-2928 64 9 , , PUNCT iajs-2928 64 10 one one PRON iajs-2928 64 11 observes observe VERB iajs-2928 64 12 that that DET iajs-2928 64 13 i. i. NOUN iajs-2928 64 14 in in ADP iajs-2928 64 15 definition definition NOUN iajs-2928 64 16 1.3 1.3 NUM iajs-2928 64 17 , , PUNCT iajs-2928 64 18 if if SCONJ iajs-2928 64 19 𝑝 𝑝 ADP iajs-2928 64 20 = = SYM iajs-2928 64 21 1 1 NUM iajs-2928 64 22 , , PUNCT iajs-2928 64 23 the the DET iajs-2928 64 24 definition definition NOUN iajs-2928 64 25 of of ADP iajs-2928 64 26 𝐸-convex 𝐸-convex PROPN iajs-2928 64 27 set set NOUN iajs-2928 64 28 is be AUX iajs-2928 64 29 obtained obtain VERB iajs-2928 64 30 . . PUNCT iajs-2928 65 1 also also ADV iajs-2928 65 2 , , PUNCT iajs-2928 65 3 if if SCONJ iajs-2928 65 4 𝐸 𝐸 PROPN iajs-2928 65 5 = = SYM iajs-2928 65 6 𝐼 𝐼 PROPN iajs-2928 65 7 ( ( PUNCT iajs-2928 65 8 identity identity NOUN iajs-2928 65 9 mapping mapping NOUN iajs-2928 65 10 ) ) PUNCT iajs-2928 65 11 , , PUNCT iajs-2928 65 12 then then ADV iajs-2928 65 13 𝐴 𝐴 PROPN iajs-2928 65 14 is be AUX iajs-2928 65 15 𝑝-convex 𝑝-convex PROPN iajs-2928 65 16 set set PROPN iajs-2928 65 17 ; ; PUNCT iajs-2928 65 18 ii ii X iajs-2928 65 19 . . PUNCT iajs-2928 66 1 likewise likewise ADV iajs-2928 66 2 , , PUNCT iajs-2928 66 3 from from ADP iajs-2928 66 4 definition definition NOUN iajs-2928 66 5 1.4 1.4 NUM iajs-2928 66 6 , , PUNCT iajs-2928 66 7 if if SCONJ iajs-2928 66 8 𝑝 𝑝 ADJ iajs-2928 66 9 = = SYM iajs-2928 66 10 1 1 NUM iajs-2928 66 11 we we PRON iajs-2928 66 12 have have VERB iajs-2928 66 13 𝑓 𝑓 PRON iajs-2928 66 14 is be AUX iajs-2928 66 15 𝐸-convex 𝐸-convex PROPN iajs-2928 66 16 function function NOUN iajs-2928 66 17 and and CCONJ iajs-2928 67 1 when when SCONJ iajs-2928 67 2 𝐸 𝐸 PROPN iajs-2928 67 3 = = SYM iajs-2928 67 4 𝐼 𝐼 PROPN iajs-2928 67 5 , , PUNCT iajs-2928 67 6 then then ADV iajs-2928 67 7 the the DET iajs-2928 67 8 definition definition NOUN iajs-2928 67 9 of of ADP iajs-2928 67 10 𝑓 𝑓 PRON iajs-2928 67 11 is be AUX iajs-2928 67 12 𝑝-convex 𝑝-convex NOUN iajs-2928 67 13 function function NOUN iajs-2928 67 14 is be AUX iajs-2928 67 15 obtained obtain VERB iajs-2928 67 16 . . PUNCT iajs-2928 68 1 for for ADP iajs-2928 68 2 the the DET iajs-2928 68 3 rest rest NOUN iajs-2928 68 4 of of ADP iajs-2928 68 5 the the DET iajs-2928 68 6 paper paper NOUN iajs-2928 68 7 , , PUNCT iajs-2928 68 8 the the DET iajs-2928 68 9 next next ADJ iajs-2928 68 10 remark remark NOUN iajs-2928 68 11 is be AUX iajs-2928 68 12 needed need VERB iajs-2928 68 13 . . PUNCT iajs-2928 69 1 remark remark VERB iajs-2928 69 2 1.6 1.6 NUM iajs-2928 69 3 . . PUNCT iajs-2928 70 1 1 1 NUM iajs-2928 70 2 . . X iajs-2928 71 1 the the DET iajs-2928 71 2 mapping mapping NOUN iajs-2928 71 3 𝐸(π‘₯ 𝐸(π‘₯ NOUN iajs-2928 71 4 ) ) PUNCT iajs-2928 71 5 will will AUX iajs-2928 71 6 be be AUX iajs-2928 71 7 written write VERB iajs-2928 71 8 as as ADP iajs-2928 71 9 𝐸π‘₯. 𝐸π‘₯. PROPN iajs-2928 71 10 2 2 NUM iajs-2928 71 11 . . PUNCT iajs-2928 72 1 the the DET iajs-2928 72 2 set set NOUN iajs-2928 72 3 𝐴 𝐴 PROPN iajs-2928 72 4 is be AUX iajs-2928 72 5 ( ( PUNCT iajs-2928 72 6 𝑝 𝑝 PROPN iajs-2928 72 7 , , PUNCT iajs-2928 72 8 𝐸 𝐸 PROPN iajs-2928 72 9 ) ) PUNCT iajs-2928 72 10 convex convex NOUN iajs-2928 72 11 set set NOUN iajs-2928 72 12 . . PUNCT iajs-2928 73 1 2 2 X iajs-2928 73 2 . . X iajs-2928 73 3 quasi quasi NOUN iajs-2928 73 4 semi semi ADJ iajs-2928 73 5 and and CCONJ iajs-2928 73 6 pseudo pseudo NOUN iajs-2928 73 7 semi semi ADJ iajs-2928 73 8 ( ( PUNCT iajs-2928 73 9 𝑝 𝑝 NOUN iajs-2928 73 10 , , PUNCT iajs-2928 73 11 𝐸)-convex 𝐸)-convex NOUN iajs-2928 73 12 functions function NOUN iajs-2928 73 13 in in ADP iajs-2928 73 14 this this DET iajs-2928 73 15 section section NOUN iajs-2928 73 16 , , PUNCT iajs-2928 73 17 a a DET iajs-2928 73 18 new new ADJ iajs-2928 73 19 class class NOUN iajs-2928 73 20 of of ADP iajs-2928 73 21 functions function NOUN iajs-2928 73 22 , , PUNCT iajs-2928 73 23 which which PRON iajs-2928 73 24 includes include VERB iajs-2928 73 25 quasi quasi NOUN iajs-2928 73 26 semi semi ADJ iajs-2928 73 27 ( ( PUNCT iajs-2928 73 28 𝑝 𝑝 NOUN iajs-2928 73 29 , , PUNCT iajs-2928 73 30 𝐸)-convex 𝐸)-convex NUM iajs-2928 73 31 and and CCONJ iajs-2928 73 32 pseudo pseudo NOUN iajs-2928 73 33 semi semi ADJ iajs-2928 73 34 ( ( PUNCT iajs-2928 73 35 𝑝 𝑝 NOUN iajs-2928 73 36 , , PUNCT iajs-2928 73 37 𝐸)-convex 𝐸)-convex NOUN iajs-2928 73 38 functions function NOUN iajs-2928 73 39 , , PUNCT iajs-2928 73 40 is be AUX iajs-2928 73 41 introduced introduce VERB iajs-2928 73 42 . . PUNCT iajs-2928 74 1 this this DET iajs-2928 74 2 class class NOUN iajs-2928 74 3 generalizes generalize VERB iajs-2928 74 4 each each PRON iajs-2928 74 5 of of ADP iajs-2928 74 6 quasi quasi NOUN iajs-2928 74 7 semi semi ADV iajs-2928 74 8 𝐸-convex 𝐸-convex PROPN iajs-2928 74 9 and and CCONJ iajs-2928 74 10 pseudo pseudo NOUN iajs-2928 74 11 semi semi ADV iajs-2928 74 12 𝐸-convex 𝐸-convex PROPN iajs-2928 74 13 functions function NOUN iajs-2928 74 14 [ [ X iajs-2928 74 15 2 2 NUM iajs-2928 74 16 ] ] PUNCT iajs-2928 74 17 . . PUNCT iajs-2928 75 1 some some DET iajs-2928 75 2 properties property NOUN iajs-2928 75 3 and and CCONJ iajs-2928 75 4 related related ADJ iajs-2928 75 5 examples example NOUN iajs-2928 75 6 are be AUX iajs-2928 75 7 established establish VERB iajs-2928 75 8 for for ADP iajs-2928 75 9 this this DET iajs-2928 75 10 class class NOUN iajs-2928 75 11 . . PUNCT iajs-2928 76 1 definition definition NOUN iajs-2928 76 2 2.1 2.1 NUM iajs-2928 76 3 . . PUNCT iajs-2928 77 1 the the DET iajs-2928 77 2 function function NOUN iajs-2928 77 3 𝑓 𝑓 NOUN iajs-2928 77 4 is be AUX iajs-2928 77 5 named name VERB iajs-2928 77 6 as as ADP iajs-2928 77 7 i. i. PROPN iajs-2928 77 8 quasi quasi PROPN iajs-2928 77 9 semi semi PROPN iajs-2928 77 10 ( ( PUNCT iajs-2928 77 11 𝑝 𝑝 NOUN iajs-2928 77 12 , , PUNCT iajs-2928 77 13 𝐸)-convex 𝐸)-convex PROPN iajs-2928 77 14 on on ADP iajs-2928 77 15 𝐴 𝐴 PROPN iajs-2928 77 16 if if SCONJ iajs-2928 77 17 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 77 18 + + NUM iajs-2928 77 19 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 77 20 ) ) PUNCT iajs-2928 77 21 ≀ ≀ NOUN iajs-2928 77 22 { { PUNCT iajs-2928 77 23 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 77 24 ) ) PUNCT iajs-2928 77 25 , , PUNCT iajs-2928 77 26 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 77 27 ) ) PUNCT iajs-2928 77 28 } } PUNCT iajs-2928 77 29 , , PUNCT iajs-2928 77 30 and and CCONJ iajs-2928 77 31 𝑓 𝑓 PRON iajs-2928 77 32 is be AUX iajs-2928 77 33 strictly strictly ADV iajs-2928 77 34 quasi quasi ADJ iajs-2928 77 35 semi semi ADJ iajs-2928 77 36 ( ( PUNCT iajs-2928 77 37 𝑝 𝑝 NOUN iajs-2928 77 38 , , PUNCT iajs-2928 77 39 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 77 40 if if SCONJ iajs-2928 77 41 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 77 42 + + NUM iajs-2928 77 43 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 77 44 ) ) PUNCT iajs-2928 77 45 < < X iajs-2928 77 46 { { PUNCT iajs-2928 77 47 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 77 48 ) ) PUNCT iajs-2928 77 49 , , PUNCT iajs-2928 77 50 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 77 51 ) ) PUNCT iajs-2928 77 52 } } PUNCT iajs-2928 77 53 , , PUNCT iajs-2928 77 54 where where SCONJ iajs-2928 77 55 π‘Ÿ π‘Ÿ X iajs-2928 77 56 , , PUNCT iajs-2928 77 57 𝑠 𝑠 PROPN iajs-2928 77 58 ∈ ∈ PROPN iajs-2928 77 59 ( ( PUNCT iajs-2928 77 60 0,1 0,1 NUM iajs-2928 77 61 ) ) PUNCT iajs-2928 77 62 . . PUNCT iajs-2928 78 1 ii ii PROPN iajs-2928 78 2 . . PUNCT iajs-2928 79 1 pseudo pseudo NOUN iajs-2928 79 2 semi semi ADJ iajs-2928 79 3 ( ( PUNCT iajs-2928 79 4 𝑝 𝑝 NOUN iajs-2928 79 5 , , PUNCT iajs-2928 79 6 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 79 7 if if SCONJ iajs-2928 79 8 there there PRON iajs-2928 79 9 exist exist VERB iajs-2928 79 10 a a DET iajs-2928 79 11 strictly strictly ADV iajs-2928 79 12 positive positive ADJ iajs-2928 79 13 function function NOUN iajs-2928 79 14 𝑏 𝑏 NOUN iajs-2928 79 15 : : PUNCT iajs-2928 79 16 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 79 17 Γ— Γ— NOUN iajs-2928 79 18 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 79 19 β†’ β†’ SYM iajs-2928 79 20 𝑅 𝑅 PROPN iajs-2928 79 21 such such ADJ iajs-2928 79 22 that that SCONJ iajs-2928 79 23 if if SCONJ iajs-2928 79 24 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 79 25 ) ) PUNCT iajs-2928 79 26 < < X iajs-2928 79 27 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 79 28 ) ) PUNCT iajs-2928 79 29 π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’π‘› NOUN iajs-2928 79 30 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ PUNCT iajs-2928 79 31 + + CCONJ iajs-2928 79 32 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 79 33 ) ) PUNCT iajs-2928 79 34 ≀ ≀ NOUN iajs-2928 79 35 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 79 36 ) ) PUNCT iajs-2928 80 1 + + CCONJ iajs-2928 80 2 ( ( PUNCT iajs-2928 80 3 βˆ’π‘Ÿπ‘  βˆ’π‘Ÿπ‘  NOUN iajs-2928 80 4 ) ) PUNCT iajs-2928 80 5 𝑏(π‘₯ 𝑏(π‘₯ PROPN iajs-2928 80 6 , , PUNCT iajs-2928 80 7 𝑦 𝑦 NOUN iajs-2928 80 8 ) ) PUNCT iajs-2928 80 9 , , PUNCT iajs-2928 80 10 for for ADP iajs-2928 80 11 all all DET iajs-2928 80 12 π‘Ÿ π‘Ÿ NOUN iajs-2928 80 13 , , PUNCT iajs-2928 80 14 𝑠 𝑠 PROPN iajs-2928 80 15 ∈ ∈ PROPN iajs-2928 80 16 ( ( PUNCT iajs-2928 80 17 0,1 0,1 NUM iajs-2928 80 18 ) ) PUNCT iajs-2928 80 19 . . PUNCT iajs-2928 81 1 remark remark PROPN iajs-2928 81 2 2.2 2.2 NUM iajs-2928 81 3 . . PUNCT iajs-2928 82 1 in in ADP iajs-2928 82 2 definition definition NOUN iajs-2928 82 3 2.1(i 2.1(i NUM iajs-2928 82 4 ) ) PUNCT iajs-2928 82 5 , , PUNCT iajs-2928 82 6 if if SCONJ iajs-2928 82 7 𝑝 𝑝 ADP iajs-2928 82 8 = = NOUN iajs-2928 82 9 1 1 NUM iajs-2928 82 10 then then ADV iajs-2928 82 11 𝑓 𝑓 PRON iajs-2928 82 12 turned turn VERB iajs-2928 82 13 to to PART iajs-2928 82 14 be be AUX iajs-2928 82 15 quasi quasi NOUN iajs-2928 82 16 semi semi ADV iajs-2928 82 17 𝐸-convex 𝐸-convex PROPN iajs-2928 82 18 . . PUNCT iajs-2928 83 1 likewise likewise ADV iajs-2928 83 2 , , PUNCT iajs-2928 83 3 𝑓 𝑓 X iajs-2928 83 4 in in ADP iajs-2928 83 5 definition definition NOUN iajs-2928 83 6 2.1(ii 2.1(ii NUM iajs-2928 83 7 ) ) PUNCT iajs-2928 83 8 becomes become VERB iajs-2928 83 9 pseudo pseudo NOUN iajs-2928 83 10 semi semi ADV iajs-2928 83 11 𝐸-convex 𝐸-convex PROPN iajs-2928 83 12 function function NOUN iajs-2928 83 13 . . PUNCT iajs-2928 84 1 quasi quasi NOUN iajs-2928 84 2 semi semi ADJ iajs-2928 84 3 and and CCONJ iajs-2928 84 4 pseudo pseudo NOUN iajs-2928 84 5 semi semi ADJ iajs-2928 84 6 ( ( PUNCT iajs-2928 84 7 𝑝 𝑝 NOUN iajs-2928 84 8 , , PUNCT iajs-2928 84 9 𝐸)-convex 𝐸)-convex NUM iajs-2928 84 10 functions function NOUN iajs-2928 84 11 are be AUX iajs-2928 84 12 not not PART iajs-2928 84 13 necessarily necessarily ADV iajs-2928 84 14 ( ( PUNCT iajs-2928 84 15 𝑝 𝑝 NOUN iajs-2928 84 16 , , PUNCT iajs-2928 84 17 𝐸)-convex 𝐸)-convex PRON iajs-2928 84 18 function function VERB iajs-2928 84 19 as as ADP iajs-2928 84 20 the the DET iajs-2928 84 21 following follow VERB iajs-2928 84 22 example example NOUN iajs-2928 84 23 shows show NOUN iajs-2928 84 24 . . PUNCT iajs-2928 85 1 example example NOUN iajs-2928 85 2 2.3 2.3 NUM iajs-2928 85 3 . . PUNCT iajs-2928 86 1 let let VERB iajs-2928 86 2 𝑓 𝑓 PRON iajs-2928 86 3 , , PUNCT iajs-2928 86 4 𝐸 𝐸 PROPN iajs-2928 86 5 : : PUNCT iajs-2928 86 6 𝑅 𝑅 PROPN iajs-2928 86 7 β†’ β†’ SYM iajs-2928 86 8 𝑅 𝑅 PROPN iajs-2928 86 9 π‘ π‘’π‘β„Ž π‘ π‘’π‘β„Ž NOUN iajs-2928 86 10 π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘Žπ‘‘ ADJ iajs-2928 86 11 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 86 12 ) ) PUNCT iajs-2928 87 1 = = PRON iajs-2928 87 2 { { PUNCT iajs-2928 88 1 βˆ’3 βˆ’3 ADV iajs-2928 88 2 𝑖𝑓 𝑖𝑓 INTJ iajs-2928 88 3 π‘₯ π‘₯ NOUN iajs-2928 88 4 = = SYM iajs-2928 88 5 0 0 NUM iajs-2928 88 6 1 1 NUM iajs-2928 88 7 𝑖𝑓 𝑖𝑓 ADP iajs-2928 88 8 π‘₯ π‘₯ PROPN iajs-2928 88 9 β‰  β‰  PROPN iajs-2928 88 10 0 0 PUNCT iajs-2928 88 11 and and CCONJ iajs-2928 88 12 𝐸π‘₯ 𝐸π‘₯ PROPN iajs-2928 88 13 = = PUNCT iajs-2928 88 14 { { PUNCT iajs-2928 88 15 0 0 NUM iajs-2928 88 16 𝑖𝑓 𝑖𝑓 NOUN iajs-2928 89 1 π‘₯ π‘₯ NOUN iajs-2928 89 2 = = SYM iajs-2928 89 3 0 0 NUM iajs-2928 89 4 4 4 NUM iajs-2928 89 5 𝑖𝑓 𝑖𝑓 ADP iajs-2928 89 6 π‘₯ π‘₯ PROPN iajs-2928 89 7 β‰  β‰  PROPN iajs-2928 89 8 0 0 NUM iajs-2928 89 9 . . PUNCT iajs-2928 90 1 ihjpas ihjpas PROPN iajs-2928 90 2 . . PUNCT iajs-2928 91 1 36(1)2023 36(1)2023 NUM iajs-2928 91 2 358 358 NUM iajs-2928 91 3 let let VERB iajs-2928 91 4 π‘₯ π‘₯ PRON iajs-2928 91 5 , , PUNCT iajs-2928 91 6 𝑦 𝑦 PROPN iajs-2928 91 7 ∈ ∈ PROPN iajs-2928 91 8 𝑅 𝑅 PROPN iajs-2928 91 9 , , PUNCT iajs-2928 91 10 𝑝 𝑝 PROPN iajs-2928 91 11 ∈ ∈ PROPN iajs-2928 91 12 ( ( PUNCT iajs-2928 91 13 0,1 0,1 NOUN iajs-2928 91 14 ] ] PUNCT iajs-2928 91 15 and and CCONJ iajs-2928 91 16 π‘Ÿ π‘Ÿ NOUN iajs-2928 91 17 , , PUNCT iajs-2928 91 18 𝑠 𝑠 PROPN iajs-2928 91 19 ∈ ∈ PROPN iajs-2928 92 1 [ [ X iajs-2928 92 2 0,1 0,1 NUM iajs-2928 92 3 ] ] X iajs-2928 92 4 π‘ π‘’π‘β„Ž π‘ π‘’π‘β„Ž NOUN iajs-2928 92 5 π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘Žπ‘‘ VERB iajs-2928 92 6 π‘Ÿπ‘ π‘Ÿπ‘ X iajs-2928 93 1 + + CCONJ iajs-2928 93 2 𝑠𝑝 𝑠𝑝 NOUN iajs-2928 93 3 = = SYM iajs-2928 93 4 1 1 NUM iajs-2928 93 5 . . PUNCT iajs-2928 94 1 first first ADV iajs-2928 94 2 , , PUNCT iajs-2928 94 3 we we PRON iajs-2928 94 4 show show VERB iajs-2928 94 5 that that SCONJ iajs-2928 94 6 𝑓 𝑓 PRON iajs-2928 94 7 is be AUX iajs-2928 94 8 quasi quasi NOUN iajs-2928 94 9 semi semi ADJ iajs-2928 94 10 ( ( PUNCT iajs-2928 94 11 𝑝 𝑝 NOUN iajs-2928 94 12 , , PUNCT iajs-2928 94 13 𝐸)-convex 𝐸)-convex NUM iajs-2928 94 14 and and CCONJ iajs-2928 94 15 pseudo pseudo NOUN iajs-2928 94 16 semi semi ADJ iajs-2928 94 17 ( ( PUNCT iajs-2928 94 18 𝑝 𝑝 NOUN iajs-2928 94 19 , , PUNCT iajs-2928 94 20 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 94 21 function function NOUN iajs-2928 94 22 . . PUNCT iajs-2928 95 1 for for ADP iajs-2928 95 2 showing show VERB iajs-2928 95 3 𝑓 𝑓 PRON iajs-2928 95 4 is be AUX iajs-2928 95 5 quasi quasi NOUN iajs-2928 95 6 semi semi ADJ iajs-2928 95 7 ( ( PUNCT iajs-2928 95 8 𝑝 𝑝 NOUN iajs-2928 95 9 , , PUNCT iajs-2928 95 10 𝐸)convex 𝐸)convex PROPN iajs-2928 95 11 , , PUNCT iajs-2928 95 12 we we PRON iajs-2928 95 13 consider consider VERB iajs-2928 95 14 three three NUM iajs-2928 95 15 cases case NOUN iajs-2928 95 16 : : PUNCT iajs-2928 95 17 case case NOUN iajs-2928 95 18 1 1 NUM iajs-2928 95 19 : : PUNCT iajs-2928 95 20 if if SCONJ iajs-2928 95 21 π‘₯ π‘₯ PRON iajs-2928 95 22 = = SYM iajs-2928 95 23 𝑦 𝑦 SYM iajs-2928 95 24 = = SYM iajs-2928 95 25 0 0 NUM iajs-2928 95 26 , , PUNCT iajs-2928 95 27 we we PRON iajs-2928 95 28 get get VERB iajs-2928 95 29 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 95 30 + + CCONJ iajs-2928 95 31 𝑠𝐸𝑦 𝑠𝐸𝑦 NUM iajs-2928 95 32 ) ) PUNCT iajs-2928 96 1 = = SYM iajs-2928 96 2 𝑓(0 𝑓(0 NOUN iajs-2928 96 3 ) ) PUNCT iajs-2928 96 4 = = PUNCT iajs-2928 97 1 βˆ’3 βˆ’3 NOUN iajs-2928 97 2 = = PUNCT iajs-2928 97 3 { { PUNCT iajs-2928 97 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 97 5 ) ) PUNCT iajs-2928 97 6 , , PUNCT iajs-2928 97 7 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 97 8 ) ) PUNCT iajs-2928 97 9 } } PUNCT iajs-2928 97 10 . . PUNCT iajs-2928 98 1 case case NOUN iajs-2928 98 2 2 2 NUM iajs-2928 98 3 : : PUNCT iajs-2928 98 4 if if SCONJ iajs-2928 98 5 π‘₯ π‘₯ PRON iajs-2928 98 6 β‰  β‰  PROPN iajs-2928 98 7 0 0 NUM iajs-2928 98 8 , , PUNCT iajs-2928 98 9 𝑦 𝑦 NOUN iajs-2928 98 10 β‰  β‰  PROPN iajs-2928 98 11 0 0 NUM iajs-2928 98 12 , , PUNCT iajs-2928 98 13 we we PRON iajs-2928 98 14 get get VERB iajs-2928 98 15 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 98 16 + + CCONJ iajs-2928 98 17 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 98 18 ) ) PUNCT iajs-2928 98 19 = = SYM iajs-2928 99 1 𝑓(4π‘Ÿ 𝑓(4π‘Ÿ X iajs-2928 99 2 + + NUM iajs-2928 99 3 4𝑠 4𝑠 NUM iajs-2928 99 4 ) ) PUNCT iajs-2928 100 1 = = SYM iajs-2928 100 2 1 1 NUM iajs-2928 100 3 = = SYM iajs-2928 100 4 { { PUNCT iajs-2928 100 5 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 100 6 ) ) PUNCT iajs-2928 100 7 , , PUNCT iajs-2928 100 8 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 100 9 ) ) PUNCT iajs-2928 100 10 } } PUNCT iajs-2928 100 11 . . PUNCT iajs-2928 101 1 case case NOUN iajs-2928 101 2 3 3 NUM iajs-2928 101 3 : : PUNCT iajs-2928 101 4 if if SCONJ iajs-2928 101 5 π‘₯ π‘₯ PROPN iajs-2928 101 6 = = SYM iajs-2928 101 7 0 0 NUM iajs-2928 101 8 , , PUNCT iajs-2928 101 9 𝑦 𝑦 NOUN iajs-2928 101 10 β‰  β‰  PROPN iajs-2928 101 11 0 0 NUM iajs-2928 101 12 , , PUNCT iajs-2928 101 13 we we PRON iajs-2928 101 14 get get VERB iajs-2928 101 15 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 101 16 + + CCONJ iajs-2928 101 17 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 101 18 ) ) PUNCT iajs-2928 101 19 = = PUNCT iajs-2928 102 1 𝑓(4𝑠 𝑓(4𝑠 VERB iajs-2928 102 2 ) ) PUNCT iajs-2928 102 3 = = PRON iajs-2928 102 4 { { PUNCT iajs-2928 102 5 βˆ’3 βˆ’3 ADV iajs-2928 102 6 if if SCONJ iajs-2928 102 7 𝑠 𝑠 PROPN iajs-2928 102 8 = = SYM iajs-2928 102 9 0 0 PROPN iajs-2928 102 10 1 1 NUM iajs-2928 102 11 if if SCONJ iajs-2928 102 12 𝑠 𝑠 PROPN iajs-2928 102 13 β‰  β‰  PROPN iajs-2928 102 14 0 0 NUM iajs-2928 102 15 ( ( PUNCT iajs-2928 102 16 1 1 NUM iajs-2928 102 17 ) ) PUNCT iajs-2928 102 18 ≀ ≀ NUM iajs-2928 102 19 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ iajs-2928 102 20 { { PUNCT iajs-2928 102 21 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 102 22 ) ) PUNCT iajs-2928 102 23 , , PUNCT iajs-2928 102 24 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 102 25 ) ) PUNCT iajs-2928 102 26 } } PUNCT iajs-2928 103 1 = = SYM iajs-2928 103 2 { { PUNCT iajs-2928 103 3 βˆ’3,1 βˆ’3,1 NOUN iajs-2928 103 4 } } PUNCT iajs-2928 103 5 = = SYM iajs-2928 103 6 1 1 X iajs-2928 103 7 . . PUNCT iajs-2928 104 1 in in ADP iajs-2928 104 2 all all DET iajs-2928 104 3 cases case NOUN iajs-2928 104 4 , , PUNCT iajs-2928 104 5 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ PUNCT iajs-2928 104 6 + + CCONJ iajs-2928 104 7 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 104 8 ) ) PUNCT iajs-2928 104 9 ≀ ≀ NOUN iajs-2928 104 10 π‘šπ‘Žπ‘₯{𝑓(π‘₯ π‘šπ‘Žπ‘₯{𝑓(π‘₯ NOUN iajs-2928 104 11 ) ) PUNCT iajs-2928 104 12 , , PUNCT iajs-2928 104 13 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 104 14 ) ) PUNCT iajs-2928 104 15 } } PUNCT iajs-2928 104 16 , , PUNCT iajs-2928 104 17 and and CCONJ iajs-2928 104 18 hence hence ADV iajs-2928 104 19 𝑓 𝑓 PRON iajs-2928 104 20 is be AUX iajs-2928 104 21 quasi quasi NOUN iajs-2928 104 22 semi semi ADJ iajs-2928 104 23 ( ( PUNCT iajs-2928 104 24 𝑝 𝑝 NOUN iajs-2928 104 25 , , PUNCT iajs-2928 104 26 𝐸)-convex 𝐸)-convex X iajs-2928 104 27 . . PUNCT iajs-2928 105 1 from from ADP iajs-2928 105 2 case case NOUN iajs-2928 105 3 3 3 NUM iajs-2928 105 4 , , PUNCT iajs-2928 105 5 we we PRON iajs-2928 105 6 get get VERB iajs-2928 105 7 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 105 8 ) ) PUNCT iajs-2928 105 9 = = PUNCT iajs-2928 106 1 βˆ’3 βˆ’3 X iajs-2928 106 2 < < X iajs-2928 106 3 1 1 NUM iajs-2928 106 4 = = SYM iajs-2928 106 5 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 106 6 ) ) PUNCT iajs-2928 106 7 . . PUNCT iajs-2928 107 1 thus thus ADV iajs-2928 107 2 , , PUNCT iajs-2928 107 3 from from ADP iajs-2928 107 4 ( ( PUNCT iajs-2928 107 5 1 1 NUM iajs-2928 107 6 ) ) PUNCT iajs-2928 107 7 , , PUNCT iajs-2928 107 8 one one PRON iajs-2928 107 9 can can AUX iajs-2928 107 10 choose choose VERB iajs-2928 107 11 a a DET iajs-2928 107 12 strictly strictly ADV iajs-2928 107 13 positive positive ADJ iajs-2928 107 14 function function NOUN iajs-2928 107 15 𝑏(π‘₯ 𝑏(π‘₯ NOUN iajs-2928 107 16 , , PUNCT iajs-2928 107 17 𝑦 𝑦 NOUN iajs-2928 107 18 ) ) PUNCT iajs-2928 107 19 such such ADJ iajs-2928 107 20 that that PRON iajs-2928 107 21 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 107 22 + + CCONJ iajs-2928 107 23 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 107 24 ) ) PUNCT iajs-2928 107 25 ≀ ≀ NOUN iajs-2928 107 26 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 107 27 ) ) PUNCT iajs-2928 108 1 + + CCONJ iajs-2928 108 2 ( ( PUNCT iajs-2928 108 3 βˆ’π‘Ÿπ‘ )𝑏(π‘₯ βˆ’π‘Ÿπ‘ )𝑏(π‘₯ NOUN iajs-2928 108 4 , , PUNCT iajs-2928 108 5 𝑦 𝑦 NOUN iajs-2928 108 6 ) ) PUNCT iajs-2928 108 7 ≀ ≀ NOUN iajs-2928 108 8 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 108 9 ) ) PUNCT iajs-2928 108 10 = = SYM iajs-2928 108 11 1 1 X iajs-2928 108 12 . . PUNCT iajs-2928 108 13 thus thus ADV iajs-2928 108 14 , , PUNCT iajs-2928 108 15 𝑓 𝑓 PRON iajs-2928 108 16 is be AUX iajs-2928 108 17 pseudo pseudo NOUN iajs-2928 108 18 semi semi ADJ iajs-2928 108 19 ( ( PUNCT iajs-2928 108 20 𝑝 𝑝 NOUN iajs-2928 108 21 , , PUNCT iajs-2928 108 22 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 108 23 function function NOUN iajs-2928 108 24 . . PUNCT iajs-2928 109 1 finally finally ADV iajs-2928 109 2 , , PUNCT iajs-2928 109 3 to to PART iajs-2928 109 4 show show VERB iajs-2928 109 5 that that SCONJ iajs-2928 109 6 𝑓 𝑓 PRON iajs-2928 109 7 is be AUX iajs-2928 109 8 not not PART iajs-2928 109 9 ( ( PUNCT iajs-2928 109 10 1 1 NUM iajs-2928 109 11 2 2 NUM iajs-2928 109 12 , , PUNCT iajs-2928 109 13 𝐸)-convex 𝐸)-convex X iajs-2928 109 14 , , PUNCT iajs-2928 109 15 take take VERB iajs-2928 109 16 π‘₯ π‘₯ PRON iajs-2928 109 17 β‰  β‰  PROPN iajs-2928 109 18 0 0 NUM iajs-2928 109 19 , , PUNCT iajs-2928 109 20 𝑦 𝑦 NOUN iajs-2928 109 21 β‰  β‰  PROPN iajs-2928 109 22 0 0 NUM iajs-2928 109 23 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN iajs-2928 109 24 𝑝 𝑝 PROPN iajs-2928 109 25 = = NOUN iajs-2928 109 26 1 1 NUM iajs-2928 109 27 2 2 NUM iajs-2928 109 28 with with ADP iajs-2928 109 29 𝑠 𝑠 PROPN iajs-2928 109 30 = = SYM iajs-2928 109 31 π‘Ÿ π‘Ÿ NOUN iajs-2928 109 32 = = SYM iajs-2928 109 33 1 1 NUM iajs-2928 109 34 4 4 NUM iajs-2928 109 35 . . PUNCT iajs-2928 110 1 then then ADV iajs-2928 110 2 , , PUNCT iajs-2928 110 3 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ AUX iajs-2928 110 4 + + CCONJ iajs-2928 110 5 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 110 6 ) ) PUNCT iajs-2928 110 7 = = SYM iajs-2928 110 8 1 1 X iajs-2928 110 9 > > X iajs-2928 110 10 π‘Ÿπ‘“(𝐸π‘₯ π‘Ÿπ‘“(𝐸π‘₯ PROPN iajs-2928 110 11 ) ) PUNCT iajs-2928 111 1 + + CCONJ iajs-2928 111 2 𝑠𝑓(𝐸𝑦 𝑠𝑓(𝐸𝑦 ADJ iajs-2928 111 3 ) ) PUNCT iajs-2928 111 4 = = SYM iajs-2928 111 5 π‘Ÿπ‘“(4 π‘Ÿπ‘“(4 NUM iajs-2928 111 6 ) ) PUNCT iajs-2928 111 7 + + PUNCT iajs-2928 111 8 𝑠𝑓(4 𝑠𝑓(4 NOUN iajs-2928 111 9 ) ) PUNCT iajs-2928 111 10 = = SYM iajs-2928 112 1 π‘Ÿ π‘Ÿ NOUN iajs-2928 112 2 + + CCONJ iajs-2928 112 3 𝑠 𝑠 X iajs-2928 112 4 = = SYM iajs-2928 112 5 1 1 NUM iajs-2928 112 6 2 2 NUM iajs-2928 112 7 as as SCONJ iajs-2928 112 8 required require VERB iajs-2928 112 9 . . PUNCT iajs-2928 113 1 the the DET iajs-2928 113 2 next next ADJ iajs-2928 113 3 example example NOUN iajs-2928 113 4 provides provide VERB iajs-2928 113 5 ( ( PUNCT iajs-2928 113 6 𝑝 𝑝 NOUN iajs-2928 113 7 , , PUNCT iajs-2928 113 8 𝐸)-convex 𝐸)-convex PRON iajs-2928 113 9 function function VERB iajs-2928 113 10 which which PRON iajs-2928 113 11 is be AUX iajs-2928 113 12 neither neither CCONJ iajs-2928 113 13 quasi quasi ADJ iajs-2928 113 14 semi semi ADJ iajs-2928 113 15 ( ( PUNCT iajs-2928 113 16 𝑝 𝑝 NOUN iajs-2928 113 17 , , PUNCT iajs-2928 113 18 𝐸)-convex 𝐸)-convex X iajs-2928 113 19 nor nor CCONJ iajs-2928 113 20 pseudo pseudo NOUN iajs-2928 113 21 semi semi ADJ iajs-2928 113 22 ( ( PUNCT iajs-2928 113 23 𝑝 𝑝 NOUN iajs-2928 113 24 , , PUNCT iajs-2928 113 25 𝐸)-convex 𝐸)-convex PROPN iajs-2928 113 26 . . PUNCT iajs-2928 113 27 example example NOUN iajs-2928 113 28 2.4 2.4 NUM iajs-2928 113 29 . . PUNCT iajs-2928 114 1 let let VERB iajs-2928 114 2 𝐴 𝐴 PROPN iajs-2928 114 3 = = PUNCT iajs-2928 115 1 [ [ X iajs-2928 115 2 βˆ’5 βˆ’5 NOUN iajs-2928 115 3 , , PUNCT iajs-2928 115 4 βˆ’βˆž βˆ’βˆž NOUN iajs-2928 115 5 ) ) PUNCT iajs-2928 115 6 Γ— Γ— NOUN iajs-2928 116 1 [ [ X iajs-2928 116 2 βˆ’5 βˆ’5 NOUN iajs-2928 116 3 , , PUNCT iajs-2928 116 4 βˆ’βˆž βˆ’βˆž NOUN iajs-2928 116 5 ) ) PUNCT iajs-2928 116 6 βŠ† βŠ† NUM iajs-2928 116 7 𝑅2 𝑅2 NOUN iajs-2928 116 8 and and CCONJ iajs-2928 116 9 𝐸 𝐸 PROPN iajs-2928 116 10 : : PUNCT iajs-2928 116 11 𝑅2β†’ 𝑅2β†’ PUNCT iajs-2928 116 12 𝑅2 𝑅2 VERB iajs-2928 116 13 such such ADJ iajs-2928 116 14 that that SCONJ iajs-2928 116 15 𝐸(π‘₯1 𝐸(π‘₯1 ADJ iajs-2928 116 16 , , PUNCT iajs-2928 116 17 π‘₯2 π‘₯2 NOUN iajs-2928 116 18 ) ) PUNCT iajs-2928 117 1 = = PRON iajs-2928 117 2 { { PUNCT iajs-2928 117 3 ( ( PUNCT iajs-2928 117 4 ( ( PUNCT iajs-2928 117 5 π‘₯1 π‘₯1 NOUN iajs-2928 117 6 + + NOUN iajs-2928 117 7 1)2 1)2 NUM iajs-2928 117 8 , , PUNCT iajs-2928 117 9 ( ( PUNCT iajs-2928 117 10 π‘₯2 π‘₯2 NOUN iajs-2928 117 11 + + CCONJ iajs-2928 117 12 1)2 1)2 NUM iajs-2928 117 13 ) ) PUNCT iajs-2928 117 14 if if SCONJ iajs-2928 117 15 π‘₯1 π‘₯1 NOUN iajs-2928 117 16 , , PUNCT iajs-2928 117 17 π‘₯2 π‘₯2 NOUN iajs-2928 117 18 < < X iajs-2928 117 19 0 0 NUM iajs-2928 117 20 ( ( PUNCT iajs-2928 117 21 0,0 0,0 NUM iajs-2928 117 22 ) ) PUNCT iajs-2928 117 23 o. o. PROPN iajs-2928 117 24 w. w. PROPN iajs-2928 117 25 define define VERB iajs-2928 117 26 𝑓 𝑓 PRON iajs-2928 117 27 : : PUNCT iajs-2928 117 28 𝑅2 𝑅2 ADP iajs-2928 117 29 β†’ β†’ SYM iajs-2928 117 30 𝑅 𝑅 NOUN iajs-2928 117 31 such such ADJ iajs-2928 117 32 that that SCONJ iajs-2928 117 33 𝑓(π‘₯1 𝑓(π‘₯1 ADJ iajs-2928 117 34 , , PUNCT iajs-2928 117 35 π‘₯2 π‘₯2 NOUN iajs-2928 117 36 ) ) PUNCT iajs-2928 117 37 = = PRON iajs-2928 117 38 { { PUNCT iajs-2928 117 39 π‘₯1+π‘₯2 π‘₯1+π‘₯2 NUM iajs-2928 117 40 3 3 NUM iajs-2928 117 41 if if SCONJ iajs-2928 117 42 π‘₯1 π‘₯1 NOUN iajs-2928 117 43 , , PUNCT iajs-2928 117 44 π‘₯2 π‘₯2 NOUN iajs-2928 117 45 < < X iajs-2928 117 46 0 0 NUM iajs-2928 117 47 0 0 NUM iajs-2928 117 48 o. o. PROPN iajs-2928 117 49 w. w. PROPN iajs-2928 117 50 first first PROPN iajs-2928 117 51 , , PUNCT iajs-2928 117 52 we we PRON iajs-2928 117 53 show show VERB iajs-2928 117 54 that that SCONJ iajs-2928 117 55 𝐴 𝐴 PROPN iajs-2928 117 56 is be AUX iajs-2928 117 57 ( ( PUNCT iajs-2928 117 58 𝑝 𝑝 NOUN iajs-2928 117 59 , , PUNCT iajs-2928 117 60 𝐸)-convex 𝐸)-convex PRON iajs-2928 117 61 set set VERB iajs-2928 117 62 . . PUNCT iajs-2928 118 1 let let VERB iajs-2928 118 2 π‘₯ π‘₯ VERB iajs-2928 118 3 = = SYM iajs-2928 118 4 ( ( PUNCT iajs-2928 118 5 π‘₯1 π‘₯1 NOUN iajs-2928 118 6 , , PUNCT iajs-2928 118 7 π‘₯2 π‘₯2 NOUN iajs-2928 118 8 ) ) PUNCT iajs-2928 118 9 , , PUNCT iajs-2928 118 10 𝑦 𝑦 NOUN iajs-2928 118 11 = = SYM iajs-2928 118 12 ( ( PUNCT iajs-2928 118 13 𝑦1 𝑦1 PROPN iajs-2928 118 14 , , PUNCT iajs-2928 118 15 𝑦2 𝑦2 PROPN iajs-2928 118 16 ) ) PUNCT iajs-2928 118 17 ∈ ∈ PROPN iajs-2928 118 18 𝐴. 𝐴. PROPN iajs-2928 118 19 if if SCONJ iajs-2928 118 20 π‘₯1 π‘₯1 NOUN iajs-2928 118 21 , , PUNCT iajs-2928 118 22 π‘₯2 π‘₯2 NOUN iajs-2928 118 23 , , PUNCT iajs-2928 118 24 𝑦1 𝑦1 PROPN iajs-2928 118 25 , , PUNCT iajs-2928 118 26 𝑦2 𝑦2 PROPN iajs-2928 118 27 < < X iajs-2928 118 28 0 0 PROPN iajs-2928 118 29 then then ADV iajs-2928 118 30 π‘ŸπΈ(π‘₯1 π‘ŸπΈ(π‘₯1 PROPN iajs-2928 118 31 , , PUNCT iajs-2928 118 32 π‘₯2 π‘₯2 NOUN iajs-2928 118 33 ) ) PUNCT iajs-2928 119 1 + + CCONJ iajs-2928 119 2 𝑠𝐸(𝑦1 𝑠𝐸(𝑦1 PROPN iajs-2928 119 3 , , PUNCT iajs-2928 119 4 𝑦2 𝑦2 PROPN iajs-2928 119 5 ) ) PUNCT iajs-2928 119 6 = = SYM iajs-2928 120 1 ( ( PUNCT iajs-2928 120 2 π‘Ÿ(π‘₯1 π‘Ÿ(π‘₯1 X iajs-2928 120 3 + + CCONJ iajs-2928 120 4 1)2 1)2 NUM iajs-2928 120 5 + + NUM iajs-2928 120 6 𝑠(𝑦1 𝑠(𝑦1 X iajs-2928 120 7 + + CCONJ iajs-2928 120 8 1)2 1)2 NUM iajs-2928 120 9 , , PUNCT iajs-2928 120 10 π‘Ÿ(π‘₯2 π‘Ÿ(π‘₯2 NOUN iajs-2928 121 1 + + CCONJ iajs-2928 122 1 1)2 1)2 NUM iajs-2928 122 2 + + NUM iajs-2928 122 3 𝑠(𝑦2 𝑠(𝑦2 VERB iajs-2928 122 4 + + ADV iajs-2928 122 5 1)2 1)2 NUM iajs-2928 122 6 ) ) PUNCT iajs-2928 122 7 ∈ ∈ PROPN iajs-2928 123 1 [ [ X iajs-2928 123 2 0 0 NUM iajs-2928 123 3 , , PUNCT iajs-2928 123 4 + + NOUN iajs-2928 123 5 ∞ ∞ NOUN iajs-2928 123 6 ) ) PUNCT iajs-2928 123 7 Γ— Γ— NOUN iajs-2928 124 1 [ [ X iajs-2928 124 2 0 0 NUM iajs-2928 124 3 , , PUNCT iajs-2928 124 4 + + NOUN iajs-2928 124 5 ∞ ∞ NOUN iajs-2928 124 6 ) ) PUNCT iajs-2928 124 7 βŠ† βŠ† NUM iajs-2928 124 8 𝐴. 𝐴. NOUN iajs-2928 124 9 similarly similarly ADV iajs-2928 124 10 , , PUNCT iajs-2928 124 11 if if SCONJ iajs-2928 124 12 𝐸π‘₯ 𝐸π‘₯ PROPN iajs-2928 124 13 = = SYM iajs-2928 124 14 𝐸𝑦 𝐸𝑦 PROPN iajs-2928 124 15 = = SYM iajs-2928 124 16 ( ( PUNCT iajs-2928 124 17 0,0 0,0 NOUN iajs-2928 124 18 ) ) PUNCT iajs-2928 124 19 then then ADV iajs-2928 124 20 π‘ŸπΈ(π‘₯1 π‘ŸπΈ(π‘₯1 PROPN iajs-2928 124 21 , , PUNCT iajs-2928 124 22 π‘₯2 π‘₯2 NOUN iajs-2928 124 23 ) ) PUNCT iajs-2928 124 24 + + CCONJ iajs-2928 124 25 𝑠𝐸(𝑦1 𝑠𝐸(𝑦1 PROPN iajs-2928 124 26 , , PUNCT iajs-2928 124 27 𝑦2 𝑦2 PROPN iajs-2928 124 28 ) ) PUNCT iajs-2928 124 29 = = SYM iajs-2928 124 30 ( ( PUNCT iajs-2928 124 31 0,0 0,0 NUM iajs-2928 124 32 ) ) PUNCT iajs-2928 124 33 ∈ ∈ PROPN iajs-2928 124 34 𝐴. 𝐴. PROPN iajs-2928 124 35 thus thus ADV iajs-2928 124 36 , , PUNCT iajs-2928 124 37 𝐴 𝐴 PROPN iajs-2928 124 38 is be AUX iajs-2928 124 39 ( ( PUNCT iajs-2928 124 40 𝑝 𝑝 NOUN iajs-2928 124 41 , , PUNCT iajs-2928 124 42 𝐸)-convex 𝐸)-convex PRON iajs-2928 124 43 set set VERB iajs-2928 124 44 . . PUNCT iajs-2928 125 1 to to PART iajs-2928 125 2 show show VERB iajs-2928 125 3 that that SCONJ iajs-2928 125 4 𝑓 𝑓 PRON iajs-2928 125 5 is be AUX iajs-2928 125 6 ( ( PUNCT iajs-2928 125 7 𝑝 𝑝 NOUN iajs-2928 125 8 , , PUNCT iajs-2928 125 9 𝐸)-convex 𝐸)-convex PRON iajs-2928 125 10 function function VERB iajs-2928 125 11 on on ADP iajs-2928 125 12 𝐴 𝐴 PROPN iajs-2928 125 13 , , PUNCT iajs-2928 125 14 let let VERB iajs-2928 125 15 π‘₯ π‘₯ X iajs-2928 125 16 = = SYM iajs-2928 125 17 ( ( PUNCT iajs-2928 125 18 π‘₯1 π‘₯1 NOUN iajs-2928 125 19 , , PUNCT iajs-2928 125 20 π‘₯2 π‘₯2 NOUN iajs-2928 125 21 ) ) PUNCT iajs-2928 125 22 , , PUNCT iajs-2928 125 23 𝑦 𝑦 NOUN iajs-2928 125 24 = = SYM iajs-2928 125 25 ( ( PUNCT iajs-2928 125 26 𝑦1 𝑦1 PROPN iajs-2928 125 27 , , PUNCT iajs-2928 125 28 𝑦2 𝑦2 PROPN iajs-2928 125 29 ) ) PUNCT iajs-2928 125 30 ∈ ∈ PROPN iajs-2928 125 31 𝐴. 𝐴. PROPN iajs-2928 126 1 if if SCONJ iajs-2928 126 2 π‘₯1 π‘₯1 NOUN iajs-2928 126 3 , , PUNCT iajs-2928 126 4 π‘₯2 π‘₯2 NOUN iajs-2928 126 5 , , PUNCT iajs-2928 126 6 𝑦1 𝑦1 PROPN iajs-2928 126 7 , , PUNCT iajs-2928 126 8 𝑦2 𝑦2 NOUN iajs-2928 126 9 < < X iajs-2928 126 10 0 0 PROPN iajs-2928 126 11 , , PUNCT iajs-2928 126 12 then then ADV iajs-2928 126 13 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ PUNCT iajs-2928 126 14 + + CCONJ iajs-2928 126 15 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 126 16 ) ) PUNCT iajs-2928 126 17 = = SYM iajs-2928 126 18 0 0 PUNCT iajs-2928 126 19 = = SYM iajs-2928 126 20 π‘Ÿπ‘“(𝐸π‘₯ π‘Ÿπ‘“(𝐸π‘₯ PROPN iajs-2928 126 21 ) ) PUNCT iajs-2928 127 1 + + CCONJ iajs-2928 127 2 𝑠𝑓(𝐸𝑦 𝑠𝑓(𝐸𝑦 PROPN iajs-2928 127 3 ) ) PUNCT iajs-2928 127 4 . . PUNCT iajs-2928 128 1 similarly similarly ADV iajs-2928 128 2 , , PUNCT iajs-2928 128 3 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ PUNCT iajs-2928 129 1 + + CCONJ iajs-2928 129 2 𝑠𝐸𝑦 𝑠𝐸𝑦 NUM iajs-2928 129 3 ) ) PUNCT iajs-2928 129 4 = = SYM iajs-2928 129 5 π‘Ÿπ‘“(𝐸π‘₯ π‘Ÿπ‘“(𝐸π‘₯ PROPN iajs-2928 129 6 ) ) PUNCT iajs-2928 129 7 + + CCONJ iajs-2928 129 8 𝑠𝑓(𝐸𝑦 𝑠𝑓(𝐸𝑦 ADJ iajs-2928 129 9 ) ) PUNCT iajs-2928 129 10 βˆ€π‘₯ βˆ€π‘₯ NOUN iajs-2928 129 11 , , PUNCT iajs-2928 129 12 𝑦 𝑦 NOUN iajs-2928 129 13 ∈ ∈ NOUN iajs-2928 129 14 𝐴. 𝐴. PROPN iajs-2928 129 15 hence hence ADV iajs-2928 129 16 , , PUNCT iajs-2928 129 17 𝑓 𝑓 PRON iajs-2928 129 18 is be AUX iajs-2928 129 19 ( ( PUNCT iajs-2928 129 20 𝑝 𝑝 NOUN iajs-2928 129 21 , , PUNCT iajs-2928 129 22 𝐸)-convex 𝐸)-convex PRON iajs-2928 129 23 function function VERB iajs-2928 129 24 as as SCONJ iajs-2928 129 25 required require VERB iajs-2928 129 26 . . PUNCT iajs-2928 130 1 now now ADV iajs-2928 130 2 , , PUNCT iajs-2928 130 3 take take VERB iajs-2928 130 4 π‘₯ π‘₯ X iajs-2928 130 5 = = PUNCT iajs-2928 130 6 ( ( PUNCT iajs-2928 130 7 βˆ’1 βˆ’1 NOUN iajs-2928 130 8 2 2 NUM iajs-2928 130 9 , , PUNCT iajs-2928 130 10 βˆ’1 βˆ’1 NOUN iajs-2928 130 11 2 2 NUM iajs-2928 130 12 ) ) PUNCT iajs-2928 130 13 , , PUNCT iajs-2928 130 14 𝑦 𝑦 NOUN iajs-2928 130 15 = = PUNCT iajs-2928 130 16 ( ( PUNCT iajs-2928 130 17 βˆ’1 βˆ’1 NOUN iajs-2928 130 18 4 4 NUM iajs-2928 130 19 , , PUNCT iajs-2928 130 20 βˆ’1 βˆ’1 NOUN iajs-2928 130 21 4 4 NUM iajs-2928 130 22 ) ) PUNCT iajs-2928 130 23 , , PUNCT iajs-2928 130 24 π‘Ÿ π‘Ÿ X iajs-2928 130 25 = = SYM iajs-2928 130 26 𝑠 𝑠 PROPN iajs-2928 130 27 = = SYM iajs-2928 130 28 1 1 NUM iajs-2928 130 29 4 4 NUM iajs-2928 130 30 and and CCONJ iajs-2928 130 31 𝑝 𝑝 NOUN iajs-2928 130 32 = = SYM iajs-2928 130 33 1 1 NUM iajs-2928 130 34 2 2 NUM iajs-2928 130 35 . . PUNCT iajs-2928 131 1 then then ADV iajs-2928 131 2 , , PUNCT iajs-2928 131 3 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ AUX iajs-2928 131 4 + + CCONJ iajs-2928 131 5 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 131 6 ) ) PUNCT iajs-2928 131 7 = = SYM iajs-2928 132 1 𝑓 𝑓 PROPN iajs-2928 132 2 ( ( PUNCT iajs-2928 132 3 π‘Ÿ π‘Ÿ NOUN iajs-2928 132 4 ( ( PUNCT iajs-2928 132 5 1 1 NUM iajs-2928 132 6 4 4 NUM iajs-2928 132 7 , , PUNCT iajs-2928 132 8 1 1 NUM iajs-2928 132 9 4 4 NUM iajs-2928 132 10 ) ) PUNCT iajs-2928 133 1 + + CCONJ iajs-2928 133 2 𝑠 𝑠 X iajs-2928 133 3 ( ( PUNCT iajs-2928 133 4 9 9 NUM iajs-2928 133 5 16 16 NUM iajs-2928 133 6 , , PUNCT iajs-2928 133 7 9 9 NUM iajs-2928 133 8 16 16 NUM iajs-2928 133 9 ) ) PUNCT iajs-2928 133 10 ) ) PUNCT iajs-2928 134 1 = = SYM iajs-2928 134 2 0 0 PUNCT iajs-2928 134 3 > > X iajs-2928 134 4 max max PROPN iajs-2928 134 5 { { PUNCT iajs-2928 134 6 βˆ’ βˆ’ PROPN iajs-2928 134 7 1 1 NUM iajs-2928 134 8 3 3 NUM iajs-2928 134 9 , , PUNCT iajs-2928 134 10 βˆ’ βˆ’ PROPN iajs-2928 134 11 1 1 NUM iajs-2928 134 12 6 6 NUM iajs-2928 134 13 } } PUNCT iajs-2928 134 14 = = PUNCT iajs-2928 134 15 βˆ’ βˆ’ PROPN iajs-2928 134 16 1 1 NUM iajs-2928 134 17 6 6 NUM iajs-2928 134 18 . . PUNCT iajs-2928 135 1 ihjpas ihjpas PROPN iajs-2928 135 2 . . PUNCT iajs-2928 136 1 36(1)2023 36(1)2023 NUM iajs-2928 136 2 359 359 NUM iajs-2928 136 3 also also ADV iajs-2928 136 4 , , PUNCT iajs-2928 136 5 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 136 6 ) ) PUNCT iajs-2928 136 7 < < X iajs-2928 136 8 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 136 9 ) ) PUNCT iajs-2928 136 10 and and CCONJ iajs-2928 136 11 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 136 12 + + CCONJ iajs-2928 136 13 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 136 14 ) ) PUNCT iajs-2928 136 15 = = PUNCT iajs-2928 136 16 0 0 PUNCT iajs-2928 136 17 > > PUNCT iajs-2928 136 18 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 136 19 ) ) PUNCT iajs-2928 137 1 + + CCONJ iajs-2928 137 2 ( ( PUNCT iajs-2928 137 3 βˆ’π‘Ÿπ‘ )𝑏(π‘₯ βˆ’π‘Ÿπ‘ )𝑏(π‘₯ NOUN iajs-2928 137 4 , , PUNCT iajs-2928 137 5 𝑦 𝑦 NOUN iajs-2928 137 6 ) ) PUNCT iajs-2928 137 7 = = SYM iajs-2928 138 1 βˆ’ βˆ’ PROPN iajs-2928 138 2 1 1 NUM iajs-2928 138 3 6 6 NUM iajs-2928 138 4 + + CCONJ iajs-2928 138 5 ( ( PUNCT iajs-2928 138 6 βˆ’ βˆ’ PROPN iajs-2928 138 7 1 1 NUM iajs-2928 138 8 16 16 NUM iajs-2928 138 9 ) ) PUNCT iajs-2928 138 10 𝑏(π‘₯ 𝑏(π‘₯ PROPN iajs-2928 138 11 , , PUNCT iajs-2928 138 12 𝑦 𝑦 NOUN iajs-2928 138 13 ) ) PUNCT iajs-2928 138 14 , , PUNCT iajs-2928 138 15 for for ADP iajs-2928 138 16 a a DET iajs-2928 138 17 strictly strictly ADV iajs-2928 138 18 positive positive ADJ iajs-2928 138 19 function function NOUN iajs-2928 138 20 𝑏(π‘₯ 𝑏(π‘₯ NOUN iajs-2928 138 21 , , PUNCT iajs-2928 138 22 𝑦 𝑦 NOUN iajs-2928 138 23 ) ) PUNCT iajs-2928 138 24 . . PUNCT iajs-2928 139 1 hence hence ADV iajs-2928 139 2 , , PUNCT iajs-2928 139 3 𝑓 𝑓 PRON iajs-2928 139 4 is be AUX iajs-2928 139 5 neither neither CCONJ iajs-2928 139 6 quasi quasi NOUN iajs-2928 139 7 semi semi ADJ iajs-2928 139 8 ( ( PUNCT iajs-2928 139 9 1 1 NUM iajs-2928 139 10 2 2 NUM iajs-2928 139 11 , , PUNCT iajs-2928 139 12 𝐸)-convex 𝐸)-convex X iajs-2928 139 13 nor nor CCONJ iajs-2928 139 14 pseudo pseudo NOUN iajs-2928 139 15 semi semi ADJ iajs-2928 139 16 ( ( PUNCT iajs-2928 139 17 1 1 NUM iajs-2928 139 18 2 2 NUM iajs-2928 139 19 , , PUNCT iajs-2928 139 20 𝐸)-convex 𝐸)-convex X iajs-2928 139 21 . . PUNCT iajs-2928 140 1 the the DET iajs-2928 140 2 relation relation NOUN iajs-2928 140 3 between between ADP iajs-2928 140 4 pseudo pseudo NOUN iajs-2928 140 5 semi semi ADJ iajs-2928 140 6 ( ( PUNCT iajs-2928 140 7 𝑝 𝑝 PROPN iajs-2928 140 8 , , PUNCT iajs-2928 140 9 𝐸)convex 𝐸)convex PROPN iajs-2928 140 10 function function NOUN iajs-2928 140 11 and and CCONJ iajs-2928 140 12 quasi quasi NOUN iajs-2928 140 13 semi semi ADJ iajs-2928 140 14 ( ( PUNCT iajs-2928 140 15 𝑝 𝑝 NOUN iajs-2928 140 16 , , PUNCT iajs-2928 140 17 𝐸)-convex 𝐸)-convex NOUN iajs-2928 140 18 functions function NOUN iajs-2928 140 19 are be AUX iajs-2928 140 20 given give VERB iajs-2928 140 21 in in ADP iajs-2928 140 22 the the DET iajs-2928 140 23 next next ADJ iajs-2928 140 24 proposition proposition NOUN iajs-2928 140 25 and and CCONJ iajs-2928 140 26 example example NOUN iajs-2928 140 27 . . PUNCT iajs-2928 141 1 proposition proposition NOUN iajs-2928 141 2 2.5 2.5 NUM iajs-2928 141 3 . . PUNCT iajs-2928 142 1 every every DET iajs-2928 142 2 pseudo pseudo NOUN iajs-2928 142 3 semi semi X iajs-2928 142 4 ( ( PUNCT iajs-2928 142 5 𝑝 𝑝 PROPN iajs-2928 142 6 , , PUNCT iajs-2928 142 7 𝐸)convex 𝐸)convex PROPN iajs-2928 142 8 function function NOUN iajs-2928 142 9 on on ADP iajs-2928 142 10 𝐴 𝐴 PROPN iajs-2928 142 11 is be AUX iajs-2928 142 12 quasi quasi NOUN iajs-2928 142 13 semi semi ADJ iajs-2928 142 14 ( ( PUNCT iajs-2928 142 15 𝑝 𝑝 PROPN iajs-2928 142 16 , , PUNCT iajs-2928 142 17 𝐸)convex 𝐸)convex NOUN iajs-2928 142 18 . . PUNCT iajs-2928 143 1 proof proof NOUN iajs-2928 143 2 . . PUNCT iajs-2928 144 1 let let VERB iajs-2928 144 2 π‘₯ π‘₯ PRON iajs-2928 144 3 , , PUNCT iajs-2928 144 4 𝑦 𝑦 NOUN iajs-2928 144 5 ∈ ∈ PROPN iajs-2928 144 6 𝐴 𝐴 PROPN iajs-2928 144 7 such such ADJ iajs-2928 144 8 that that SCONJ iajs-2928 144 9 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 144 10 ) ) PUNCT iajs-2928 144 11 < < X iajs-2928 144 12 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 144 13 ) ) PUNCT iajs-2928 144 14 . . PUNCT iajs-2928 145 1 since since SCONJ iajs-2928 145 2 𝑓 𝑓 PROPN iajs-2928 145 3 is be AUX iajs-2928 145 4 pseudo pseudo NOUN iajs-2928 145 5 semi semi ADJ iajs-2928 145 6 ( ( PUNCT iajs-2928 145 7 𝑝 𝑝 NOUN iajs-2928 145 8 , , PUNCT iajs-2928 145 9 𝐸)convex 𝐸)convex PROPN iajs-2928 145 10 , , PUNCT iajs-2928 145 11 then then ADV iajs-2928 145 12 we we PRON iajs-2928 145 13 have have VERB iajs-2928 145 14 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ NOUN iajs-2928 145 15 + + CCONJ iajs-2928 145 16 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 145 17 ) ) PUNCT iajs-2928 145 18 ≀ ≀ NOUN iajs-2928 145 19 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 145 20 ) ) PUNCT iajs-2928 146 1 + + CCONJ iajs-2928 146 2 ( ( PUNCT iajs-2928 146 3 βˆ’π‘Ÿπ‘ )𝑏(π‘₯ βˆ’π‘Ÿπ‘ )𝑏(π‘₯ NOUN iajs-2928 146 4 , , PUNCT iajs-2928 146 5 𝑦 𝑦 NOUN iajs-2928 146 6 ) ) PUNCT iajs-2928 146 7 ≀ ≀ NOUN iajs-2928 146 8 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 146 9 ) ) PUNCT iajs-2928 146 10 = = SYM iajs-2928 146 11 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ iajs-2928 146 12 { { PUNCT iajs-2928 146 13 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 146 14 ) ) PUNCT iajs-2928 146 15 , , PUNCT iajs-2928 146 16 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 146 17 ) ) PUNCT iajs-2928 146 18 } } PUNCT iajs-2928 146 19 . . PUNCT iajs-2928 147 1 β–  β–  PUNCT iajs-2928 147 2 example example NOUN iajs-2928 147 3 2.6 2.6 NUM iajs-2928 147 4 . . PUNCT iajs-2928 148 1 let let VERB iajs-2928 148 2 𝐴 𝐴 PROPN iajs-2928 148 3 = = PUNCT iajs-2928 148 4 { { PUNCT iajs-2928 148 5 π‘₯ π‘₯ X iajs-2928 148 6 = = SYM iajs-2928 148 7 ( ( PUNCT iajs-2928 148 8 π‘₯1 π‘₯1 NOUN iajs-2928 148 9 , , PUNCT iajs-2928 148 10 … … PUNCT iajs-2928 148 11 , , PUNCT iajs-2928 148 12 π‘₯𝑛 π‘₯𝑛 NOUN iajs-2928 148 13 ) ) PUNCT iajs-2928 148 14 ∈ ∈ PROPN iajs-2928 149 1 𝑅𝑛 𝑅𝑛 VERB iajs-2928 149 2 : : PUNCT iajs-2928 149 3 βˆ‘ βˆ‘ PUNCT iajs-2928 149 4 |π‘₯𝑖| |π‘₯𝑖| NOUN iajs-2928 149 5 1 1 NUM iajs-2928 149 6 2𝑛 2𝑛 NUM iajs-2928 149 7 𝑖=1 𝑖=1 PUNCT iajs-2928 149 8 ≀ ≀ ADV iajs-2928 149 9 1 1 NUM iajs-2928 149 10 } } PUNCT iajs-2928 149 11 , , PUNCT iajs-2928 149 12 and and CCONJ iajs-2928 149 13 𝐸 𝐸 PROPN iajs-2928 149 14 : : PUNCT iajs-2928 149 15 𝑅𝑛→ 𝑅𝑛→ PROPN iajs-2928 149 16 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 149 17 such such DET iajs-2928 149 18 that that PRON iajs-2928 149 19 𝐸(π‘₯ 𝐸(π‘₯ NOUN iajs-2928 149 20 ) ) PUNCT iajs-2928 149 21 = = SYM iajs-2928 149 22 𝐸(π‘₯1 𝐸(π‘₯1 ADJ iajs-2928 149 23 , , PUNCT iajs-2928 149 24 … … PUNCT iajs-2928 149 25 , , PUNCT iajs-2928 149 26 π‘₯𝑛 π‘₯𝑛 NOUN iajs-2928 149 27 ) ) PUNCT iajs-2928 149 28 = = PUNCT iajs-2928 150 1 ( ( PUNCT iajs-2928 150 2 π‘₯1 π‘₯1 NOUN iajs-2928 150 3 2 2 NUM iajs-2928 150 4 , , PUNCT iajs-2928 150 5 … … PUNCT iajs-2928 150 6 , , PUNCT iajs-2928 150 7 π‘₯𝑛 π‘₯𝑛 PROPN iajs-2928 150 8 2 2 NUM iajs-2928 150 9 ) ) PUNCT iajs-2928 150 10 for for ADP iajs-2928 150 11 all all PRON iajs-2928 150 12 π‘₯ π‘₯ DET iajs-2928 150 13 ∈ ∈ NOUN iajs-2928 150 14 𝑅𝑛. 𝑅𝑛. NOUN iajs-2928 150 15 define define VERB iajs-2928 150 16 𝑓 𝑓 PRON iajs-2928 150 17 : : PUNCT iajs-2928 150 18 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 150 19 β†’ β†’ SYM iajs-2928 150 20 𝑅 𝑅 NOUN iajs-2928 150 21 π‘ π‘’π‘β„Ž π‘ π‘’π‘β„Ž NOUN iajs-2928 150 22 π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘Žπ‘‘ ADJ iajs-2928 150 23 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 150 24 ) ) PUNCT iajs-2928 151 1 = = PRON iajs-2928 151 2 { { PUNCT iajs-2928 151 3 βˆ’1 βˆ’1 NOUN iajs-2928 151 4 if if SCONJ iajs-2928 151 5 π‘₯𝑖 π‘₯𝑖 NUM iajs-2928 151 6 = = SYM iajs-2928 151 7 0 0 PUNCT iajs-2928 152 1 βˆ€π‘– βˆ€π‘– PROPN iajs-2928 152 2 = = SYM iajs-2928 152 3 1 1 NUM iajs-2928 152 4 , , PUNCT iajs-2928 152 5 … … PUNCT iajs-2928 152 6 , , PUNCT iajs-2928 152 7 𝑛 𝑛 PROPN iajs-2928 152 8 0 0 NUM iajs-2928 152 9 o. o. PROPN iajs-2928 152 10 w. w. PROPN iajs-2928 152 11 from from ADP iajs-2928 152 12 [ [ X iajs-2928 152 13 9 9 NUM iajs-2928 152 14 , , PUNCT iajs-2928 152 15 example example NOUN iajs-2928 152 16 2.4 2.4 NUM iajs-2928 152 17 ] ] PUNCT iajs-2928 152 18 , , PUNCT iajs-2928 152 19 the the DET iajs-2928 152 20 set set NOUN iajs-2928 152 21 𝐴 𝐴 PROPN iajs-2928 152 22 is be AUX iajs-2928 152 23 ( ( PUNCT iajs-2928 152 24 1 1 NUM iajs-2928 152 25 2 2 NUM iajs-2928 152 26 , , PUNCT iajs-2928 152 27 𝐸)-convex 𝐸)-convex X iajs-2928 152 28 . . PUNCT iajs-2928 153 1 next next ADV iajs-2928 153 2 , , PUNCT iajs-2928 153 3 we we PRON iajs-2928 153 4 show show VERB iajs-2928 153 5 that that SCONJ iajs-2928 153 6 𝑓 𝑓 PRON iajs-2928 153 7 is be AUX iajs-2928 153 8 quasi quasi NOUN iajs-2928 153 9 semi semi ADJ iajs-2928 153 10 ( ( PUNCT iajs-2928 153 11 1 1 NUM iajs-2928 153 12 2 2 NUM iajs-2928 153 13 , , PUNCT iajs-2928 153 14 𝐸)convex 𝐸)convex PROPN iajs-2928 153 15 function function NOUN iajs-2928 153 16 on on ADP iajs-2928 153 17 𝐴. 𝐴. PROPN iajs-2928 153 18 to to ADP iajs-2928 153 19 this this DET iajs-2928 153 20 end end NOUN iajs-2928 153 21 , , PUNCT iajs-2928 153 22 let let VERB iajs-2928 153 23 π‘₯ π‘₯ PRON iajs-2928 153 24 , , PUNCT iajs-2928 153 25 𝑦 𝑦 NOUN iajs-2928 153 26 ∈ ∈ PROPN iajs-2928 153 27 𝐴 𝐴 PROPN iajs-2928 153 28 and and CCONJ iajs-2928 153 29 we we PRON iajs-2928 153 30 consider consider VERB iajs-2928 153 31 three three NUM iajs-2928 153 32 cases case NOUN iajs-2928 153 33 : : PUNCT iajs-2928 153 34 case case NOUN iajs-2928 153 35 1 1 NUM iajs-2928 153 36 : : PUNCT iajs-2928 154 1 if if SCONJ iajs-2928 154 2 π‘₯𝑖 π‘₯𝑖 PRON iajs-2928 154 3 = = PUNCT iajs-2928 154 4 𝑦𝑖 𝑦𝑖 PROPN iajs-2928 154 5 = = NOUN iajs-2928 154 6 0 0 PUNCT iajs-2928 155 1 βˆ€π‘– βˆ€π‘– PROPN iajs-2928 155 2 = = SYM iajs-2928 155 3 1 1 NUM iajs-2928 155 4 , , PUNCT iajs-2928 155 5 … … PUNCT iajs-2928 155 6 , , PUNCT iajs-2928 155 7 𝑛 𝑛 NOUN iajs-2928 155 8 , , PUNCT iajs-2928 155 9 then then ADV iajs-2928 155 10 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ PUNCT iajs-2928 155 11 + + CCONJ iajs-2928 155 12 𝑠𝐸𝑦 𝑠𝐸𝑦 NUM iajs-2928 155 13 ) ) PUNCT iajs-2928 155 14 = = SYM iajs-2928 156 1 𝑓(0 𝑓(0 PROPN iajs-2928 156 2 , , PUNCT iajs-2928 156 3 … … PUNCT iajs-2928 156 4 , , PUNCT iajs-2928 156 5 0 0 NUM iajs-2928 156 6 ) ) PUNCT iajs-2928 156 7 = = SYM iajs-2928 156 8 βˆ’1 βˆ’1 NOUN iajs-2928 156 9 = = NOUN iajs-2928 156 10 { { PUNCT iajs-2928 156 11 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 156 12 ) ) PUNCT iajs-2928 156 13 , , PUNCT iajs-2928 156 14 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 156 15 ) ) PUNCT iajs-2928 156 16 } } PUNCT iajs-2928 156 17 . . PUNCT iajs-2928 157 1 case case NOUN iajs-2928 157 2 2 2 NUM iajs-2928 157 3 : : PUNCT iajs-2928 158 1 if if SCONJ iajs-2928 158 2 π‘₯𝑖 π‘₯𝑖 PROPN iajs-2928 158 3 β‰  β‰  PROPN iajs-2928 158 4 0 0 NUM iajs-2928 159 1 βˆ€π‘– βˆ€π‘– PRON iajs-2928 159 2 = = SYM iajs-2928 159 3 1 1 NUM iajs-2928 159 4 , , PUNCT iajs-2928 159 5 … … PUNCT iajs-2928 159 6 , , PUNCT iajs-2928 159 7 𝑛 𝑛 PROPN iajs-2928 159 8 and and CCONJ iajs-2928 159 9 𝑦𝑖 𝑦𝑖 NUM iajs-2928 159 10 β‰  β‰  PROPN iajs-2928 159 11 0 0 NUM iajs-2928 159 12 for for ADP iajs-2928 159 13 some some PRON iajs-2928 159 14 𝑖 𝑖 NOUN iajs-2928 159 15 = = NOUN iajs-2928 159 16 1 1 NUM iajs-2928 159 17 , , PUNCT iajs-2928 159 18 … … PUNCT iajs-2928 159 19 , , PUNCT iajs-2928 159 20 𝑛. 𝑛. NOUN iajs-2928 159 21 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ PUNCT iajs-2928 159 22 + + SYM iajs-2928 159 23 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 159 24 ) ) PUNCT iajs-2928 159 25 = = PRON iajs-2928 159 26 { { PUNCT iajs-2928 159 27 βˆ’1 βˆ’1 NOUN iajs-2928 159 28 if if SCONJ iajs-2928 159 29 π‘Ÿ π‘Ÿ NOUN iajs-2928 159 30 π‘₯𝑖 π‘₯𝑖 ADP iajs-2928 159 31 2 2 NUM iajs-2928 159 32 + + CCONJ iajs-2928 159 33 𝑠 𝑠 PART iajs-2928 159 34 𝑦𝑖 𝑦𝑖 ADP iajs-2928 159 35 2 2 NUM iajs-2928 159 36 = = SYM iajs-2928 159 37 0 0 PUNCT iajs-2928 160 1 βˆ€π‘– βˆ€π‘– PROPN iajs-2928 160 2 = = SYM iajs-2928 160 3 1 1 NUM iajs-2928 160 4 , , PUNCT iajs-2928 160 5 … … PUNCT iajs-2928 160 6 , , PUNCT iajs-2928 160 7 𝑛 𝑛 PRON iajs-2928 160 8 0 0 NUM iajs-2928 160 9 o. o. PROPN iajs-2928 160 10 w. w. PROPN iajs-2928 160 11 ≀ ≀ PROPN iajs-2928 161 1 π‘šπ‘Žπ‘₯{𝑓(π‘₯ π‘šπ‘Žπ‘₯{𝑓(π‘₯ NOUN iajs-2928 161 2 ) ) PUNCT iajs-2928 161 3 , , PUNCT iajs-2928 161 4 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 161 5 ) ) PUNCT iajs-2928 161 6 } } PUNCT iajs-2928 161 7 = = SYM iajs-2928 161 8 0 0 X iajs-2928 161 9 . . PUNCT iajs-2928 162 1 case case NOUN iajs-2928 162 2 3 3 NUM iajs-2928 162 3 : : PUNCT iajs-2928 163 1 if if SCONJ iajs-2928 163 2 π‘₯𝑖 π‘₯𝑖 PROPN iajs-2928 163 3 β‰  β‰  PROPN iajs-2928 163 4 0 0 NUM iajs-2928 163 5 and and CCONJ iajs-2928 163 6 𝑦𝑖 𝑦𝑖 X iajs-2928 163 7 = = NOUN iajs-2928 163 8 0 0 NUM iajs-2928 163 9 for for ADP iajs-2928 163 10 some some PRON iajs-2928 163 11 𝑖 𝑖 NOUN iajs-2928 163 12 = = NOUN iajs-2928 163 13 1 1 NUM iajs-2928 163 14 , , PUNCT iajs-2928 163 15 … … PUNCT iajs-2928 163 16 , , PUNCT iajs-2928 163 17 𝑛. 𝑛. NOUN iajs-2928 163 18 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ PUNCT iajs-2928 163 19 + + SYM iajs-2928 163 20 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 163 21 ) ) PUNCT iajs-2928 163 22 = = SYM iajs-2928 163 23 0 0 PUNCT iajs-2928 163 24 = = SYM iajs-2928 163 25 π‘šπ‘Žπ‘₯{𝑓(π‘₯ π‘šπ‘Žπ‘₯{𝑓(π‘₯ NOUN iajs-2928 163 26 ) ) PUNCT iajs-2928 163 27 , , PUNCT iajs-2928 163 28 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 163 29 ) ) PUNCT iajs-2928 163 30 } } PUNCT iajs-2928 163 31 . . PUNCT iajs-2928 164 1 from from ADP iajs-2928 164 2 all all DET iajs-2928 164 3 cases case NOUN iajs-2928 164 4 , , PUNCT iajs-2928 164 5 we we PRON iajs-2928 164 6 have have VERB iajs-2928 164 7 𝑓 𝑓 PRON iajs-2928 164 8 is be AUX iajs-2928 164 9 quasi quasi NOUN iajs-2928 164 10 semi semi ADJ iajs-2928 164 11 ( ( PUNCT iajs-2928 164 12 1 1 NUM iajs-2928 164 13 2 2 NUM iajs-2928 164 14 , , PUNCT iajs-2928 164 15 𝐸)-convex 𝐸)-convex PRON iajs-2928 164 16 function function VERB iajs-2928 164 17 on on ADP iajs-2928 164 18 𝐴. 𝐴. PROPN iajs-2928 164 19 to to PART iajs-2928 164 20 show show VERB iajs-2928 164 21 𝑓 𝑓 PRON iajs-2928 164 22 is be AUX iajs-2928 164 23 not not PART iajs-2928 164 24 pseudo pseudo NOUN iajs-2928 164 25 semi semi ADV iajs-2928 164 26 ( ( PUNCT iajs-2928 164 27 1 1 NUM iajs-2928 164 28 2 2 NUM iajs-2928 164 29 , , PUNCT iajs-2928 164 30 𝐸)-convex 𝐸)-convex PRON iajs-2928 164 31 function function VERB iajs-2928 164 32 on on ADP iajs-2928 164 33 𝐴 𝐴 PROPN iajs-2928 164 34 , , PUNCT iajs-2928 164 35 𝑑ake 𝑑ake NOUN iajs-2928 164 36 π‘₯ π‘₯ NOUN iajs-2928 164 37 = = SYM iajs-2928 164 38 ( ( PUNCT iajs-2928 164 39 0 0 NUM iajs-2928 164 40 , , PUNCT iajs-2928 164 41 … … PUNCT iajs-2928 164 42 , , PUNCT iajs-2928 164 43 1 1 NUM iajs-2928 164 44 ) ) PUNCT iajs-2928 164 45 , , PUNCT iajs-2928 164 46 𝑦 𝑦 NOUN iajs-2928 164 47 = = SYM iajs-2928 164 48 ( ( PUNCT iajs-2928 164 49 0 0 NUM iajs-2928 164 50 , , PUNCT iajs-2928 164 51 … … PUNCT iajs-2928 164 52 , , PUNCT iajs-2928 164 53 0 0 NUM iajs-2928 164 54 ) ) PUNCT iajs-2928 164 55 such such ADJ iajs-2928 164 56 that that SCONJ iajs-2928 164 57 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 164 58 ) ) PUNCT iajs-2928 164 59 < < X iajs-2928 164 60 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 164 61 ) ) PUNCT iajs-2928 164 62 . . PUNCT iajs-2928 165 1 let let VERB iajs-2928 165 2 π‘Ÿ π‘Ÿ NOUN iajs-2928 165 3 = = SYM iajs-2928 165 4 𝑠 𝑠 PROPN iajs-2928 165 5 = = SYM iajs-2928 165 6 1 1 NUM iajs-2928 165 7 4 4 NUM iajs-2928 165 8 then then ADV iajs-2928 165 9 there there PRON iajs-2928 165 10 exist exist VERB iajs-2928 165 11 strictly strictly ADV iajs-2928 165 12 positive positive ADJ iajs-2928 165 13 function function NOUN iajs-2928 165 14 𝑏(π‘₯ 𝑏(π‘₯ NOUN iajs-2928 165 15 , , PUNCT iajs-2928 165 16 𝑦 𝑦 NOUN iajs-2928 165 17 ) ) PUNCT iajs-2928 165 18 = = SYM iajs-2928 166 1 3 3 NUM iajs-2928 166 2 such such ADJ iajs-2928 166 3 that that PRON iajs-2928 166 4 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 166 5 + + CCONJ iajs-2928 166 6 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 166 7 ) ) PUNCT iajs-2928 166 8 = = SYM iajs-2928 167 1 𝑓(π‘ŸπΈ(0 𝑓(π‘ŸπΈ(0 X iajs-2928 167 2 , , PUNCT iajs-2928 167 3 … … PUNCT iajs-2928 167 4 , , PUNCT iajs-2928 167 5 1 1 NUM iajs-2928 167 6 ) ) PUNCT iajs-2928 167 7 , , PUNCT iajs-2928 168 1 + + PROPN iajs-2928 168 2 𝑠𝐸(0 𝑠𝐸(0 NUM iajs-2928 168 3 , , PUNCT iajs-2928 168 4 … … PUNCT iajs-2928 168 5 , , PUNCT iajs-2928 168 6 0 0 NUM iajs-2928 168 7 ) ) PUNCT iajs-2928 168 8 ) ) PUNCT iajs-2928 169 1 = = SYM iajs-2928 169 2 𝑓 𝑓 X iajs-2928 169 3 ( ( PUNCT iajs-2928 169 4 0 0 NUM iajs-2928 169 5 , , PUNCT iajs-2928 169 6 … … PUNCT iajs-2928 169 7 , , PUNCT iajs-2928 169 8 1 1 NUM iajs-2928 169 9 2 2 NUM iajs-2928 169 10 𝑠 𝑠 NOUN iajs-2928 169 11 ) ) PUNCT iajs-2928 169 12 = = PUNCT iajs-2928 169 13 0 0 PUNCT iajs-2928 169 14 > > PUNCT iajs-2928 169 15 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 169 16 ) ) PUNCT iajs-2928 170 1 + + CCONJ iajs-2928 170 2 ( ( PUNCT iajs-2928 170 3 βˆ’π‘Ÿπ‘ )𝑏(π‘₯ βˆ’π‘Ÿπ‘ )𝑏(π‘₯ NOUN iajs-2928 170 4 , , PUNCT iajs-2928 170 5 𝑦 𝑦 NOUN iajs-2928 170 6 ) ) PUNCT iajs-2928 170 7 = = SYM iajs-2928 170 8 βˆ’1 βˆ’1 NOUN iajs-2928 170 9 βˆ’ βˆ’ NOUN iajs-2928 170 10 3 3 NUM iajs-2928 170 11 16 16 NUM iajs-2928 170 12 = = SYM iajs-2928 170 13 βˆ’ βˆ’ PROPN iajs-2928 170 14 19 19 NUM iajs-2928 170 15 16 16 NUM iajs-2928 170 16 hence hence ADV iajs-2928 170 17 , , PUNCT iajs-2928 170 18 𝑓is 𝑓is X iajs-2928 170 19 not not PART iajs-2928 170 20 pseudo pseudo NOUN iajs-2928 170 21 semi semi ADJ iajs-2928 170 22 ( ( PUNCT iajs-2928 170 23 𝑝 𝑝 NOUN iajs-2928 170 24 , , PUNCT iajs-2928 170 25 𝐸)-convex 𝐸)-convex X iajs-2928 170 26 . . PUNCT iajs-2928 171 1 ihjpas ihjpas PROPN iajs-2928 171 2 . . PUNCT iajs-2928 172 1 36(1)2023 36(1)2023 NUM iajs-2928 172 2 360 360 NUM iajs-2928 172 3 proposition proposition NOUN iajs-2928 172 4 2.7 2.7 NUM iajs-2928 172 5 . . PUNCT iajs-2928 173 1 the the DET iajs-2928 173 2 function function NOUN iajs-2928 173 3 𝑓 𝑓 PROPN iajs-2928 173 4 is be AUX iajs-2928 173 5 quasi quasi NOUN iajs-2928 173 6 semi semi ADJ iajs-2928 173 7 ( ( PUNCT iajs-2928 173 8 𝑝 𝑝 NOUN iajs-2928 173 9 , , PUNCT iajs-2928 173 10 𝐸)-convex 𝐸)-convex PROPN iajs-2928 173 11 on on ADP iajs-2928 173 12 𝐴 𝐴 PROPN iajs-2928 173 13 if if SCONJ iajs-2928 174 1 and and CCONJ iajs-2928 174 2 only only ADV iajs-2928 174 3 if if SCONJ iajs-2928 174 4 the the DET iajs-2928 174 5 level level NOUN iajs-2928 174 6 set set VERB iajs-2928 174 7 𝐾𝑛 𝐾𝑛 PROPN iajs-2928 174 8 = = PUNCT iajs-2928 174 9 { { PUNCT iajs-2928 174 10 π‘₯ π‘₯ NOUN iajs-2928 174 11 ∈ ∈ PROPN iajs-2928 174 12 𝐴 𝐴 PROPN iajs-2928 174 13 : : PUNCT iajs-2928 174 14 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 174 15 ) ) PUNCT iajs-2928 174 16 ≀ ≀ NOUN iajs-2928 174 17 𝑛 𝑛 X iajs-2928 174 18 } } PUNCT iajs-2928 174 19 is be AUX iajs-2928 174 20 ( ( PUNCT iajs-2928 174 21 𝑝 𝑝 NOUN iajs-2928 174 22 , , PUNCT iajs-2928 174 23 𝐸)-convex 𝐸)-convex PRON iajs-2928 174 24 set set VERB iajs-2928 174 25 for for ADP iajs-2928 174 26 all all DET iajs-2928 174 27 𝑛 𝑛 PRON iajs-2928 174 28 ∈ ∈ NOUN iajs-2928 174 29 𝑅. 𝑅. NOUN iajs-2928 174 30 proof proof NOUN iajs-2928 174 31 . . PUNCT iajs-2928 175 1 let let VERB iajs-2928 175 2 𝑓 𝑓 PRON iajs-2928 175 3 is be AUX iajs-2928 175 4 quasi quasi NOUN iajs-2928 175 5 semi semi ADJ iajs-2928 175 6 ( ( PUNCT iajs-2928 175 7 𝑝 𝑝 NOUN iajs-2928 175 8 , , PUNCT iajs-2928 175 9 𝐸)-convex 𝐸)-convex PROPN iajs-2928 175 10 on on ADP iajs-2928 175 11 ( ( PUNCT iajs-2928 175 12 𝑝 𝑝 PROPN iajs-2928 175 13 , , PUNCT iajs-2928 175 14 𝐸)-convex 𝐸)-convex PRON iajs-2928 175 15 set set VERB iajs-2928 175 16 𝐴. 𝐴. PROPN iajs-2928 175 17 then then ADV iajs-2928 175 18 , , PUNCT iajs-2928 175 19 for for ADP iajs-2928 175 20 any any DET iajs-2928 175 21 π‘₯ π‘₯ PROPN iajs-2928 175 22 , , PUNCT iajs-2928 175 23 𝑦 𝑦 NOUN iajs-2928 175 24 ∈ ∈ NOUN iajs-2928 175 25 𝐾𝑛 𝐾𝑛 PROPN iajs-2928 175 26 , , PUNCT iajs-2928 175 27 we we PRON iajs-2928 175 28 have have VERB iajs-2928 175 29 π‘ŸπΈπ‘₯ π‘ŸπΈπ‘₯ NUM iajs-2928 175 30 + + NUM iajs-2928 175 31 𝑠𝐸𝑦 𝑠𝐸𝑦 PROPN iajs-2928 175 32 ∈ ∈ PROPN iajs-2928 175 33 𝐴 𝐴 PROPN iajs-2928 175 34 , , PUNCT iajs-2928 175 35 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 175 36 ) ) PUNCT iajs-2928 175 37 ≀ ≀ NUM iajs-2928 175 38 𝑛 𝑛 NOUN iajs-2928 175 39 , , PUNCT iajs-2928 175 40 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 175 41 ) ) PUNCT iajs-2928 175 42 ≀ ≀ NUM iajs-2928 175 43 𝑛 𝑛 NOUN iajs-2928 175 44 , , PUNCT iajs-2928 175 45 and and CCONJ iajs-2928 175 46 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 175 47 + + NUM iajs-2928 175 48 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 175 49 ) ) PUNCT iajs-2928 175 50 ≀ ≀ NOUN iajs-2928 175 51 π‘šπ‘Žπ‘₯{𝑓(π‘₯ π‘šπ‘Žπ‘₯{𝑓(π‘₯ NOUN iajs-2928 175 52 ) ) PUNCT iajs-2928 175 53 , , PUNCT iajs-2928 175 54 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 175 55 ) ) PUNCT iajs-2928 175 56 } } PUNCT iajs-2928 175 57 ≀ ≀ NOUN iajs-2928 175 58 𝑛. 𝑛. NOUN iajs-2928 175 59 it it PRON iajs-2928 175 60 follows follow VERB iajs-2928 175 61 that that SCONJ iajs-2928 175 62 π‘ŸπΈπ‘₯ π‘ŸπΈπ‘₯ PRON iajs-2928 175 63 + + X iajs-2928 175 64 𝑠𝐸𝑦 𝑠𝐸𝑦 PROPN iajs-2928 175 65 ∈ ∈ PROPN iajs-2928 175 66 𝐾𝑛. 𝐾𝑛. NOUN iajs-2928 175 67 conversely conversely ADV iajs-2928 175 68 suppose suppose VERB iajs-2928 175 69 that that SCONJ iajs-2928 175 70 𝐾𝑛 𝐾𝑛 PROPN iajs-2928 175 71 is be AUX iajs-2928 175 72 ( ( PUNCT iajs-2928 175 73 𝑝 𝑝 NOUN iajs-2928 175 74 , , PUNCT iajs-2928 175 75 𝐸)-convex 𝐸)-convex PROPN iajs-2928 175 76 for for ADP iajs-2928 175 77 all all DET iajs-2928 175 78 𝑛 𝑛 DET iajs-2928 175 79 ∈ ∈ NOUN iajs-2928 175 80 𝑅. 𝑅. NOUN iajs-2928 175 81 let let VERB iajs-2928 175 82 𝑛 𝑛 PRON iajs-2928 175 83 = = PUNCT iajs-2928 175 84 π‘šπ‘Žπ‘₯{𝑓(π‘₯ π‘šπ‘Žπ‘₯{𝑓(π‘₯ NOUN iajs-2928 175 85 ) ) PUNCT iajs-2928 175 86 , , PUNCT iajs-2928 175 87 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 175 88 ) ) PUNCT iajs-2928 175 89 } } PUNCT iajs-2928 175 90 . . PUNCT iajs-2928 176 1 since since SCONJ iajs-2928 176 2 𝐴 𝐴 PROPN iajs-2928 176 3 is be AUX iajs-2928 176 4 ( ( PUNCT iajs-2928 176 5 𝑝 𝑝 NOUN iajs-2928 176 6 , , PUNCT iajs-2928 176 7 𝐸)-convex 𝐸)-convex PRON iajs-2928 176 8 set set VERB iajs-2928 176 9 then then ADV iajs-2928 176 10 π‘ŸπΈπ‘₯ π‘ŸπΈπ‘₯ NUM iajs-2928 176 11 + + NUM iajs-2928 176 12 𝑠𝐸𝑦 𝑠𝐸𝑦 PROPN iajs-2928 176 13 ∈ ∈ PROPN iajs-2928 176 14 𝐴 𝐴 PROPN iajs-2928 176 15 and and CCONJ iajs-2928 176 16 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ AUX iajs-2928 176 17 + + NUM iajs-2928 176 18 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 176 19 ) ) PUNCT iajs-2928 176 20 ≀ ≀ NOUN iajs-2928 176 21 𝑛 𝑛 ADP iajs-2928 176 22 = = SYM iajs-2928 176 23 π‘šπ‘Žπ‘₯{𝑓(π‘₯ π‘šπ‘Žπ‘₯{𝑓(π‘₯ NOUN iajs-2928 176 24 ) ) PUNCT iajs-2928 176 25 , , PUNCT iajs-2928 176 26 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 176 27 ) ) PUNCT iajs-2928 176 28 } } PUNCT iajs-2928 176 29 . . PUNCT iajs-2928 177 1 hence hence ADV iajs-2928 177 2 , , PUNCT iajs-2928 177 3 𝑓 𝑓 PRON iajs-2928 177 4 is be AUX iajs-2928 177 5 quasi quasi NOUN iajs-2928 177 6 semi semi ADJ iajs-2928 177 7 ( ( PUNCT iajs-2928 177 8 𝑝 𝑝 NOUN iajs-2928 177 9 , , PUNCT iajs-2928 177 10 𝐸)-convex 𝐸)-convex PROPN iajs-2928 177 11 on on ADP iajs-2928 177 12 𝐴. 𝐴. PROPN iajs-2928 177 13 β–  β–  PUNCT iajs-2928 177 14 proposition proposition NOUN iajs-2928 177 15 2.8 2.8 NUM iajs-2928 177 16 . . PUNCT iajs-2928 178 1 if if SCONJ iajs-2928 178 2 𝑓 𝑓 PRON iajs-2928 178 3 is be AUX iajs-2928 178 4 pseudo pseudo NOUN iajs-2928 178 5 semi semi ADJ iajs-2928 178 6 ( ( PUNCT iajs-2928 178 7 𝑝 𝑝 NOUN iajs-2928 178 8 , , PUNCT iajs-2928 178 9 𝐸)-convex 𝐸)-convex PRON iajs-2928 178 10 function function VERB iajs-2928 178 11 on on ADP iajs-2928 178 12 𝐴 𝐴 PROPN iajs-2928 178 13 then then ADV iajs-2928 178 14 the the DET iajs-2928 178 15 level level NOUN iajs-2928 178 16 set set VERB iajs-2928 178 17 𝐾𝑛 𝐾𝑛 PROPN iajs-2928 178 18 is be AUX iajs-2928 178 19 ( ( PUNCT iajs-2928 178 20 𝑝 𝑝 PROPN iajs-2928 178 21 , , PUNCT iajs-2928 178 22 𝐸)convex 𝐸)convex PROPN iajs-2928 178 23 set set PROPN iajs-2928 178 24 . . PUNCT iajs-2928 179 1 proof proof NOUN iajs-2928 179 2 . . PUNCT iajs-2928 180 1 let let VERB iajs-2928 180 2 π‘₯ π‘₯ PRON iajs-2928 180 3 , , PUNCT iajs-2928 180 4 𝑦 𝑦 NOUN iajs-2928 180 5 ∈ ∈ X iajs-2928 181 1 𝐾𝑛. 𝐾𝑛. NOUN iajs-2928 181 2 we we PRON iajs-2928 181 3 show show VERB iajs-2928 181 4 that that SCONJ iajs-2928 181 5 π‘ŸπΈπ‘₯ π‘ŸπΈπ‘₯ PRON iajs-2928 181 6 + + NUM iajs-2928 181 7 𝑠𝐸𝑦 𝑠𝐸𝑦 PROPN iajs-2928 181 8 ∈ ∈ PROPN iajs-2928 181 9 𝐾𝑛. 𝐾𝑛. PROPN iajs-2928 181 10 now now ADV iajs-2928 181 11 , , PUNCT iajs-2928 181 12 since since SCONJ iajs-2928 181 13 𝑓 𝑓 PRON iajs-2928 181 14 is be AUX iajs-2928 181 15 pseudo pseudo NOUN iajs-2928 181 16 semi semi ADJ iajs-2928 181 17 ( ( PUNCT iajs-2928 181 18 𝑝 𝑝 NOUN iajs-2928 181 19 , , PUNCT iajs-2928 181 20 𝐸)-convex 𝐸)-convex PROPN iajs-2928 181 21 . . PUNCT iajs-2928 182 1 then then ADV iajs-2928 182 2 , , PUNCT iajs-2928 182 3 we we PRON iajs-2928 182 4 have have VERB iajs-2928 182 5 a a DET iajs-2928 182 6 strictly strictly ADV iajs-2928 182 7 positive positive ADJ iajs-2928 182 8 function function NOUN iajs-2928 182 9 𝑏(π‘₯ 𝑏(π‘₯ NOUN iajs-2928 182 10 , , PUNCT iajs-2928 182 11 𝑦 𝑦 NOUN iajs-2928 182 12 ) ) PUNCT iajs-2928 182 13 such such ADJ iajs-2928 182 14 that that SCONJ iajs-2928 182 15 if if SCONJ iajs-2928 182 16 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 182 17 ) ) PUNCT iajs-2928 182 18 < < X iajs-2928 182 19 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 182 20 ) ) PUNCT iajs-2928 182 21 then then ADV iajs-2928 182 22 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ PUNCT iajs-2928 182 23 + + CCONJ iajs-2928 182 24 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 182 25 ) ) PUNCT iajs-2928 182 26 ≀ ≀ NOUN iajs-2928 182 27 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 182 28 ) ) PUNCT iajs-2928 183 1 + + CCONJ iajs-2928 183 2 ( ( PUNCT iajs-2928 183 3 βˆ’π‘Ÿπ‘ )𝑏(π‘₯ βˆ’π‘Ÿπ‘ )𝑏(π‘₯ NOUN iajs-2928 183 4 , , PUNCT iajs-2928 183 5 𝑦 𝑦 NOUN iajs-2928 183 6 ) ) PUNCT iajs-2928 183 7 ≀ ≀ NOUN iajs-2928 183 8 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 183 9 ) ) PUNCT iajs-2928 183 10 . . PUNCT iajs-2928 184 1 hence hence ADV iajs-2928 184 2 , , PUNCT iajs-2928 184 3 𝐾𝛼 𝐾𝛼 PROPN iajs-2928 184 4 is be AUX iajs-2928 184 5 ( ( PUNCT iajs-2928 184 6 𝑝 𝑝 NOUN iajs-2928 184 7 , , PUNCT iajs-2928 184 8 𝐸)-convex 𝐸)-convex PRON iajs-2928 184 9 set set VERB iajs-2928 184 10 . . PUNCT iajs-2928 185 1 β–  β–  PUNCT iajs-2928 185 2 the the DET iajs-2928 185 3 converse converse NOUN iajs-2928 185 4 of of ADP iajs-2928 185 5 the the DET iajs-2928 185 6 proceeding proceeding NOUN iajs-2928 185 7 proposition proposition NOUN iajs-2928 185 8 does do AUX iajs-2928 185 9 not not PART iajs-2928 185 10 satisfy satisfy VERB iajs-2928 185 11 as as SCONJ iajs-2928 185 12 it it PRON iajs-2928 185 13 is be AUX iajs-2928 185 14 clarified clarify VERB iajs-2928 185 15 in in ADP iajs-2928 185 16 the the DET iajs-2928 185 17 next next ADJ iajs-2928 185 18 example example NOUN iajs-2928 185 19 . . PUNCT iajs-2928 186 1 example example NOUN iajs-2928 186 2 2.9 2.9 NUM iajs-2928 186 3 . . PUNCT iajs-2928 187 1 let let VERB iajs-2928 187 2 𝑓 𝑓 PRON iajs-2928 187 3 , , PUNCT iajs-2928 187 4 𝐸 𝐸 PROPN iajs-2928 187 5 : : PUNCT iajs-2928 187 6 𝑅 𝑅 PROPN iajs-2928 187 7 β†’ β†’ SYM iajs-2928 187 8 𝑅 𝑅 PROPN iajs-2928 187 9 such such ADJ iajs-2928 187 10 that that SCONJ iajs-2928 187 11 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 187 12 ) ) PUNCT iajs-2928 188 1 = = PRON iajs-2928 188 2 { { PUNCT iajs-2928 188 3 1 1 NUM iajs-2928 188 4 if if SCONJ iajs-2928 188 5 π‘₯ π‘₯ PRON iajs-2928 188 6 ∈ ∈ PROPN iajs-2928 189 1 [ [ X iajs-2928 189 2 0 0 NUM iajs-2928 189 3 , , PUNCT iajs-2928 189 4 ∞ ∞ NUM iajs-2928 189 5 ) ) PUNCT iajs-2928 189 6 βˆ’1 βˆ’1 NOUN iajs-2928 189 7 if if SCONJ iajs-2928 189 8 π‘₯ π‘₯ PRON iajs-2928 189 9 ∈ ∈ PROPN iajs-2928 190 1 [ [ X iajs-2928 190 2 βˆ’βˆž βˆ’βˆž NOUN iajs-2928 190 3 , , PUNCT iajs-2928 190 4 0 0 NUM iajs-2928 190 5 ) ) PUNCT iajs-2928 190 6 and and CCONJ iajs-2928 190 7 𝐸(π‘₯ 𝐸(π‘₯ NOUN iajs-2928 190 8 ) ) PUNCT iajs-2928 190 9 = = PRON iajs-2928 190 10 { { PUNCT iajs-2928 191 1 π‘₯ π‘₯ NOUN iajs-2928 191 2 2 2 NUM iajs-2928 191 3 if if SCONJ iajs-2928 191 4 π‘₯ π‘₯ PROPN iajs-2928 191 5 β‰₯ β‰₯ NOUN iajs-2928 191 6 0 0 NUM iajs-2928 192 1 βˆ’π‘₯2 βˆ’π‘₯2 VERB iajs-2928 192 2 if if SCONJ iajs-2928 192 3 π‘₯ π‘₯ X iajs-2928 192 4 < < X iajs-2928 192 5 0 0 X iajs-2928 193 1 then then ADV iajs-2928 193 2 , , PUNCT iajs-2928 193 3 for for ADP iajs-2928 193 4 any any DET iajs-2928 193 5 𝑛 𝑛 PRON iajs-2928 193 6 ∈ ∈ PROPN iajs-2928 193 7 𝑅 𝑅 PROPN iajs-2928 193 8 , , PUNCT iajs-2928 193 9 the the DET iajs-2928 193 10 level level NOUN iajs-2928 193 11 set set VERB iajs-2928 193 12 𝐾𝑛 𝐾𝑛 PROPN iajs-2928 193 13 = = PUNCT iajs-2928 193 14 { { PUNCT iajs-2928 193 15 [ [ X iajs-2928 193 16 βˆ’βˆž βˆ’βˆž NOUN iajs-2928 193 17 , , PUNCT iajs-2928 193 18 0 0 NUM iajs-2928 193 19 ) ) PUNCT iajs-2928 193 20 if if SCONJ iajs-2928 193 21 𝑛 𝑛 PRON iajs-2928 193 22 ∈ ∈ PROPN iajs-2928 193 23 [ [ X iajs-2928 193 24 0,1 0,1 NUM iajs-2928 193 25 ) ) PUNCT iajs-2928 193 26 ℝ ℝ NOUN iajs-2928 193 27 if if SCONJ iajs-2928 193 28 𝑛 𝑛 PRON iajs-2928 193 29 β‰₯ β‰₯ NOUN iajs-2928 193 30 1 1 NUM iajs-2928 193 31 to to PART iajs-2928 193 32 show show VERB iajs-2928 193 33 that that SCONJ iajs-2928 193 34 𝐾𝑛 𝐾𝑛 PROPN iajs-2928 193 35 is be AUX iajs-2928 193 36 ( ( PUNCT iajs-2928 193 37 𝑝 𝑝 NOUN iajs-2928 193 38 , , PUNCT iajs-2928 193 39 𝐸)-convex 𝐸)-convex PRON iajs-2928 193 40 set set VERB iajs-2928 193 41 , , PUNCT iajs-2928 193 42 we we PRON iajs-2928 193 43 consider consider VERB iajs-2928 193 44 the the DET iajs-2928 193 45 following follow VERB iajs-2928 193 46 cases case NOUN iajs-2928 193 47 : : PUNCT iajs-2928 193 48 case case NOUN iajs-2928 193 49 1 1 NUM iajs-2928 193 50 : : PUNCT iajs-2928 193 51 if if SCONJ iajs-2928 193 52 𝐾𝑛= 𝐾𝑛= PROPN iajs-2928 193 53 𝑅 𝑅 PROPN iajs-2928 193 54 ( ( PUNCT iajs-2928 193 55 i.e. i.e. X iajs-2928 193 56 , , PUNCT iajs-2928 193 57 𝑛 𝑛 DET iajs-2928 193 58 β‰₯ β‰₯ NOUN iajs-2928 193 59 1 1 NUM iajs-2928 193 60 ) ) PUNCT iajs-2928 193 61 then then ADV iajs-2928 193 62 , , PUNCT iajs-2928 193 63 π‘ŸπΈπ‘₯ π‘ŸπΈπ‘₯ PRON iajs-2928 193 64 + + NUM iajs-2928 193 65 𝑠𝐸𝑦 𝑠𝐸𝑦 PART iajs-2928 193 66 ∈ ∈ PROPN iajs-2928 193 67 𝐾𝑛 𝐾𝑛 NOUN iajs-2928 193 68 for for ADP iajs-2928 193 69 all all PRON iajs-2928 193 70 π‘₯ π‘₯ PROPN iajs-2928 193 71 , , PUNCT iajs-2928 193 72 𝑦 𝑦 NOUN iajs-2928 193 73 ∈ ∈ NOUN iajs-2928 193 74 𝐾𝑛. 𝐾𝑛. NOUN iajs-2928 193 75 hence hence NOUN iajs-2928 193 76 , , PUNCT iajs-2928 193 77 𝐾𝑛 𝐾𝑛 PROPN iajs-2928 193 78 is be AUX iajs-2928 193 79 ( ( PUNCT iajs-2928 193 80 𝑝 𝑝 NOUN iajs-2928 193 81 , , PUNCT iajs-2928 193 82 𝐸)-convex 𝐸)-convex PRON iajs-2928 193 83 set set VERB iajs-2928 193 84 . . PUNCT iajs-2928 194 1 case case NOUN iajs-2928 194 2 2 2 NUM iajs-2928 194 3 : : PUNCT iajs-2928 194 4 if if SCONJ iajs-2928 194 5 𝐾𝑛= 𝐾𝑛= PROPN iajs-2928 194 6 [ [ X iajs-2928 194 7 βˆ’βˆž βˆ’βˆž NOUN iajs-2928 194 8 , , PUNCT iajs-2928 194 9 0 0 NUM iajs-2928 194 10 ) ) PUNCT iajs-2928 194 11 for for ADP iajs-2928 194 12 𝑛 𝑛 DET iajs-2928 194 13 ∈ ∈ PROPN iajs-2928 194 14 [ [ X iajs-2928 194 15 0,1 0,1 NUM iajs-2928 194 16 ) ) PUNCT iajs-2928 194 17 . . PUNCT iajs-2928 195 1 then then ADV iajs-2928 195 2 , , PUNCT iajs-2928 195 3 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ PUNCT iajs-2928 195 4 + + CCONJ iajs-2928 195 5 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 195 6 ) ) PUNCT iajs-2928 195 7 = = SYM iajs-2928 196 1 𝑓(βˆ’π‘Ÿπ‘₯2 𝑓(βˆ’π‘Ÿπ‘₯2 PRON iajs-2928 196 2 βˆ’ βˆ’ NOUN iajs-2928 196 3 𝑠𝑦2 𝑠𝑦2 PROPN iajs-2928 196 4 ) ) PUNCT iajs-2928 196 5 = = SYM iajs-2928 196 6 βˆ’1 βˆ’1 NOUN iajs-2928 196 7 ≀ ≀ PUNCT iajs-2928 196 8 𝑛 𝑛 PROPN iajs-2928 196 9 for for ADP iajs-2928 196 10 all all DET iajs-2928 196 11 π‘₯ π‘₯ PROPN iajs-2928 196 12 , , PUNCT iajs-2928 196 13 𝑦 𝑦 NOUN iajs-2928 196 14 ∈ ∈ PROPN iajs-2928 196 15 𝐾𝑛. 𝐾𝑛. PROPN iajs-2928 197 1 thus thus ADV iajs-2928 197 2 , , PUNCT iajs-2928 197 3 π‘ŸπΈπ‘₯ π‘ŸπΈπ‘₯ PRON iajs-2928 197 4 + + NUM iajs-2928 197 5 𝑠𝐸𝑦 𝑠𝐸𝑦 PART iajs-2928 197 6 ∈ ∈ PROPN iajs-2928 197 7 𝐾𝑛 𝐾𝑛 NOUN iajs-2928 197 8 which which PRON iajs-2928 197 9 yields yield VERB iajs-2928 197 10 the the DET iajs-2928 197 11 ( ( PUNCT iajs-2928 197 12 𝑝 𝑝 NOUN iajs-2928 197 13 , , PUNCT iajs-2928 197 14 𝐸)-convexity 𝐸)-convexity PROPN iajs-2928 197 15 of of ADP iajs-2928 197 16 𝐾𝑛. 𝐾𝑛. PROPN iajs-2928 197 17 from from ADP iajs-2928 197 18 both both DET iajs-2928 197 19 cases case NOUN iajs-2928 197 20 , , PUNCT iajs-2928 197 21 we we PRON iajs-2928 197 22 obtain obtain VERB iajs-2928 197 23 the the DET iajs-2928 197 24 ( ( PUNCT iajs-2928 197 25 𝑝 𝑝 NOUN iajs-2928 197 26 , , PUNCT iajs-2928 197 27 𝐸)-convexity 𝐸)-convexity PROPN iajs-2928 197 28 of of ADP iajs-2928 197 29 𝐾𝑛. 𝐾𝑛. PROPN iajs-2928 197 30 to to PART iajs-2928 197 31 confirm confirm VERB iajs-2928 197 32 that that SCONJ iajs-2928 197 33 𝑓 𝑓 PRON iajs-2928 197 34 is be AUX iajs-2928 197 35 not not PART iajs-2928 197 36 pseudo pseudo NOUN iajs-2928 197 37 semi semi ADV iajs-2928 197 38 ( ( PUNCT iajs-2928 197 39 𝑝 𝑝 NOUN iajs-2928 197 40 , , PUNCT iajs-2928 197 41 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 197 42 function function NOUN iajs-2928 197 43 . . PUNCT iajs-2928 198 1 let let VERB iajs-2928 198 2 π‘₯ π‘₯ PRON iajs-2928 198 3 = = SYM iajs-2928 198 4 βˆ’1 βˆ’1 NOUN iajs-2928 198 5 , , PUNCT iajs-2928 198 6 𝑦 𝑦 NOUN iajs-2928 198 7 = = SYM iajs-2928 198 8 1 1 NUM iajs-2928 198 9 , , PUNCT iajs-2928 198 10 π‘Ÿ π‘Ÿ NOUN iajs-2928 198 11 = = SYM iajs-2928 198 12 𝑠 𝑠 PROPN iajs-2928 198 13 = = SYM iajs-2928 198 14 1 1 NUM iajs-2928 198 15 4 4 NUM iajs-2928 198 16 , , PUNCT iajs-2928 198 17 and and CCONJ iajs-2928 198 18 𝑝 𝑝 X iajs-2928 198 19 = = SYM iajs-2928 198 20 1 1 NUM iajs-2928 198 21 2 2 NUM iajs-2928 198 22 then then ADV iajs-2928 198 23 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 198 24 ) ) PUNCT iajs-2928 198 25 < < X iajs-2928 198 26 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 198 27 ) ) PUNCT iajs-2928 198 28 and and CCONJ iajs-2928 198 29 there there PRON iajs-2928 198 30 exists exist VERB iajs-2928 198 31 𝑏(π‘₯ 𝑏(π‘₯ PROPN iajs-2928 198 32 , , PUNCT iajs-2928 198 33 𝑦 𝑦 NOUN iajs-2928 198 34 ) ) PUNCT iajs-2928 198 35 = = SYM iajs-2928 199 1 3 3 NUM iajs-2928 199 2 > > SYM iajs-2928 199 3 0 0 NUM iajs-2928 199 4 such such ADJ iajs-2928 199 5 that that DET iajs-2928 199 6 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 199 7 + + CCONJ iajs-2928 199 8 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 199 9 ) ) PUNCT iajs-2928 199 10 = = SYM iajs-2928 200 1 𝑓 𝑓 PROPN iajs-2928 200 2 ( ( PUNCT iajs-2928 200 3 βˆ’ βˆ’ PROPN iajs-2928 200 4 1 1 NUM iajs-2928 200 5 4 4 NUM iajs-2928 200 6 π‘₯2 π‘₯2 NOUN iajs-2928 200 7 + + CCONJ iajs-2928 200 8 1 1 NUM iajs-2928 200 9 4 4 NUM iajs-2928 200 10 𝑦2 𝑦2 NOUN iajs-2928 200 11 ) ) PUNCT iajs-2928 200 12 = = PUNCT iajs-2928 201 1 𝑓(0 𝑓(0 NOUN iajs-2928 201 2 ) ) PUNCT iajs-2928 201 3 = = SYM iajs-2928 202 1 1 1 NUM iajs-2928 202 2 > > PUNCT iajs-2928 202 3 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 202 4 ) ) PUNCT iajs-2928 203 1 + + CCONJ iajs-2928 203 2 ( ( PUNCT iajs-2928 203 3 βˆ’π‘Ÿπ‘ )𝑏(π‘₯ βˆ’π‘Ÿπ‘ )𝑏(π‘₯ NOUN iajs-2928 203 4 , , PUNCT iajs-2928 203 5 𝑦 𝑦 NOUN iajs-2928 203 6 ) ) PUNCT iajs-2928 203 7 = = SYM iajs-2928 203 8 𝑓(1 𝑓(1 NOUN iajs-2928 203 9 ) ) PUNCT iajs-2928 203 10 + + CCONJ iajs-2928 203 11 ( ( PUNCT iajs-2928 203 12 βˆ’ βˆ’ PROPN iajs-2928 203 13 1 1 NUM iajs-2928 203 14 16 16 NUM iajs-2928 203 15 ) ) PUNCT iajs-2928 203 16 ( ( PUNCT iajs-2928 203 17 3 3 X iajs-2928 203 18 ) ) PUNCT iajs-2928 203 19 = = SYM iajs-2928 203 20 13 13 NUM iajs-2928 203 21 16 16 NUM iajs-2928 203 22 ihjpas ihjpa NOUN iajs-2928 203 23 . . PUNCT iajs-2928 204 1 36(1)2023 36(1)2023 NUM iajs-2928 204 2 361 361 NUM iajs-2928 204 3 hence hence ADV iajs-2928 204 4 , , PUNCT iajs-2928 204 5 𝑓 𝑓 PRON iajs-2928 204 6 is be AUX iajs-2928 204 7 not not PART iajs-2928 204 8 a a DET iajs-2928 204 9 pseudo pseudo NOUN iajs-2928 204 10 semi semi ADJ iajs-2928 204 11 ( ( PUNCT iajs-2928 204 12 𝑝 𝑝 NOUN iajs-2928 204 13 , , PUNCT iajs-2928 204 14 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 204 15 function function NOUN iajs-2928 204 16 . . PUNCT iajs-2928 205 1 3 3 X iajs-2928 205 2 . . X iajs-2928 206 1 some some DET iajs-2928 206 2 properties property NOUN iajs-2928 206 3 of of ADP iajs-2928 206 4 quasi quasi NOUN iajs-2928 206 5 semi semi ADJ iajs-2928 206 6 and and CCONJ iajs-2928 206 7 pseudo pseudo NOUN iajs-2928 206 8 semi semi ADJ iajs-2928 206 9 ( ( PUNCT iajs-2928 206 10 𝑝 𝑝 NOUN iajs-2928 206 11 , , PUNCT iajs-2928 206 12 𝐸)-convex 𝐸)-convex NOUN iajs-2928 206 13 functions function NOUN iajs-2928 206 14 in in ADP iajs-2928 206 15 this this DET iajs-2928 206 16 section section NOUN iajs-2928 206 17 , , PUNCT iajs-2928 206 18 we we PRON iajs-2928 206 19 discuss discuss VERB iajs-2928 206 20 some some DET iajs-2928 206 21 properties property NOUN iajs-2928 206 22 of of ADP iajs-2928 206 23 quasi quasi NOUN iajs-2928 206 24 semi semi ADJ iajs-2928 206 25 ( ( PUNCT iajs-2928 206 26 𝑝 𝑝 NOUN iajs-2928 206 27 , , PUNCT iajs-2928 206 28 𝐸)-convex 𝐸)-convex NUM iajs-2928 206 29 and and CCONJ iajs-2928 206 30 pseudo pseudo NOUN iajs-2928 206 31 semi semi ADJ iajs-2928 206 32 ( ( PUNCT iajs-2928 206 33 𝑝 𝑝 PROPN iajs-2928 206 34 , , PUNCT iajs-2928 206 35 𝐸)convex 𝐸)convex NOUN iajs-2928 206 36 functions function NOUN iajs-2928 206 37 . . PUNCT iajs-2928 207 1 we we PRON iajs-2928 207 2 start start VERB iajs-2928 207 3 first first ADV iajs-2928 207 4 by by ADP iajs-2928 207 5 showing show VERB iajs-2928 207 6 that that SCONJ iajs-2928 207 7 the the DET iajs-2928 207 8 increasing increase VERB iajs-2928 207 9 quasi quasi NOUN iajs-2928 207 10 semi semi ADJ iajs-2928 207 11 ( ( PUNCT iajs-2928 207 12 𝑝 𝑝 PROPN iajs-2928 207 13 , , PUNCT iajs-2928 207 14 𝐸)-convex 𝐸)-convex NOUN iajs-2928 207 15 functions function NOUN iajs-2928 207 16 ( ( PUNCT iajs-2928 207 17 respectively respectively ADV iajs-2928 207 18 , , PUNCT iajs-2928 207 19 strictly strictly ADV iajs-2928 207 20 increasing increase VERB iajs-2928 207 21 pseudo pseudo NOUN iajs-2928 207 22 quasi quasi NOUN iajs-2928 207 23 semi semi ADJ iajs-2928 207 24 ( ( PUNCT iajs-2928 207 25 𝑝 𝑝 NOUN iajs-2928 207 26 , , PUNCT iajs-2928 207 27 𝐸)-convex 𝐸)-convex NUM iajs-2928 207 28 ) ) PUNCT iajs-2928 207 29 functions function NOUN iajs-2928 207 30 defined define VERB iajs-2928 207 31 on on ADP iajs-2928 207 32 𝐴 𝐴 PROPN iajs-2928 207 33 βŠ† βŠ† PROPN iajs-2928 207 34 𝑅 𝑅 PROPN iajs-2928 207 35 are be AUX iajs-2928 207 36 closed close VERB iajs-2928 207 37 under under ADP iajs-2928 207 38 addition addition NOUN iajs-2928 207 39 and and CCONJ iajs-2928 207 40 nonnegative nonnegative VERB iajs-2928 207 41 scalar scalar ADJ iajs-2928 207 42 multiplication multiplication NOUN iajs-2928 207 43 . . PUNCT iajs-2928 208 1 proposition proposition NOUN iajs-2928 208 2 3.1 3.1 NUM iajs-2928 208 3 . . PUNCT iajs-2928 209 1 let let VERB iajs-2928 209 2 𝑓 𝑓 PRON iajs-2928 209 3 , , PUNCT iajs-2928 209 4 𝑔 𝑔 ADJ iajs-2928 209 5 : : PUNCT iajs-2928 209 6 𝐴 𝐴 PROPN iajs-2928 209 7 βŠ† βŠ† NUM iajs-2928 209 8 𝑅 𝑅 PROPN iajs-2928 209 9 ⟢ ⟢ NOUN iajs-2928 209 10 𝑅 𝑅 PROPN iajs-2928 209 11 are be AUX iajs-2928 209 12 two two NUM iajs-2928 209 13 increasing increase VERB iajs-2928 209 14 quasi quasi NOUN iajs-2928 209 15 semi semi ADJ iajs-2928 209 16 ( ( PUNCT iajs-2928 209 17 𝑝 𝑝 PROPN iajs-2928 209 18 , , PUNCT iajs-2928 209 19 𝐸)-convex 𝐸)-convex NOUN iajs-2928 209 20 functions function NOUN iajs-2928 209 21 on on ADP iajs-2928 209 22 𝐴. 𝐴. PROPN iajs-2928 209 23 then then ADV iajs-2928 209 24 , , PUNCT iajs-2928 209 25 𝛼𝑓 𝛼𝑓 PROPN iajs-2928 209 26 + + CCONJ iajs-2928 209 27 𝛽𝑔 𝛽𝑔 NOUN iajs-2928 209 28 is be AUX iajs-2928 209 29 increasing increase VERB iajs-2928 209 30 quasi quasi ADJ iajs-2928 209 31 semi semi ADJ iajs-2928 209 32 ( ( PUNCT iajs-2928 209 33 𝑝 𝑝 NOUN iajs-2928 209 34 , , PUNCT iajs-2928 209 35 𝐸)-convex 𝐸)-convex PRON iajs-2928 209 36 function function VERB iajs-2928 209 37 for for ADP iajs-2928 209 38 all all DET iajs-2928 209 39 𝛼 𝛼 PROPN iajs-2928 209 40 , , PUNCT iajs-2928 209 41 𝛽 𝛽 PRON iajs-2928 209 42 β‰₯ β‰₯ NOUN iajs-2928 209 43 0 0 NUM iajs-2928 209 44 . . PUNCT iajs-2928 210 1 proof proof NOUN iajs-2928 210 2 . . PUNCT iajs-2928 211 1 let let VERB iajs-2928 211 2 π‘₯ π‘₯ PRON iajs-2928 211 3 , , PUNCT iajs-2928 211 4 𝑦 𝑦 PROPN iajs-2928 211 5 ∈ ∈ PROPN iajs-2928 211 6 𝐴 𝐴 PROPN iajs-2928 211 7 then then ADV iajs-2928 211 8 either either CCONJ iajs-2928 211 9 π‘₯ π‘₯ ADP iajs-2928 211 10 ≀ ≀ NUM iajs-2928 211 11 𝑦 𝑦 NUM iajs-2928 211 12 or or CCONJ iajs-2928 211 13 𝑦 𝑦 NOUN iajs-2928 211 14 ≀ ≀ NOUN iajs-2928 211 15 π‘₯. π‘₯. VERB iajs-2928 211 16 if if SCONJ iajs-2928 211 17 π‘₯ π‘₯ PRON iajs-2928 211 18 ≀ ≀ NUM iajs-2928 211 19 𝑦 𝑦 NOUN iajs-2928 211 20 and and CCONJ iajs-2928 211 21 𝑓 𝑓 PROPN iajs-2928 211 22 and and CCONJ iajs-2928 211 23 𝑔 𝑔 PROPN iajs-2928 211 24 are be AUX iajs-2928 211 25 increasing increase VERB iajs-2928 211 26 functions function NOUN iajs-2928 211 27 , , PUNCT iajs-2928 211 28 then then ADV iajs-2928 211 29 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 211 30 ) ) PUNCT iajs-2928 211 31 ≀ ≀ NOUN iajs-2928 211 32 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 211 33 ) ) PUNCT iajs-2928 211 34 and and CCONJ iajs-2928 211 35 𝑔(π‘₯ 𝑔(π‘₯ NUM iajs-2928 211 36 ) ) PUNCT iajs-2928 212 1 ≀ ≀ NUM iajs-2928 212 2 𝑔(𝑦 𝑔(𝑦 NOUN iajs-2928 212 3 ) ) PUNCT iajs-2928 212 4 which which PRON iajs-2928 212 5 yield yield VERB iajs-2928 212 6 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ PRON iajs-2928 212 7 { { PUNCT iajs-2928 212 8 ( ( PUNCT iajs-2928 212 9 𝛼𝑓 𝛼𝑓 PROPN iajs-2928 212 10 + + X iajs-2928 212 11 𝛽𝑔)(π‘₯ 𝛽𝑔)(π‘₯ NOUN iajs-2928 212 12 ) ) PUNCT iajs-2928 212 13 , , PUNCT iajs-2928 212 14 ( ( PUNCT iajs-2928 212 15 𝛼𝑓 𝛼𝑓 VERB iajs-2928 212 16 + + ADJ iajs-2928 212 17 𝛽𝑔)(𝑦 𝛽𝑔)(𝑦 NOUN iajs-2928 212 18 ) ) PUNCT iajs-2928 212 19 } } PUNCT iajs-2928 212 20 = = SYM iajs-2928 212 21 ( ( PUNCT iajs-2928 212 22 𝛼𝑓 𝛼𝑓 X iajs-2928 212 23 + + ADJ iajs-2928 212 24 𝛽𝑔)(𝑦 𝛽𝑔)(𝑦 ADJ iajs-2928 212 25 ) ) PUNCT iajs-2928 212 26 ( ( PUNCT iajs-2928 212 27 2 2 X iajs-2928 212 28 ) ) PUNCT iajs-2928 212 29 hence hence ADV iajs-2928 212 30 , , PUNCT iajs-2928 212 31 𝛼𝑓 𝛼𝑓 PROPN iajs-2928 212 32 + + CCONJ iajs-2928 212 33 𝛽𝑔 𝛽𝑔 NOUN iajs-2928 212 34 is be AUX iajs-2928 212 35 increasing increase VERB iajs-2928 212 36 function function NOUN iajs-2928 212 37 . . PUNCT iajs-2928 213 1 let let VERB iajs-2928 213 2 𝑧 𝑧 VERB iajs-2928 213 3 = = VERB iajs-2928 213 4 π‘ŸπΈπ‘₯ π‘ŸπΈπ‘₯ NUM iajs-2928 213 5 + + NUM iajs-2928 213 6 𝑠𝐸𝑦 𝑠𝐸𝑦 PROPN iajs-2928 213 7 ∈ ∈ PROPN iajs-2928 213 8 𝐴. 𝐴. NOUN iajs-2928 213 9 then then ADV iajs-2928 213 10 ( ( PUNCT iajs-2928 213 11 𝛼𝑓 𝛼𝑓 X iajs-2928 213 12 + + X iajs-2928 213 13 𝛽𝑔 𝛽𝑔 NOUN iajs-2928 213 14 ) ) PUNCT iajs-2928 213 15 ( ( PUNCT iajs-2928 213 16 𝑧 𝑧 NOUN iajs-2928 213 17 ) ) PUNCT iajs-2928 213 18 = = SYM iajs-2928 214 1 𝛼𝑓(𝑧 𝛼𝑓(𝑧 X iajs-2928 214 2 ) ) PUNCT iajs-2928 214 3 + + CCONJ iajs-2928 214 4 𝛽𝑔(𝑧 𝛽𝑔(𝑧 NOUN iajs-2928 214 5 ) ) PUNCT iajs-2928 214 6 ≀ ≀ NUM iajs-2928 214 7 𝛼 𝛼 NUM iajs-2928 214 8 π‘šπ‘Žπ‘₯{𝑓(π‘₯ π‘šπ‘Žπ‘₯{𝑓(π‘₯ NOUN iajs-2928 214 9 ) ) PUNCT iajs-2928 214 10 , , PUNCT iajs-2928 214 11 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 214 12 ) ) PUNCT iajs-2928 214 13 } } PUNCT iajs-2928 215 1 + + CCONJ iajs-2928 215 2 𝛽 𝛽 NOUN iajs-2928 215 3 π‘šπ‘Žπ‘₯{𝑔(π‘₯ π‘šπ‘Žπ‘₯{𝑔(π‘₯ NUM iajs-2928 215 4 ) ) PUNCT iajs-2928 215 5 , , PUNCT iajs-2928 215 6 𝑔(𝑦 𝑔(𝑦 PROPN iajs-2928 215 7 ) ) PUNCT iajs-2928 215 8 } } PUNCT iajs-2928 215 9 = = SYM iajs-2928 215 10 𝛼 𝛼 NOUN iajs-2928 215 11 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 215 12 ) ) PUNCT iajs-2928 215 13 + + NUM iajs-2928 215 14 𝛽𝑔(𝑦 𝛽𝑔(𝑦 NOUN iajs-2928 215 15 ) ) PUNCT iajs-2928 215 16 = = SYM iajs-2928 215 17 ( ( PUNCT iajs-2928 215 18 𝛼𝑓 𝛼𝑓 X iajs-2928 215 19 + + ADJ iajs-2928 215 20 𝛽𝑔)(𝑦 𝛽𝑔)(𝑦 ADJ iajs-2928 215 21 ) ) PUNCT iajs-2928 215 22 = = SYM iajs-2928 216 1 π‘šπ‘Žπ‘₯{(𝛼𝑓 π‘šπ‘Žπ‘₯{(𝛼𝑓 NOUN iajs-2928 216 2 + + SYM iajs-2928 216 3 𝛽𝑔)(π‘₯ 𝛽𝑔)(π‘₯ NOUN iajs-2928 216 4 ) ) PUNCT iajs-2928 216 5 , , PUNCT iajs-2928 216 6 ( ( PUNCT iajs-2928 216 7 𝛼𝑓 𝛼𝑓 VERB iajs-2928 216 8 + + ADJ iajs-2928 216 9 𝛽𝑔)(𝑦 𝛽𝑔)(𝑦 NOUN iajs-2928 216 10 ) ) PUNCT iajs-2928 216 11 } } PUNCT iajs-2928 216 12 , , PUNCT iajs-2928 216 13 where where SCONJ iajs-2928 216 14 the the DET iajs-2928 216 15 last last ADJ iajs-2928 216 16 conclusion conclusion NOUN iajs-2928 216 17 follows follow VERB iajs-2928 216 18 from from ADP iajs-2928 216 19 ( ( PUNCT iajs-2928 216 20 2 2 NUM iajs-2928 216 21 ) ) PUNCT iajs-2928 216 22 . . PUNCT iajs-2928 217 1 hence hence ADV iajs-2928 217 2 , , PUNCT iajs-2928 217 3 𝛼𝑓 𝛼𝑓 PROPN iajs-2928 217 4 + + CCONJ iajs-2928 217 5 𝛽𝑔 𝛽𝑔 NOUN iajs-2928 217 6 is be AUX iajs-2928 217 7 increasing increase VERB iajs-2928 217 8 quasi quasi ADJ iajs-2928 217 9 semi semi ADJ iajs-2928 217 10 ( ( PUNCT iajs-2928 217 11 𝑝 𝑝 PROPN iajs-2928 217 12 , , PUNCT iajs-2928 217 13 𝐸)convex 𝐸)convex PROPN iajs-2928 217 14 function function NOUN iajs-2928 217 15 . . PUNCT iajs-2928 218 1 if if SCONJ iajs-2928 218 2 𝑦 𝑦 NOUN iajs-2928 218 3 ≀ ≀ X iajs-2928 218 4 π‘₯ π‘₯ PRON iajs-2928 218 5 , , PUNCT iajs-2928 218 6 we we PRON iajs-2928 218 7 proceed proceed VERB iajs-2928 218 8 similarly similarly ADV iajs-2928 218 9 to to PART iajs-2928 218 10 obtain obtain VERB iajs-2928 218 11 the the DET iajs-2928 218 12 required required ADJ iajs-2928 218 13 conclusion conclusion NOUN iajs-2928 218 14 . . PUNCT iajs-2928 219 1 β–  β–  PUNCT iajs-2928 219 2 proposition proposition NOUN iajs-2928 219 3 3.2 3.2 NUM iajs-2928 219 4 . . PUNCT iajs-2928 220 1 let let VERB iajs-2928 220 2 𝑓 𝑓 PRON iajs-2928 220 3 , , PUNCT iajs-2928 220 4 𝑔 𝑔 ADJ iajs-2928 220 5 : : PUNCT iajs-2928 220 6 𝐴 𝐴 PROPN iajs-2928 220 7 βŠ† βŠ† NUM iajs-2928 220 8 𝑅 𝑅 PROPN iajs-2928 220 9 ⟢ ⟢ NOUN iajs-2928 220 10 𝑅 𝑅 PROPN iajs-2928 220 11 are be AUX iajs-2928 220 12 strictly strictly ADV iajs-2928 220 13 increasing increase VERB iajs-2928 220 14 pseudo pseudo NOUN iajs-2928 220 15 semi semi ADJ iajs-2928 220 16 ( ( PUNCT iajs-2928 220 17 𝑝 𝑝 NOUN iajs-2928 220 18 , , PUNCT iajs-2928 220 19 𝐸)-convex 𝐸)-convex PROPN iajs-2928 220 20 on on ADP iajs-2928 220 21 𝐴.then 𝐴.then ADV iajs-2928 220 22 , , PUNCT iajs-2928 220 23 for for ADP iajs-2928 220 24 all all DET iajs-2928 220 25 𝛼 𝛼 PROPN iajs-2928 220 26 , , PUNCT iajs-2928 220 27 𝛽 𝛽 PRON iajs-2928 220 28 β‰₯ β‰₯ NOUN iajs-2928 220 29 0 0 NUM iajs-2928 220 30 , , PUNCT iajs-2928 220 31 𝛼𝑓 𝛼𝑓 X iajs-2928 220 32 + + CCONJ iajs-2928 220 33 𝛽𝑔 𝛽𝑔 NOUN iajs-2928 220 34 is be AUX iajs-2928 220 35 strictly strictly ADV iajs-2928 220 36 increasing increase VERB iajs-2928 220 37 pseudo pseudo NOUN iajs-2928 220 38 semi semi ADJ iajs-2928 220 39 ( ( PUNCT iajs-2928 220 40 𝑝 𝑝 NOUN iajs-2928 220 41 , , PUNCT iajs-2928 220 42 𝐸)-convex 𝐸)-convex PROPN iajs-2928 220 43 on on ADP iajs-2928 220 44 𝐴. 𝐴. PROPN iajs-2928 220 45 proof proof NOUN iajs-2928 220 46 . . PUNCT iajs-2928 221 1 from from ADP iajs-2928 221 2 the the DET iajs-2928 221 3 definition definition NOUN iajs-2928 221 4 of of ADP iajs-2928 221 5 𝑓 𝑓 PRON iajs-2928 221 6 and and CCONJ iajs-2928 221 7 𝑔 𝑔 PROPN iajs-2928 221 8 we we PRON iajs-2928 221 9 have have VERB iajs-2928 221 10 , , PUNCT iajs-2928 221 11 if if SCONJ iajs-2928 221 12 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 221 13 ) ) PUNCT iajs-2928 221 14 < < X iajs-2928 221 15 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 221 16 ) ) PUNCT iajs-2928 221 17 and and CCONJ iajs-2928 221 18 𝑔(π‘₯ 𝑔(π‘₯ NUM iajs-2928 221 19 ) ) PUNCT iajs-2928 221 20 < < X iajs-2928 221 21 𝑔(𝑦 𝑔(𝑦 NOUN iajs-2928 221 22 ) ) PUNCT iajs-2928 221 23 then then ADV iajs-2928 221 24 then then ADV iajs-2928 221 25 there there PRON iajs-2928 221 26 exist exist VERB iajs-2928 221 27 𝑏1 𝑏1 NOUN iajs-2928 221 28 , , PUNCT iajs-2928 221 29 𝑏2 𝑏2 NOUN iajs-2928 221 30 : : PUNCT iajs-2928 221 31 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 221 32 Γ— Γ— NOUN iajs-2928 221 33 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 221 34 β†’ β†’ SYM iajs-2928 221 35 𝑅 𝑅 PROPN iajs-2928 221 36 such such ADJ iajs-2928 221 37 that that PRON iajs-2928 221 38 ( ( PUNCT iajs-2928 221 39 𝛼𝑓 𝛼𝑓 PROPN iajs-2928 221 40 + + X iajs-2928 221 41 𝛽𝑔)(π‘₯ 𝛽𝑔)(π‘₯ NOUN iajs-2928 221 42 ) ) PUNCT iajs-2928 221 43 < < X iajs-2928 221 44 ( ( PUNCT iajs-2928 221 45 𝛼𝑓 𝛼𝑓 VERB iajs-2928 221 46 + + ADJ iajs-2928 221 47 𝛽𝑔)(𝑦 𝛽𝑔)(𝑦 NOUN iajs-2928 221 48 ) ) PUNCT iajs-2928 221 49 , , PUNCT iajs-2928 221 50 ( ( PUNCT iajs-2928 221 51 3 3 X iajs-2928 221 52 ) ) PUNCT iajs-2928 221 53 thus thus ADV iajs-2928 221 54 , , PUNCT iajs-2928 221 55 𝛼𝑓 𝛼𝑓 X iajs-2928 221 56 + + CCONJ iajs-2928 221 57 𝛽𝑔 𝛽𝑔 NOUN iajs-2928 221 58 is be AUX iajs-2928 221 59 strictly strictly ADV iajs-2928 221 60 increasing increase VERB iajs-2928 221 61 . . PUNCT iajs-2928 222 1 let let VERB iajs-2928 222 2 𝑧 𝑧 VERB iajs-2928 222 3 = = VERB iajs-2928 222 4 π‘ŸπΈπ‘₯ π‘ŸπΈπ‘₯ NUM iajs-2928 222 5 + + NUM iajs-2928 222 6 𝑠𝐸𝑦 𝑠𝐸𝑦 PROPN iajs-2928 222 7 ∈ ∈ PROPN iajs-2928 222 8 𝐴 𝐴 PROPN iajs-2928 222 9 then then ADV iajs-2928 222 10 𝑓(𝑧 𝑓(𝑧 NUM iajs-2928 222 11 ) ) PUNCT iajs-2928 222 12 ≀ ≀ NOUN iajs-2928 222 13 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 222 14 ) ) PUNCT iajs-2928 223 1 + + CCONJ iajs-2928 223 2 ( ( PUNCT iajs-2928 223 3 βˆ’π‘Ÿπ‘  βˆ’π‘Ÿπ‘  NOUN iajs-2928 223 4 ) ) PUNCT iajs-2928 223 5 𝑏1(π‘₯ 𝑏1(π‘₯ NOUN iajs-2928 223 6 , , PUNCT iajs-2928 223 7 𝑦 𝑦 NOUN iajs-2928 223 8 ) ) PUNCT iajs-2928 223 9 and and CCONJ iajs-2928 223 10 𝑔(𝑧 𝑔(𝑧 NOUN iajs-2928 223 11 ) ) PUNCT iajs-2928 223 12 ≀ ≀ NOUN iajs-2928 223 13 𝑔(𝑦 𝑔(𝑦 NOUN iajs-2928 223 14 ) ) PUNCT iajs-2928 224 1 + + CCONJ iajs-2928 224 2 ( ( PUNCT iajs-2928 224 3 βˆ’π‘Ÿπ‘  βˆ’π‘Ÿπ‘  NOUN iajs-2928 224 4 ) ) PUNCT iajs-2928 224 5 𝑏2(π‘₯ 𝑏2(π‘₯ PROPN iajs-2928 224 6 , , PUNCT iajs-2928 224 7 𝑦 𝑦 NOUN iajs-2928 224 8 ) ) PUNCT iajs-2928 224 9 . . PUNCT iajs-2928 225 1 now now ADV iajs-2928 225 2 , , PUNCT iajs-2928 225 3 ( ( PUNCT iajs-2928 225 4 𝛼𝑓 𝛼𝑓 X iajs-2928 225 5 + + X iajs-2928 225 6 𝛽𝑔)(𝑧 𝛽𝑔)(𝑧 PROPN iajs-2928 225 7 ) ) PUNCT iajs-2928 225 8 = = PUNCT iajs-2928 225 9 𝛼 𝛼 PROPN iajs-2928 225 10 𝑓 𝑓 PROPN iajs-2928 225 11 ( ( PUNCT iajs-2928 225 12 𝑧 𝑧 NOUN iajs-2928 225 13 ) ) PUNCT iajs-2928 225 14 + + NUM iajs-2928 225 15 𝛽𝑔(𝑧 𝛽𝑔(𝑧 NOUN iajs-2928 225 16 ) ) PUNCT iajs-2928 225 17 ≀ ≀ NUM iajs-2928 225 18 𝛼 𝛼 X iajs-2928 225 19 ( ( PUNCT iajs-2928 225 20 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 225 21 ) ) PUNCT iajs-2928 225 22 + + CCONJ iajs-2928 225 23 ( ( PUNCT iajs-2928 225 24 βˆ’π‘Ÿπ‘ )𝑏1(π‘₯ βˆ’π‘Ÿπ‘ )𝑏1(π‘₯ PROPN iajs-2928 225 25 , , PUNCT iajs-2928 225 26 𝑦 𝑦 NOUN iajs-2928 225 27 ) ) PUNCT iajs-2928 225 28 ) ) PUNCT iajs-2928 226 1 + + PUNCT iajs-2928 227 1 𝛽(𝑔(𝑦 𝛽(𝑔(𝑦 NOUN iajs-2928 227 2 ) ) PUNCT iajs-2928 228 1 + + CCONJ iajs-2928 228 2 ( ( PUNCT iajs-2928 228 3 βˆ’π‘Ÿπ‘ )𝑏2(π‘₯ βˆ’π‘Ÿπ‘ )𝑏2(π‘₯ PROPN iajs-2928 228 4 , , PUNCT iajs-2928 228 5 𝑦 𝑦 NOUN iajs-2928 228 6 ) ) PUNCT iajs-2928 228 7 ) ) PUNCT iajs-2928 229 1 = = SYM iajs-2928 229 2 ( ( PUNCT iajs-2928 229 3 𝛼𝑓 𝛼𝑓 X iajs-2928 229 4 + + ADJ iajs-2928 229 5 𝛽𝑔)(𝑦 𝛽𝑔)(𝑦 ADJ iajs-2928 229 6 ) ) PUNCT iajs-2928 230 1 + + CCONJ iajs-2928 230 2 ( ( PUNCT iajs-2928 230 3 βˆ’π‘Ÿπ‘ )[𝑏1(π‘₯ βˆ’π‘Ÿπ‘ )[𝑏1(π‘₯ NUM iajs-2928 230 4 , , PUNCT iajs-2928 230 5 𝑦 𝑦 NOUN iajs-2928 230 6 ) ) PUNCT iajs-2928 230 7 + + CCONJ iajs-2928 231 1 𝑏2(π‘₯ 𝑏2(π‘₯ PROPN iajs-2928 231 2 , , PUNCT iajs-2928 231 3 𝑦 𝑦 NOUN iajs-2928 231 4 ) ) PUNCT iajs-2928 231 5 ] ] PUNCT iajs-2928 232 1 = = PUNCT iajs-2928 232 2 ( ( PUNCT iajs-2928 232 3 𝛼𝑓 𝛼𝑓 X iajs-2928 232 4 + + ADJ iajs-2928 232 5 𝛽𝑔)(𝑦 𝛽𝑔)(𝑦 ADJ iajs-2928 232 6 ) ) PUNCT iajs-2928 232 7 + + CCONJ iajs-2928 232 8 ( ( PUNCT iajs-2928 232 9 βˆ’π‘Ÿπ‘  βˆ’π‘Ÿπ‘  NOUN iajs-2928 232 10 ) ) PUNCT iajs-2928 232 11 𝑏(π‘₯ 𝑏(π‘₯ PROPN iajs-2928 232 12 , , PUNCT iajs-2928 232 13 𝑦 𝑦 NOUN iajs-2928 232 14 ) ) PUNCT iajs-2928 232 15 , , PUNCT iajs-2928 232 16 ( ( PUNCT iajs-2928 232 17 4 4 X iajs-2928 232 18 ) ) PUNCT iajs-2928 233 1 where where SCONJ iajs-2928 233 2 𝑏(π‘₯ 𝑏(π‘₯ NOUN iajs-2928 233 3 , , PUNCT iajs-2928 233 4 𝑦 𝑦 NOUN iajs-2928 233 5 ) ) PUNCT iajs-2928 233 6 = = SYM iajs-2928 233 7 𝑏1(π‘₯ 𝑏1(π‘₯ PROPN iajs-2928 233 8 , , PUNCT iajs-2928 233 9 𝑦 𝑦 NOUN iajs-2928 233 10 ) ) PUNCT iajs-2928 233 11 + + CCONJ iajs-2928 233 12 𝑏2(π‘₯ 𝑏2(π‘₯ PROPN iajs-2928 233 13 , , PUNCT iajs-2928 233 14 𝑦 𝑦 NOUN iajs-2928 233 15 ) ) PUNCT iajs-2928 233 16 . . PUNCT iajs-2928 234 1 since since SCONJ iajs-2928 234 2 𝑏1 𝑏1 NOUN iajs-2928 234 3 and and CCONJ iajs-2928 234 4 𝑏2 𝑏2 NOUN iajs-2928 234 5 are be AUX iajs-2928 234 6 strictly strictly ADV iajs-2928 234 7 positive positive ADJ iajs-2928 234 8 functions function NOUN iajs-2928 234 9 , , PUNCT iajs-2928 234 10 then then ADV iajs-2928 234 11 𝑏(π‘₯ 𝑏(π‘₯ PROPN iajs-2928 234 12 , , PUNCT iajs-2928 234 13 𝑦 𝑦 NOUN iajs-2928 234 14 ) ) PUNCT iajs-2928 234 15 is be AUX iajs-2928 234 16 strictly strictly ADV iajs-2928 234 17 positive positive ADJ iajs-2928 234 18 . . PUNCT iajs-2928 235 1 from from ADP iajs-2928 235 2 ( ( PUNCT iajs-2928 235 3 3 3 NUM iajs-2928 235 4 ) ) PUNCT iajs-2928 235 5 and and CCONJ iajs-2928 235 6 ( ( PUNCT iajs-2928 235 7 4 4 NUM iajs-2928 235 8 ) ) PUNCT iajs-2928 235 9 , , PUNCT iajs-2928 235 10 we we PRON iajs-2928 235 11 obtain obtain VERB iajs-2928 235 12 the the DET iajs-2928 235 13 required required ADJ iajs-2928 235 14 conclusion conclusion NOUN iajs-2928 235 15 . . PUNCT iajs-2928 236 1 β–  β–  PUNCT iajs-2928 236 2 next next ADV iajs-2928 236 3 , , PUNCT iajs-2928 236 4 we we PRON iajs-2928 236 5 show show VERB iajs-2928 236 6 the the DET iajs-2928 236 7 supremum supremum ADJ iajs-2928 236 8 property property NOUN iajs-2928 236 9 of of ADP iajs-2928 236 10 an an DET iajs-2928 236 11 arbitrary arbitrary ADJ iajs-2928 236 12 non non ADJ iajs-2928 236 13 - - ADJ iajs-2928 236 14 empty empty ADJ iajs-2928 236 15 finite finite ADJ iajs-2928 236 16 collection collection NOUN iajs-2928 236 17 of of ADP iajs-2928 236 18 quasi quasi NOUN iajs-2928 236 19 semi semi ADJ iajs-2928 236 20 ( ( PUNCT iajs-2928 236 21 𝑝 𝑝 PROPN iajs-2928 236 22 , , PUNCT iajs-2928 236 23 𝐸)-convex 𝐸)-convex NOUN iajs-2928 236 24 functions function NOUN iajs-2928 236 25 . . PUNCT iajs-2928 237 1 proposition proposition NOUN iajs-2928 237 2 3.3 3.3 NUM iajs-2928 237 3 . . PUNCT iajs-2928 238 1 let let VERB iajs-2928 238 2 𝑓𝑖 𝑓𝑖 VERB iajs-2928 238 3 : : PUNCT iajs-2928 238 4 𝑅 𝑅 PROPN iajs-2928 238 5 ⟢ ⟢ NOUN iajs-2928 238 6 𝑅 𝑅 PROPN iajs-2928 238 7 be be AUX iajs-2928 238 8 bounded bound VERB iajs-2928 238 9 from from ADP iajs-2928 238 10 above above ADP iajs-2928 238 11 increasing increase VERB iajs-2928 238 12 quasi quasi NOUN iajs-2928 238 13 semi semi ADJ iajs-2928 238 14 ( ( PUNCT iajs-2928 238 15 𝑝 𝑝 PROPN iajs-2928 238 16 , , PUNCT iajs-2928 238 17 𝐸)-convex 𝐸)-convex NOUN iajs-2928 238 18 functions function NOUN iajs-2928 238 19 for for ADP iajs-2928 238 20 each each DET iajs-2928 238 21 𝑖 𝑖 SYM iajs-2928 238 22 ∈ ∈ PROPN iajs-2928 238 23 𝛬 𝛬 NOUN iajs-2928 238 24 = = PUNCT iajs-2928 238 25 { { PUNCT iajs-2928 238 26 1 1 NUM iajs-2928 238 27 , , PUNCT iajs-2928 238 28 … … PUNCT iajs-2928 238 29 , , PUNCT iajs-2928 238 30 𝑛 𝑛 NOUN iajs-2928 238 31 } } PUNCT iajs-2928 238 32 . . PUNCT iajs-2928 239 1 define define NOUN iajs-2928 239 2 , , PUNCT iajs-2928 239 3 𝑓 𝑓 X iajs-2928 239 4 : : PUNCT iajs-2928 239 5 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 239 6 ⟢ ⟢ NOUN iajs-2928 239 7 𝑅 𝑅 PROPN iajs-2928 239 8 such such ADJ iajs-2928 239 9 that that SCONJ iajs-2928 239 10 𝑓 𝑓 DET iajs-2928 239 11 = = SYM iajs-2928 239 12 π‘ π‘’π‘π‘–βˆˆπ›¬π‘“π‘– π‘ π‘’π‘π‘–βˆˆπ›¬π‘“π‘– PROPN iajs-2928 239 13 . . PUNCT iajs-2928 240 1 then then ADV iajs-2928 240 2 𝑓 𝑓 PRON iajs-2928 240 3 is be AUX iajs-2928 240 4 quasi quasi NOUN iajs-2928 240 5 semi semi ADJ iajs-2928 240 6 ( ( PUNCT iajs-2928 240 7 𝑝 𝑝 NOUN iajs-2928 240 8 , , PUNCT iajs-2928 240 9 𝐸)-convex 𝐸)-convex X iajs-2928 240 10 . . PUNCT iajs-2928 241 1 proof proof NOUN iajs-2928 241 2 . . PUNCT iajs-2928 242 1 let let VERB iajs-2928 242 2 π‘₯ π‘₯ PRON iajs-2928 242 3 , , PUNCT iajs-2928 242 4 𝑦 𝑦 NOUN iajs-2928 242 5 ∈ ∈ PROPN iajs-2928 242 6 𝑅 𝑅 PROPN iajs-2928 242 7 such such ADJ iajs-2928 242 8 that that SCONJ iajs-2928 242 9 π‘₯ π‘₯ PROPN iajs-2928 242 10 ≀ ≀ NUM iajs-2928 242 11 𝑦 𝑦 NOUN iajs-2928 242 12 and and CCONJ iajs-2928 242 13 𝑓𝑖 𝑓𝑖 PROPN iajs-2928 242 14 is be AUX iajs-2928 242 15 quasi quasi NOUN iajs-2928 242 16 semi semi ADJ iajs-2928 242 17 ( ( PUNCT iajs-2928 242 18 𝑝 𝑝 NOUN iajs-2928 242 19 , , PUNCT iajs-2928 242 20 𝐸)-convex 𝐸)-convex PROPN iajs-2928 242 21 for for ADP iajs-2928 242 22 each each DET iajs-2928 242 23 𝑖 𝑖 SYM iajs-2928 242 24 ∈ ∈ PROPN iajs-2928 242 25 𝛬 𝛬 NOUN iajs-2928 242 26 = = PUNCT iajs-2928 242 27 { { PUNCT iajs-2928 242 28 1 1 NUM iajs-2928 242 29 , , PUNCT iajs-2928 242 30 … … PUNCT iajs-2928 242 31 , , PUNCT iajs-2928 242 32 𝑛 𝑛 PROPN iajs-2928 242 33 } } PUNCT iajs-2928 242 34 . . PUNCT iajs-2928 243 1 then then ADV iajs-2928 243 2 , , PUNCT iajs-2928 243 3 𝑓𝑖(π‘ŸπΈπ‘₯ 𝑓𝑖(π‘ŸπΈπ‘₯ PROPN iajs-2928 243 4 + + CCONJ iajs-2928 243 5 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 243 6 ) ) PUNCT iajs-2928 243 7 ≀ ≀ NOUN iajs-2928 243 8 { { PUNCT iajs-2928 243 9 𝑓𝑖(π‘₯ 𝑓𝑖(π‘₯ NUM iajs-2928 243 10 ) ) PUNCT iajs-2928 243 11 , , PUNCT iajs-2928 243 12 𝑓𝑖(𝑦 𝑓𝑖(𝑦 X iajs-2928 243 13 ) ) PUNCT iajs-2928 243 14 } } PUNCT iajs-2928 243 15 = = SYM iajs-2928 243 16 𝑓𝑖(𝑦 𝑓𝑖(𝑦 X iajs-2928 243 17 ) ) PUNCT iajs-2928 243 18 for for ADP iajs-2928 243 19 each each DET iajs-2928 243 20 𝑖 𝑖 SYM iajs-2928 243 21 ∈ ∈ PROPN iajs-2928 243 22 𝛬. 𝛬. NOUN iajs-2928 243 23 applying apply VERB iajs-2928 243 24 the the DET iajs-2928 243 25 supremum supremum NOUN iajs-2928 243 26 for for ADP iajs-2928 243 27 the the DET iajs-2928 243 28 both both DET iajs-2928 243 29 sides side NOUN iajs-2928 243 30 of of ADP iajs-2928 243 31 the the DET iajs-2928 243 32 above above ADJ iajs-2928 243 33 inequality inequality NOUN iajs-2928 243 34 respectively respectively ADV iajs-2928 243 35 , , PUNCT iajs-2928 243 36 we we PRON iajs-2928 243 37 get get VERB iajs-2928 243 38 π‘ π‘’π‘π‘–βˆˆπ›¬π‘“π‘–(π‘ŸπΈπ‘₯ π‘ π‘’π‘π‘–βˆˆπ›¬π‘“π‘–(π‘ŸπΈπ‘₯ PROPN iajs-2928 243 39 + + CCONJ iajs-2928 243 40 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 243 41 ) ) PUNCT iajs-2928 243 42 ≀ ≀ NOUN iajs-2928 243 43 π‘ π‘’π‘π‘–βˆˆπ›¬{{𝑓𝑖(π‘₯ π‘ π‘’π‘π‘–βˆˆπ›¬{{𝑓𝑖(π‘₯ NOUN iajs-2928 243 44 ) ) PUNCT iajs-2928 243 45 , , PUNCT iajs-2928 243 46 𝑓𝑖(𝑦 𝑓𝑖(𝑦 X iajs-2928 243 47 ) ) PUNCT iajs-2928 243 48 } } PUNCT iajs-2928 243 49 } } PUNCT iajs-2928 243 50 , , PUNCT iajs-2928 243 51 ihjpas ihjpa VERB iajs-2928 243 52 . . PUNCT iajs-2928 244 1 36(1)2023 36(1)2023 NUM iajs-2928 244 2 362 362 NUM iajs-2928 244 3 𝑓((π‘ŸπΈπ‘₯ 𝑓((π‘ŸπΈπ‘₯ X iajs-2928 244 4 + + NUM iajs-2928 244 5 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 244 6 ) ) PUNCT iajs-2928 244 7 ≀ ≀ NOUN iajs-2928 244 8 π‘ π‘’π‘π‘–βˆˆπ›¬π‘“π‘–(𝑦 π‘ π‘’π‘π‘–βˆˆπ›¬π‘“π‘–(𝑦 NUM iajs-2928 244 9 ) ) PUNCT iajs-2928 244 10 = = PUNCT iajs-2928 244 11 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 244 12 ) ) PUNCT iajs-2928 244 13 = = SYM iajs-2928 244 14 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ iajs-2928 244 15 { { PUNCT iajs-2928 244 16 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 244 17 ) ) PUNCT iajs-2928 244 18 , , PUNCT iajs-2928 244 19 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 244 20 ) ) PUNCT iajs-2928 244 21 } } PUNCT iajs-2928 244 22 . . PUNCT iajs-2928 245 1 the the DET iajs-2928 245 2 last last ADJ iajs-2928 245 3 inequalities inequality NOUN iajs-2928 245 4 yields yield VERB iajs-2928 245 5 𝑓 𝑓 PRON iajs-2928 245 6 is be AUX iajs-2928 245 7 quasi quasi NOUN iajs-2928 245 8 semi semi ADJ iajs-2928 245 9 ( ( PUNCT iajs-2928 245 10 𝑝 𝑝 NOUN iajs-2928 245 11 , , PUNCT iajs-2928 245 12 𝐸)-convex 𝐸)-convex X iajs-2928 245 13 . . PUNCT iajs-2928 246 1 if if SCONJ iajs-2928 246 2 𝑦 𝑦 NOUN iajs-2928 246 3 ≀ ≀ X iajs-2928 246 4 π‘₯ π‘₯ PRON iajs-2928 246 5 , , PUNCT iajs-2928 246 6 we we PRON iajs-2928 246 7 proceed proceed VERB iajs-2928 246 8 similarly similarly ADV iajs-2928 246 9 to to PART iajs-2928 246 10 obtain obtain VERB iajs-2928 246 11 the the DET iajs-2928 246 12 required require VERB iajs-2928 246 13 conclusion conclusion NOUN iajs-2928 246 14 β–  β–  PUNCT iajs-2928 246 15 two two NUM iajs-2928 246 16 composite composite ADJ iajs-2928 246 17 properties property NOUN iajs-2928 246 18 are be AUX iajs-2928 246 19 held hold VERB iajs-2928 246 20 for for ADP iajs-2928 246 21 quasi quasi NOUN iajs-2928 246 22 semi semi ADJ iajs-2928 246 23 ( ( PUNCT iajs-2928 246 24 respectively respectively ADV iajs-2928 246 25 , , PUNCT iajs-2928 246 26 pseudo pseudo NOUN iajs-2928 246 27 semi semi ADJ iajs-2928 246 28 ) ) PUNCT iajs-2928 246 29 ( ( PUNCT iajs-2928 246 30 𝑝 𝑝 PROPN iajs-2928 246 31 , , PUNCT iajs-2928 246 32 𝐸)-convex 𝐸)-convex NOUN iajs-2928 246 33 functions function NOUN iajs-2928 246 34 as as SCONJ iajs-2928 246 35 shown show VERB iajs-2928 246 36 next next ADV iajs-2928 246 37 . . PUNCT iajs-2928 247 1 proposition proposition NOUN iajs-2928 247 2 3.4 3.4 NUM iajs-2928 247 3 . . PUNCT iajs-2928 248 1 let let VERB iajs-2928 248 2 𝑓:𝐴 𝑓:𝐴 PRON iajs-2928 248 3 βŠ† βŠ† NUM iajs-2928 248 4 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 248 5 ⟢ ⟢ PROPN iajs-2928 248 6 𝑅 𝑅 PROPN iajs-2928 248 7 and and CCONJ iajs-2928 248 8 𝐺 𝐺 PROPN iajs-2928 248 9 : : PUNCT iajs-2928 248 10 𝑅 𝑅 PROPN iajs-2928 248 11 ⟢ ⟢ NOUN iajs-2928 248 12 𝑅 𝑅 PROPN iajs-2928 248 13 is be AUX iajs-2928 248 14 an an DET iajs-2928 248 15 increasing increase VERB iajs-2928 248 16 function function NOUN iajs-2928 248 17 . . PUNCT iajs-2928 249 1 then then ADV iajs-2928 249 2 i. i. PROPN iajs-2928 249 3 if if SCONJ iajs-2928 249 4 𝑓 𝑓 PRON iajs-2928 249 5 is be AUX iajs-2928 249 6 quasi quasi NOUN iajs-2928 249 7 semi semi ADJ iajs-2928 249 8 ( ( PUNCT iajs-2928 249 9 𝑝 𝑝 NOUN iajs-2928 249 10 , , PUNCT iajs-2928 249 11 𝐸)-π‘π‘œπ‘›π‘£π‘’π‘₯ 𝐸)-π‘π‘œπ‘›π‘£π‘’π‘₯ NOUN iajs-2928 249 12 on on ADP iajs-2928 249 13 𝐴 𝐴 PROPN iajs-2928 249 14 , , PUNCT iajs-2928 249 15 then then ADV iajs-2928 249 16 π‘”π‘œπ‘“ π‘”π‘œπ‘“ PROPN iajs-2928 249 17 ∢ ∢ PROPN iajs-2928 249 18 𝐴 𝐴 PROPN iajs-2928 249 19 β†’ β†’ SYM iajs-2928 249 20 𝑅 𝑅 PROPN iajs-2928 249 21 is be AUX iajs-2928 249 22 quasi quasi NOUN iajs-2928 249 23 semi semi ADJ iajs-2928 249 24 ( ( PUNCT iajs-2928 249 25 𝑝 𝑝 NOUN iajs-2928 249 26 , , PUNCT iajs-2928 249 27 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 249 28 function function NOUN iajs-2928 249 29 . . PUNCT iajs-2928 250 1 ii ii PROPN iajs-2928 250 2 . . PUNCT iajs-2928 251 1 if if SCONJ iajs-2928 251 2 𝑓 𝑓 PRON iajs-2928 251 3 is be AUX iajs-2928 251 4 pseudo pseudo NOUN iajs-2928 251 5 semi semi ADJ iajs-2928 251 6 ( ( PUNCT iajs-2928 251 7 𝑝 𝑝 NOUN iajs-2928 251 8 , , PUNCT iajs-2928 251 9 𝐸)-π‘π‘œπ‘›π‘£π‘’π‘₯ 𝐸)-π‘π‘œπ‘›π‘£π‘’π‘₯ NOUN iajs-2928 251 10 on on ADP iajs-2928 251 11 𝐴 𝐴 PROPN iajs-2928 251 12 and and CCONJ iajs-2928 251 13 𝐺 𝐺 PROPN iajs-2928 251 14 is be AUX iajs-2928 251 15 sublinear sublinear ADJ iajs-2928 251 16 and and CCONJ iajs-2928 251 17 strictly strictly ADV iajs-2928 251 18 positive positive ADJ iajs-2928 251 19 , , PUNCT iajs-2928 251 20 then then ADV iajs-2928 251 21 π‘”π‘œπ‘“ π‘”π‘œπ‘“ PROPN iajs-2928 251 22 ∢ ∢ PROPN iajs-2928 251 23 𝐴 𝐴 PROPN iajs-2928 251 24 β†’ β†’ SYM iajs-2928 251 25 𝑅 𝑅 PROPN iajs-2928 251 26 is be AUX iajs-2928 251 27 pseudo pseudo NOUN iajs-2928 251 28 semi semi ADJ iajs-2928 251 29 ( ( PUNCT iajs-2928 251 30 𝑝 𝑝 NOUN iajs-2928 251 31 , , PUNCT iajs-2928 251 32 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 251 33 function function NOUN iajs-2928 251 34 . . PUNCT iajs-2928 252 1 proof proof NOUN iajs-2928 252 2 . . PUNCT iajs-2928 253 1 let let VERB iajs-2928 253 2 us we PRON iajs-2928 253 3 show show VERB iajs-2928 253 4 ( ( PUNCT iajs-2928 253 5 i i NOUN iajs-2928 253 6 ) ) PUNCT iajs-2928 253 7 . . PUNCT iajs-2928 254 1 let let VERB iajs-2928 254 2 π‘₯ π‘₯ PRON iajs-2928 254 3 , , PUNCT iajs-2928 254 4 𝑦 𝑦 NOUN iajs-2928 254 5 ∈ ∈ PROPN iajs-2928 254 6 𝐴 𝐴 PROPN iajs-2928 254 7 and and CCONJ iajs-2928 254 8 𝑓 𝑓 PRON iajs-2928 254 9 is be AUX iajs-2928 254 10 quasi quasi NOUN iajs-2928 254 11 semi semi ADJ iajs-2928 254 12 ( ( PUNCT iajs-2928 254 13 𝑝 𝑝 NOUN iajs-2928 254 14 , , PUNCT iajs-2928 254 15 𝐸)-convex 𝐸)-convex PROPN iajs-2928 254 16 on on ADP iajs-2928 254 17 𝐴 𝐴 PROPN iajs-2928 254 18 , , PUNCT iajs-2928 254 19 then then ADV iajs-2928 254 20 π‘ŸπΈπ‘₯ π‘ŸπΈπ‘₯ NUM iajs-2928 254 21 + + NUM iajs-2928 254 22 𝑠𝐸𝑦 𝑠𝐸𝑦 PROPN iajs-2928 254 23 ∈ ∈ PROPN iajs-2928 254 24 𝐴 𝐴 PROPN iajs-2928 254 25 and and CCONJ iajs-2928 254 26 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ VERB iajs-2928 254 27 + + NUM iajs-2928 254 28 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 254 29 ) ) PUNCT iajs-2928 254 30 ≀ ≀ NOUN iajs-2928 254 31 π‘šπ‘Žπ‘₯{𝑓(π‘₯ π‘šπ‘Žπ‘₯{𝑓(π‘₯ NOUN iajs-2928 254 32 ) ) PUNCT iajs-2928 254 33 , , PUNCT iajs-2928 254 34 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 254 35 ) ) PUNCT iajs-2928 254 36 } } PUNCT iajs-2928 254 37 . . PUNCT iajs-2928 255 1 since since SCONJ iajs-2928 255 2 𝐺 𝐺 PROPN iajs-2928 255 3 is be AUX iajs-2928 255 4 an an DET iajs-2928 255 5 increasing increase VERB iajs-2928 255 6 function function NOUN iajs-2928 255 7 then then ADV iajs-2928 255 8 , , PUNCT iajs-2928 255 9 𝐺(𝑓(π‘ŸπΈπ‘₯ 𝐺(𝑓(π‘ŸπΈπ‘₯ PROPN iajs-2928 255 10 + + NUM iajs-2928 255 11 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 255 12 ) ) PUNCT iajs-2928 255 13 ≀ ≀ NOUN iajs-2928 255 14 π‘šπ‘Žπ‘₯{𝑓(π‘₯ π‘šπ‘Žπ‘₯{𝑓(π‘₯ NOUN iajs-2928 255 15 ) ) PUNCT iajs-2928 255 16 , , PUNCT iajs-2928 255 17 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 255 18 ) ) PUNCT iajs-2928 255 19 } } PUNCT iajs-2928 255 20 ) ) PUNCT iajs-2928 255 21 . . PUNCT iajs-2928 256 1 that that PRON iajs-2928 256 2 is be AUX iajs-2928 256 3 , , PUNCT iajs-2928 256 4 ( ( PUNCT iajs-2928 256 5 πΊπ‘œπ‘“)(π‘ŸπΈπ‘₯ πΊπ‘œπ‘“)(π‘ŸπΈπ‘₯ X iajs-2928 256 6 + + CCONJ iajs-2928 256 7 𝑠𝐸𝑦 𝑠𝐸𝑦 NUM iajs-2928 256 8 ) ) PUNCT iajs-2928 256 9 ≀ ≀ NOUN iajs-2928 256 10 π‘šπ‘Žπ‘₯{𝐺(𝑓(π‘₯ π‘šπ‘Žπ‘₯{𝐺(𝑓(π‘₯ NOUN iajs-2928 256 11 ) ) PUNCT iajs-2928 256 12 ) ) PUNCT iajs-2928 256 13 , , PUNCT iajs-2928 256 14 𝐺(𝑓(𝑦 𝐺(𝑓(𝑦 PROPN iajs-2928 256 15 ) ) PUNCT iajs-2928 256 16 ) ) PUNCT iajs-2928 256 17 } } PUNCT iajs-2928 256 18 = = SYM iajs-2928 256 19 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ X iajs-2928 256 20 { { PUNCT iajs-2928 256 21 ( ( PUNCT iajs-2928 256 22 πΊπ‘œπ‘“)(π‘₯ πΊπ‘œπ‘“)(π‘₯ ADJ iajs-2928 256 23 ) ) PUNCT iajs-2928 256 24 , , PUNCT iajs-2928 256 25 ( ( PUNCT iajs-2928 256 26 πΊπ‘œπ‘“)(𝑦 πΊπ‘œπ‘“)(𝑦 NOUN iajs-2928 256 27 } } PUNCT iajs-2928 256 28 . . PUNCT iajs-2928 257 1 hence hence ADV iajs-2928 257 2 , , PUNCT iajs-2928 257 3 πΊπ‘œπ‘“ πΊπ‘œπ‘“ PROPN iajs-2928 257 4 is be AUX iajs-2928 257 5 quasi quasi NOUN iajs-2928 257 6 semi semi ADJ iajs-2928 257 7 ( ( PUNCT iajs-2928 257 8 𝑝 𝑝 NOUN iajs-2928 257 9 , , PUNCT iajs-2928 257 10 𝐸)-convex 𝐸)-convex PROPN iajs-2928 257 11 on on ADP iajs-2928 257 12 𝐴. 𝐴. PROPN iajs-2928 257 13 for for ADP iajs-2928 257 14 proving prove VERB iajs-2928 257 15 ( ( PUNCT iajs-2928 257 16 ii ii NOUN iajs-2928 257 17 ) ) PUNCT iajs-2928 257 18 , , PUNCT iajs-2928 257 19 if if SCONJ iajs-2928 257 20 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 257 21 ) ) PUNCT iajs-2928 257 22 < < X iajs-2928 257 23 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 257 24 ) ) PUNCT iajs-2928 257 25 then then ADV iajs-2928 257 26 𝑓(π‘ŸπΈπ‘₯ 𝑓(π‘ŸπΈπ‘₯ PUNCT iajs-2928 257 27 + + CCONJ iajs-2928 257 28 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 257 29 ) ) PUNCT iajs-2928 257 30 ≀ ≀ NOUN iajs-2928 257 31 𝑓(𝑦 𝑓(𝑦 ADV iajs-2928 257 32 ) ) PUNCT iajs-2928 258 1 + + CCONJ iajs-2928 258 2 ( ( PUNCT iajs-2928 258 3 βˆ’π‘Ÿπ‘ )𝑏(π‘₯ βˆ’π‘Ÿπ‘ )𝑏(π‘₯ NOUN iajs-2928 258 4 , , PUNCT iajs-2928 258 5 𝑦 𝑦 NOUN iajs-2928 258 6 ) ) PUNCT iajs-2928 258 7 . . PUNCT iajs-2928 259 1 since since SCONJ iajs-2928 259 2 𝐺 𝐺 PROPN iajs-2928 259 3 is be AUX iajs-2928 259 4 an an DET iajs-2928 259 5 increasing increase VERB iajs-2928 259 6 function function NOUN iajs-2928 259 7 , , PUNCT iajs-2928 259 8 then then ADV iajs-2928 259 9 , , PUNCT iajs-2928 259 10 using use VERB iajs-2928 259 11 the the DET iajs-2928 259 12 last last ADJ iajs-2928 259 13 expression expression NOUN iajs-2928 259 14 , , PUNCT iajs-2928 259 15 if if SCONJ iajs-2928 259 16 ( ( PUNCT iajs-2928 259 17 πΊπ‘œπ‘“)(π‘₯ πΊπ‘œπ‘“)(π‘₯ ADJ iajs-2928 259 18 ) ) PUNCT iajs-2928 259 19 < < X iajs-2928 259 20 ( ( PUNCT iajs-2928 259 21 πΊπ‘œπ‘“)(𝑦 πΊπ‘œπ‘“)(𝑦 NOUN iajs-2928 259 22 ) ) PUNCT iajs-2928 259 23 we we PRON iajs-2928 259 24 get get VERB iajs-2928 259 25 ( ( PUNCT iajs-2928 259 26 πΊπ‘œπ‘“)(π‘ŸπΈπ‘₯ πΊπ‘œπ‘“)(π‘ŸπΈπ‘₯ X iajs-2928 259 27 + + CCONJ iajs-2928 259 28 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 259 29 ) ) PUNCT iajs-2928 259 30 ≀ ≀ NOUN iajs-2928 259 31 𝐺[𝑓(𝑦 𝐺[𝑓(𝑦 NOUN iajs-2928 259 32 ) ) PUNCT iajs-2928 260 1 + + CCONJ iajs-2928 260 2 ( ( PUNCT iajs-2928 260 3 βˆ’π‘Ÿπ‘ )𝑏(π‘₯ βˆ’π‘Ÿπ‘ )𝑏(π‘₯ NOUN iajs-2928 260 4 , , PUNCT iajs-2928 260 5 𝑦 𝑦 NOUN iajs-2928 260 6 ) ) PUNCT iajs-2928 260 7 ] ] PUNCT iajs-2928 260 8 . . PUNCT iajs-2928 261 1 from from ADP iajs-2928 261 2 the the DET iajs-2928 261 3 assumption assumption NOUN iajs-2928 261 4 , , PUNCT iajs-2928 261 5 𝐺 𝐺 PROPN iajs-2928 261 6 is be AUX iajs-2928 261 7 a a DET iajs-2928 261 8 sublinear sublinear NOUN iajs-2928 261 9 mapping mapping NOUN iajs-2928 261 10 . . PUNCT iajs-2928 262 1 thus thus ADV iajs-2928 262 2 , , PUNCT iajs-2928 262 3 the the DET iajs-2928 262 4 last last ADJ iajs-2928 262 5 inequality inequality NOUN iajs-2928 262 6 yields yield VERB iajs-2928 262 7 , , PUNCT iajs-2928 262 8 ( ( PUNCT iajs-2928 262 9 πΊπ‘œπ‘“)(π‘ŸπΈπ‘₯ πΊπ‘œπ‘“)(π‘ŸπΈπ‘₯ X iajs-2928 262 10 + + CCONJ iajs-2928 262 11 𝑠𝐸𝑦 𝑠𝐸𝑦 NOUN iajs-2928 262 12 ) ) PUNCT iajs-2928 262 13 ≀ ≀ NOUN iajs-2928 262 14 ( ( PUNCT iajs-2928 262 15 πΊπ‘œπ‘“)(𝑦 πΊπ‘œπ‘“)(𝑦 NOUN iajs-2928 262 16 ) ) PUNCT iajs-2928 262 17 + + CCONJ iajs-2928 262 18 ( ( PUNCT iajs-2928 262 19 βˆ’π‘Ÿπ‘ )(πΊπ‘œπ‘)(π‘₯ βˆ’π‘Ÿπ‘ )(πΊπ‘œπ‘)(π‘₯ PROPN iajs-2928 262 20 , , PUNCT iajs-2928 262 21 𝑦 𝑦 NOUN iajs-2928 262 22 ) ) PUNCT iajs-2928 262 23 . . PUNCT iajs-2928 263 1 since since SCONJ iajs-2928 263 2 𝐺 𝐺 PROPN iajs-2928 263 3 and and CCONJ iajs-2928 263 4 𝑏 𝑏 PROPN iajs-2928 263 5 are be AUX iajs-2928 263 6 strictly strictly ADV iajs-2928 263 7 positive positive ADJ iajs-2928 263 8 functions function NOUN iajs-2928 263 9 , , PUNCT iajs-2928 263 10 then then ADV iajs-2928 263 11 ( ( PUNCT iajs-2928 263 12 πΊπ‘œπ‘)(π‘₯ πΊπ‘œπ‘)(π‘₯ NOUN iajs-2928 263 13 , , PUNCT iajs-2928 263 14 𝑦 𝑦 NOUN iajs-2928 263 15 ) ) PUNCT iajs-2928 263 16 is be AUX iajs-2928 263 17 strictly strictly ADV iajs-2928 263 18 positive positive ADJ iajs-2928 263 19 . . PUNCT iajs-2928 264 1 hence hence ADV iajs-2928 264 2 , , PUNCT iajs-2928 264 3 we we PRON iajs-2928 264 4 obtain obtain VERB iajs-2928 264 5 the the DET iajs-2928 264 6 required required ADJ iajs-2928 264 7 conclusion conclusion NOUN iajs-2928 264 8 . . PUNCT iajs-2928 265 1 β–  β–  PUNCT iajs-2928 265 2 proposition proposition NOUN iajs-2928 265 3 3.5 3.5 NUM iajs-2928 265 4 . . PUNCT iajs-2928 266 1 let let VERB iajs-2928 266 2 𝑔 𝑔 NUM iajs-2928 266 3 : : PUNCT iajs-2928 266 4 𝐴 𝐴 PROPN iajs-2928 266 5 βŠ† βŠ† NUM iajs-2928 266 6 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 266 7 β†’ β†’ PUNCT iajs-2928 266 8 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 266 9 be be AUX iajs-2928 266 10 a a DET iajs-2928 266 11 linear linear ADJ iajs-2928 266 12 mapping mapping NOUN iajs-2928 266 13 such such ADJ iajs-2928 266 14 that that SCONJ iajs-2928 266 15 πΈπ‘œπ‘” πΈπ‘œπ‘” PROPN iajs-2928 266 16 = = PRON iajs-2928 266 17 π‘”π‘œπΈ. π‘”π‘œπΈ. AUX iajs-2928 266 18 assume assume VERB iajs-2928 266 19 also also ADV iajs-2928 266 20 that that SCONJ iajs-2928 266 21 𝑓 𝑓 X iajs-2928 266 22 : : PUNCT iajs-2928 266 23 𝑉 𝑉 PROPN iajs-2928 266 24 βŠ† βŠ† NUM iajs-2928 266 25 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 266 26 β†’ β†’ SYM iajs-2928 266 27 𝑅 𝑅 PROPN iajs-2928 266 28 such such ADJ iajs-2928 266 29 that that DET iajs-2928 266 30 𝑉 𝑉 PROPN iajs-2928 266 31 = = PUNCT iajs-2928 266 32 𝑔(𝐴 𝑔(𝐴 PROPN iajs-2928 266 33 ) ) PUNCT iajs-2928 266 34 . . PUNCT iajs-2928 267 1 then then ADV iajs-2928 267 2 i. i. PROPN iajs-2928 267 3 if if SCONJ iajs-2928 267 4 𝑓 𝑓 DET iajs-2928 267 5 quasi quasi NOUN iajs-2928 267 6 semi semi ADJ iajs-2928 267 7 ( ( PUNCT iajs-2928 267 8 𝑝 𝑝 NOUN iajs-2928 267 9 , , PUNCT iajs-2928 267 10 𝐸)-convex 𝐸)-convex PRON iajs-2928 267 11 function function NOUN iajs-2928 267 12 , , PUNCT iajs-2928 267 13 then then ADV iajs-2928 267 14 π‘“π‘œπ‘” π‘“π‘œπ‘” ADV iajs-2928 267 15 : : PUNCT iajs-2928 267 16 𝐴 𝐴 PROPN iajs-2928 267 17 β†’ β†’ SYM iajs-2928 267 18 𝑅 𝑅 PROPN iajs-2928 267 19 is be AUX iajs-2928 267 20 quasi quasi NOUN iajs-2928 267 21 semi semi ADJ iajs-2928 267 22 ( ( PUNCT iajs-2928 267 23 𝑝 𝑝 NOUN iajs-2928 267 24 , , PUNCT iajs-2928 267 25 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 267 26 function function NOUN iajs-2928 267 27 . . PUNCT iajs-2928 268 1 ii ii PROPN iajs-2928 268 2 . . PUNCT iajs-2928 269 1 if if SCONJ iajs-2928 269 2 𝑓 𝑓 DET iajs-2928 269 3 pseudo pseudo NOUN iajs-2928 269 4 semi semi X iajs-2928 269 5 ( ( PUNCT iajs-2928 269 6 𝑝 𝑝 NOUN iajs-2928 269 7 , , PUNCT iajs-2928 269 8 𝐸)-convex 𝐸)-convex PRON iajs-2928 269 9 function function NOUN iajs-2928 269 10 , , PUNCT iajs-2928 269 11 then then ADV iajs-2928 269 12 π‘“π‘œπ‘” π‘“π‘œπ‘” ADV iajs-2928 269 13 : : PUNCT iajs-2928 269 14 𝐴 𝐴 PROPN iajs-2928 269 15 β†’ β†’ SYM iajs-2928 269 16 𝑅 𝑅 PROPN iajs-2928 269 17 is be AUX iajs-2928 269 18 pseudo pseudo NOUN iajs-2928 269 19 semi semi ADJ iajs-2928 269 20 ( ( PUNCT iajs-2928 269 21 𝑝 𝑝 NOUN iajs-2928 269 22 , , PUNCT iajs-2928 269 23 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 269 24 function function NOUN iajs-2928 269 25 . . PUNCT iajs-2928 270 1 proof proof NOUN iajs-2928 270 2 . . PUNCT iajs-2928 271 1 let let VERB iajs-2928 271 2 π‘₯ π‘₯ PRON iajs-2928 271 3 , , PUNCT iajs-2928 271 4 𝑦 𝑦 PROPN iajs-2928 271 5 ∈ ∈ PROPN iajs-2928 271 6 𝐴 𝐴 NOUN iajs-2928 271 7 then then ADV iajs-2928 271 8 π‘ŸπΈπ‘₯ π‘ŸπΈπ‘₯ NUM iajs-2928 271 9 + + X iajs-2928 271 10 𝑠 𝑠 NUM iajs-2928 271 11 𝐸𝑦 𝐸𝑦 PROPN iajs-2928 271 12 ∈ ∈ PROPN iajs-2928 271 13 𝐴. 𝐴. PROPN iajs-2928 271 14 for for ADP iajs-2928 271 15 proving prove VERB iajs-2928 271 16 ( ( PUNCT iajs-2928 271 17 i i NOUN iajs-2928 271 18 ) ) PUNCT iajs-2928 271 19 , , PUNCT iajs-2928 271 20 we we PRON iajs-2928 271 21 need need VERB iajs-2928 271 22 to to PART iajs-2928 271 23 show show VERB iajs-2928 271 24 that that SCONJ iajs-2928 271 25 ( ( PUNCT iajs-2928 271 26 π‘“π‘œπ‘”)(π‘ŸπΈπ‘₯ π‘“π‘œπ‘”)(π‘ŸπΈπ‘₯ SYM iajs-2928 271 27 + + NOUN iajs-2928 271 28 𝑠 𝑠 NUM iajs-2928 271 29 𝐸𝑦 𝐸𝑦 PROPN iajs-2928 271 30 ) ) PUNCT iajs-2928 271 31 ≀ ≀ NUM iajs-2928 271 32 π‘šπ‘Žπ‘₯{𝑓(𝑔(π‘₯ π‘šπ‘Žπ‘₯{𝑓(𝑔(π‘₯ PROPN iajs-2928 271 33 ) ) PUNCT iajs-2928 271 34 ) ) PUNCT iajs-2928 271 35 , , PUNCT iajs-2928 271 36 𝑓(𝑔(𝑦 𝑓(𝑔(𝑦 NOUN iajs-2928 271 37 ) ) PUNCT iajs-2928 271 38 ) ) PUNCT iajs-2928 271 39 } } PUNCT iajs-2928 271 40 . . PUNCT iajs-2928 272 1 now now ADV iajs-2928 272 2 , , PUNCT iajs-2928 272 3 from from ADP iajs-2928 272 4 the the DET iajs-2928 272 5 linearity linearity NOUN iajs-2928 272 6 of of ADP iajs-2928 272 7 𝑔 𝑔 PROPN iajs-2928 272 8 and and CCONJ iajs-2928 272 9 the the DET iajs-2928 272 10 fact fact NOUN iajs-2928 272 11 that that SCONJ iajs-2928 272 12 πΈπ‘œπ‘” πΈπ‘œπ‘” PROPN iajs-2928 272 13 = = PUNCT iajs-2928 272 14 π‘”π‘œπΈ π‘”π‘œπΈ PROPN iajs-2928 272 15 , , PUNCT iajs-2928 272 16 we we PRON iajs-2928 272 17 have have AUX iajs-2928 272 18 ( ( PUNCT iajs-2928 272 19 π‘“π‘œπ‘”)(π‘ŸπΈπ‘₯ π‘“π‘œπ‘”)(π‘ŸπΈπ‘₯ X iajs-2928 272 20 + + PROPN iajs-2928 272 21 𝑠 𝑠 PRON iajs-2928 272 22 𝐸𝑦 𝐸𝑦 PROPN iajs-2928 272 23 ) ) PUNCT iajs-2928 272 24 = = SYM iajs-2928 272 25 𝑓(π‘Ÿ 𝑓(π‘Ÿ NOUN iajs-2928 272 26 𝑔(𝐸π‘₯ 𝑔(𝐸π‘₯ NOUN iajs-2928 272 27 ) ) PUNCT iajs-2928 272 28 + + CCONJ iajs-2928 272 29 𝑠 𝑠 DET iajs-2928 272 30 𝑔(𝐸𝑦 𝑔(𝐸𝑦 NOUN iajs-2928 272 31 ) ) PUNCT iajs-2928 272 32 ) ) PUNCT iajs-2928 273 1 = = SYM iajs-2928 273 2 𝑓(π‘Ÿ 𝑓(π‘Ÿ X iajs-2928 273 3 ( ( PUNCT iajs-2928 273 4 π‘”π‘œπΈ)(π‘₯ π‘”π‘œπΈ)(π‘₯ ADJ iajs-2928 273 5 ) ) PUNCT iajs-2928 273 6 + + CCONJ iajs-2928 273 7 𝑠(π‘”π‘œπΈ)(𝑦 𝑠(π‘”π‘œπΈ)(𝑦 NOUN iajs-2928 273 8 ) ) PUNCT iajs-2928 273 9 ) ) PUNCT iajs-2928 273 10 = = PUNCT iajs-2928 273 11 𝑓(π‘Ÿ(πΈπ‘œπ‘”)(π‘₯ 𝑓(π‘Ÿ(πΈπ‘œπ‘”)(π‘₯ ADJ iajs-2928 273 12 ) ) PUNCT iajs-2928 273 13 + + ADJ iajs-2928 273 14 𝑠(πΈπ‘œπ‘”)(𝑦 𝑠(πΈπ‘œπ‘”)(𝑦 NOUN iajs-2928 273 15 ) ) PUNCT iajs-2928 273 16 ) ) PUNCT iajs-2928 274 1 = = SYM iajs-2928 274 2 𝑓 𝑓 X iajs-2928 274 3 ( ( PUNCT iajs-2928 274 4 π‘Ÿ π‘Ÿ X iajs-2928 274 5 ( ( PUNCT iajs-2928 274 6 𝐸(𝑔(π‘₯ 𝐸(𝑔(π‘₯ PROPN iajs-2928 274 7 ) ) PUNCT iajs-2928 274 8 ) ) PUNCT iajs-2928 274 9 ) ) PUNCT iajs-2928 275 1 + + CCONJ iajs-2928 275 2 𝑠 𝑠 INTJ iajs-2928 275 3 ( ( PUNCT iajs-2928 275 4 𝐸(𝑔(𝑦 𝐸(𝑔(𝑦 NOUN iajs-2928 275 5 ) ) PUNCT iajs-2928 275 6 ) ) PUNCT iajs-2928 275 7 ) ) PUNCT iajs-2928 275 8 ) ) PUNCT iajs-2928 275 9 . . PUNCT iajs-2928 276 1 ( ( PUNCT iajs-2928 276 2 5 5 X iajs-2928 276 3 ) ) PUNCT iajs-2928 276 4 note note NOUN iajs-2928 276 5 that that SCONJ iajs-2928 276 6 , , PUNCT iajs-2928 276 7 since since SCONJ iajs-2928 276 8 π‘₯ π‘₯ PROPN iajs-2928 276 9 , , PUNCT iajs-2928 276 10 𝑦 𝑦 PROPN iajs-2928 276 11 ∈ ∈ PROPN iajs-2928 276 12 𝐴 𝐴 PROPN iajs-2928 276 13 then then ADV iajs-2928 276 14 𝑔(π‘₯ 𝑔(π‘₯ PROPN iajs-2928 276 15 ) ) PUNCT iajs-2928 276 16 , , PUNCT iajs-2928 276 17 𝑔(𝑦 𝑔(𝑦 PROPN iajs-2928 276 18 ) ) PUNCT iajs-2928 276 19 ∈ ∈ PROPN iajs-2928 276 20 𝑉 𝑉 PROPN iajs-2928 276 21 = = PUNCT iajs-2928 276 22 𝑔(𝐴 𝑔(𝐴 PROPN iajs-2928 276 23 ) ) PUNCT iajs-2928 276 24 . . PUNCT iajs-2928 277 1 also also ADV iajs-2928 277 2 , , PUNCT iajs-2928 277 3 we we PRON iajs-2928 277 4 have have VERB iajs-2928 277 5 𝑉 𝑉 PROPN iajs-2928 277 6 = = PUNCT iajs-2928 277 7 𝑔(𝐴 𝑔(𝐴 PROPN iajs-2928 277 8 ) ) PUNCT iajs-2928 278 1 is be AUX iajs-2928 278 2 ( ( PUNCT iajs-2928 278 3 𝑝 𝑝 PROPN iajs-2928 278 4 , , PUNCT iajs-2928 278 5 𝐸)convex 𝐸)convex PROPN iajs-2928 278 6 set set PROPN iajs-2928 278 7 ( ( PUNCT iajs-2928 278 8 see see VERB iajs-2928 278 9 [ [ X iajs-2928 278 10 9 9 NUM iajs-2928 278 11 , , PUNCT iajs-2928 278 12 proposition proposition NOUN iajs-2928 278 13 2.11 2.11 NUM iajs-2928 278 14 ] ] PUNCT iajs-2928 278 15 ) ) PUNCT iajs-2928 278 16 thus thus ADV iajs-2928 278 17 π‘Ÿ π‘Ÿ X iajs-2928 278 18 ( ( PUNCT iajs-2928 278 19 𝐸(𝑔(π‘₯ 𝐸(𝑔(π‘₯ PROPN iajs-2928 278 20 ) ) PUNCT iajs-2928 278 21 ) ) PUNCT iajs-2928 278 22 ) ) PUNCT iajs-2928 279 1 + + CCONJ iajs-2928 279 2 𝑠 𝑠 INTJ iajs-2928 279 3 ( ( PUNCT iajs-2928 279 4 𝐸(𝑔(𝑦 𝐸(𝑔(𝑦 NOUN iajs-2928 279 5 ) ) PUNCT iajs-2928 279 6 ) ) PUNCT iajs-2928 279 7 ) ) PUNCT iajs-2928 280 1 ∈ ∈ PROPN iajs-2928 280 2 𝑉. 𝑉. NOUN iajs-2928 280 3 ( ( PUNCT iajs-2928 280 4 6 6 NUM iajs-2928 280 5 ) ) PUNCT iajs-2928 280 6 ihjpas ihjpa NOUN iajs-2928 280 7 . . PUNCT iajs-2928 281 1 36(1)2023 36(1)2023 NUM iajs-2928 281 2 363 363 NUM iajs-2928 281 3 from from ADP iajs-2928 281 4 ( ( PUNCT iajs-2928 281 5 5)-(6 5)-(6 NUM iajs-2928 281 6 ) ) PUNCT iajs-2928 281 7 and and CCONJ iajs-2928 281 8 the the DET iajs-2928 281 9 fact fact NOUN iajs-2928 281 10 that that SCONJ iajs-2928 281 11 𝑓 𝑓 PRON iajs-2928 281 12 is be AUX iajs-2928 281 13 quasi quasi NOUN iajs-2928 281 14 semi semi ADJ iajs-2928 281 15 ( ( PUNCT iajs-2928 281 16 𝑝 𝑝 NOUN iajs-2928 281 17 , , PUNCT iajs-2928 281 18 𝐸)-convex 𝐸)-convex PRON iajs-2928 281 19 function function VERB iajs-2928 281 20 , , PUNCT iajs-2928 281 21 we we PRON iajs-2928 281 22 get get VERB iajs-2928 281 23 ( ( PUNCT iajs-2928 281 24 π‘“π‘œπ‘”)(π‘ŸπΈπ‘₯ π‘“π‘œπ‘”)(π‘ŸπΈπ‘₯ X iajs-2928 281 25 + + PROPN iajs-2928 281 26 𝑠 𝑠 NUM iajs-2928 281 27 𝐸𝑦 𝐸𝑦 PROPN iajs-2928 281 28 ) ) PUNCT iajs-2928 281 29 ≀ ≀ NOUN iajs-2928 281 30 π‘šπ‘Žπ‘₯{𝑓(𝑔(π‘₯ π‘šπ‘Žπ‘₯{𝑓(𝑔(π‘₯ PROPN iajs-2928 281 31 ) ) PUNCT iajs-2928 281 32 ) ) PUNCT iajs-2928 281 33 , , PUNCT iajs-2928 281 34 𝑓(𝑔(𝑦 𝑓(𝑔(𝑦 NOUN iajs-2928 281 35 ) ) PUNCT iajs-2928 281 36 ) ) PUNCT iajs-2928 281 37 } } PUNCT iajs-2928 282 1 = = SYM iajs-2928 282 2 π‘šπ‘Žπ‘₯{(π‘“π‘œπ‘”)(π‘₯ π‘šπ‘Žπ‘₯{(π‘“π‘œπ‘”)(π‘₯ NOUN iajs-2928 282 3 ) ) PUNCT iajs-2928 282 4 , , PUNCT iajs-2928 282 5 ( ( PUNCT iajs-2928 282 6 π‘“π‘œπ‘”)(𝑦 π‘“π‘œπ‘”)(𝑦 NOUN iajs-2928 282 7 ) ) PUNCT iajs-2928 282 8 } } PUNCT iajs-2928 282 9 . . PUNCT iajs-2928 283 1 hence hence ADV iajs-2928 283 2 , , PUNCT iajs-2928 283 3 π‘“π‘œπ‘” π‘“π‘œπ‘” PROPN iajs-2928 283 4 is be AUX iajs-2928 283 5 quasi quasi NOUN iajs-2928 283 6 semi semi ADJ iajs-2928 283 7 ( ( PUNCT iajs-2928 283 8 𝑝 𝑝 NOUN iajs-2928 283 9 , , PUNCT iajs-2928 283 10 𝐸)-convex 𝐸)-convex X iajs-2928 283 11 . . PUNCT iajs-2928 284 1 let let VERB iajs-2928 284 2 us we PRON iajs-2928 284 3 prove prove VERB iajs-2928 284 4 ( ( PUNCT iajs-2928 284 5 ii ii NOUN iajs-2928 284 6 ) ) PUNCT iajs-2928 284 7 , , PUNCT iajs-2928 284 8 from from ADP iajs-2928 284 9 the the DET iajs-2928 284 10 assumption assumption NOUN iajs-2928 284 11 π‘₯ π‘₯ PROPN iajs-2928 284 12 , , PUNCT iajs-2928 284 13 𝑦 𝑦 PROPN iajs-2928 284 14 ∈ ∈ PROPN iajs-2928 284 15 𝐴 𝐴 PROPN iajs-2928 284 16 , , PUNCT iajs-2928 284 17 𝑔(π‘₯ 𝑔(π‘₯ PROPN iajs-2928 284 18 ) ) PUNCT iajs-2928 284 19 , , PUNCT iajs-2928 284 20 𝑔(𝑦 𝑔(𝑦 PROPN iajs-2928 284 21 ) ) PUNCT iajs-2928 284 22 ∈ ∈ PROPN iajs-2928 284 23 𝑉 𝑉 PROPN iajs-2928 284 24 = = PUNCT iajs-2928 284 25 𝑔(𝐴 𝑔(𝐴 PROPN iajs-2928 284 26 ) ) PUNCT iajs-2928 284 27 , , PUNCT iajs-2928 284 28 and and CCONJ iajs-2928 284 29 𝑓 𝑓 PRON iajs-2928 284 30 is be AUX iajs-2928 284 31 pseudo pseudo NOUN iajs-2928 284 32 semi semi ADJ iajs-2928 284 33 ( ( PUNCT iajs-2928 284 34 𝑝 𝑝 NOUN iajs-2928 284 35 , , PUNCT iajs-2928 284 36 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 284 37 function function NOUN iajs-2928 284 38 . . PUNCT iajs-2928 285 1 hence hence ADV iajs-2928 285 2 , , PUNCT iajs-2928 285 3 𝑓(𝑔(π‘₯ 𝑓(𝑔(π‘₯ PROPN iajs-2928 285 4 ) ) PUNCT iajs-2928 285 5 ) ) PUNCT iajs-2928 286 1 < < X iajs-2928 286 2 𝑓(𝑔(𝑦 𝑓(𝑔(𝑦 NOUN iajs-2928 286 3 ) ) PUNCT iajs-2928 286 4 ) ) PUNCT iajs-2928 286 5 ( ( PUNCT iajs-2928 286 6 7 7 X iajs-2928 286 7 ) ) PUNCT iajs-2928 286 8 again again ADV iajs-2928 286 9 , , PUNCT iajs-2928 286 10 we we PRON iajs-2928 286 11 follow follow VERB iajs-2928 286 12 the the DET iajs-2928 286 13 steps step NOUN iajs-2928 286 14 of of ADP iajs-2928 286 15 the the DET iajs-2928 286 16 proof proof NOUN iajs-2928 286 17 of of ADP iajs-2928 286 18 ( ( PUNCT iajs-2928 286 19 i i NOUN iajs-2928 286 20 ) ) PUNCT iajs-2928 286 21 to to PART iajs-2928 286 22 obtain obtain VERB iajs-2928 286 23 the the DET iajs-2928 286 24 equality equality NOUN iajs-2928 286 25 ( ( PUNCT iajs-2928 286 26 5 5 NUM iajs-2928 286 27 ) ) PUNCT iajs-2928 286 28 . . PUNCT iajs-2928 287 1 namely namely ADV iajs-2928 287 2 , , PUNCT iajs-2928 287 3 ( ( PUNCT iajs-2928 287 4 π‘“π‘œπ‘”)(π‘ŸπΈπ‘₯ π‘“π‘œπ‘”)(π‘ŸπΈπ‘₯ X iajs-2928 287 5 + + NOUN iajs-2928 287 6 𝑠 𝑠 NUM iajs-2928 287 7 𝐸𝑦 𝐸𝑦 PROPN iajs-2928 287 8 ) ) PUNCT iajs-2928 287 9 = = SYM iajs-2928 288 1 𝑓 𝑓 PROPN iajs-2928 288 2 ( ( PUNCT iajs-2928 288 3 π‘Ÿ π‘Ÿ X iajs-2928 288 4 ( ( PUNCT iajs-2928 288 5 𝐸(𝑔(π‘₯ 𝐸(𝑔(π‘₯ PROPN iajs-2928 288 6 ) ) PUNCT iajs-2928 288 7 ) ) PUNCT iajs-2928 288 8 ) ) PUNCT iajs-2928 289 1 + + CCONJ iajs-2928 289 2 𝑠 𝑠 INTJ iajs-2928 289 3 ( ( PUNCT iajs-2928 289 4 𝐸(𝑔(𝑦 𝐸(𝑔(𝑦 NOUN iajs-2928 289 5 ) ) PUNCT iajs-2928 289 6 ) ) PUNCT iajs-2928 289 7 ) ) PUNCT iajs-2928 289 8 ) ) PUNCT iajs-2928 289 9 from from ADP iajs-2928 289 10 definition definition NOUN iajs-2928 289 11 of of ADP iajs-2928 289 12 𝑓 𝑓 PRON iajs-2928 289 13 there there PRON iajs-2928 289 14 exist exist VERB iajs-2928 289 15 strictly strictly ADV iajs-2928 289 16 positive positive ADJ iajs-2928 289 17 𝑏 𝑏 NOUN iajs-2928 289 18 : : PUNCT iajs-2928 289 19 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 289 20 Γ— Γ— NOUN iajs-2928 289 21 𝑅𝑛 𝑅𝑛 PROPN iajs-2928 289 22 β†’ β†’ SYM iajs-2928 289 23 𝑅 𝑅 PROPN iajs-2928 289 24 such such ADJ iajs-2928 289 25 that that SCONJ iajs-2928 289 26 the the DET iajs-2928 289 27 right right ADJ iajs-2928 289 28 - - PUNCT iajs-2928 289 29 hand hand NOUN iajs-2928 289 30 side side NOUN iajs-2928 289 31 expression expression NOUN iajs-2928 289 32 above above ADP iajs-2928 289 33 yields yield NOUN iajs-2928 289 34 ≀ ≀ NUM iajs-2928 289 35 𝑓(𝑔(𝑦 𝑓(𝑔(𝑦 NOUN iajs-2928 289 36 ) ) PUNCT iajs-2928 289 37 ) ) PUNCT iajs-2928 290 1 + + CCONJ iajs-2928 290 2 ( ( PUNCT iajs-2928 290 3 βˆ’π‘Ÿπ‘  βˆ’π‘Ÿπ‘  NOUN iajs-2928 290 4 ) ) PUNCT iajs-2928 290 5 𝑏(π‘₯ 𝑏(π‘₯ PROPN iajs-2928 290 6 , , PUNCT iajs-2928 290 7 𝑦 𝑦 NOUN iajs-2928 290 8 ) ) PUNCT iajs-2928 290 9 = = SYM iajs-2928 290 10 ( ( PUNCT iajs-2928 290 11 π‘“π‘œπ‘”)(𝑦 π‘“π‘œπ‘”)(𝑦 NOUN iajs-2928 290 12 ) ) PUNCT iajs-2928 290 13 + + CCONJ iajs-2928 290 14 ( ( PUNCT iajs-2928 290 15 βˆ’π‘Ÿπ‘ )𝑏(π‘₯ βˆ’π‘Ÿπ‘ )𝑏(π‘₯ NOUN iajs-2928 290 16 , , PUNCT iajs-2928 290 17 𝑦 𝑦 NOUN iajs-2928 290 18 ) ) PUNCT iajs-2928 290 19 ( ( PUNCT iajs-2928 290 20 8) 8) NUM iajs-2928 290 21 thus thus ADV iajs-2928 290 22 , , PUNCT iajs-2928 290 23 from from ADP iajs-2928 290 24 ( ( PUNCT iajs-2928 290 25 7 7 NUM iajs-2928 290 26 ) ) PUNCT iajs-2928 290 27 and and CCONJ iajs-2928 290 28 ( ( PUNCT iajs-2928 290 29 8) 8) NUM iajs-2928 290 30 , , PUNCT iajs-2928 290 31 π‘“π‘œπ‘” π‘“π‘œπ‘” X iajs-2928 290 32 is be AUX iajs-2928 290 33 pseudo pseudo NOUN iajs-2928 290 34 semi semi ADJ iajs-2928 290 35 ( ( PUNCT iajs-2928 290 36 𝑝 𝑝 NOUN iajs-2928 290 37 , , PUNCT iajs-2928 290 38 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 290 39 function function NOUN iajs-2928 290 40 . . PUNCT iajs-2928 291 1 β–  β–  PUNCT iajs-2928 291 2 4 4 X iajs-2928 291 3 . . PUNCT iajs-2928 291 4 applications application NOUN iajs-2928 291 5 to to ADP iajs-2928 291 6 non non ADJ iajs-2928 291 7 - - ADJ iajs-2928 291 8 linear linear ADJ iajs-2928 291 9 optimization optimization NOUN iajs-2928 291 10 programming programming NOUN iajs-2928 291 11 in in ADP iajs-2928 291 12 this this DET iajs-2928 291 13 section section NOUN iajs-2928 291 14 , , PUNCT iajs-2928 291 15 a a DET iajs-2928 291 16 non non ADJ iajs-2928 291 17 - - ADJ iajs-2928 291 18 linear linear ADJ iajs-2928 291 19 optimization optimization NOUN iajs-2928 291 20 programming programming NOUN iajs-2928 291 21 problem problem NOUN iajs-2928 291 22 denoted denote VERB iajs-2928 291 23 by by ADP iajs-2928 291 24 ( ( PUNCT iajs-2928 291 25 𝑁𝐿𝑃 𝑁𝐿𝑃 PROPN iajs-2928 291 26 ) ) PUNCT iajs-2928 291 27 and and CCONJ iajs-2928 291 28 is be AUX iajs-2928 291 29 defined define VERB iajs-2928 291 30 as as ADP iajs-2928 291 31 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 291 32 ) ) PUNCT iajs-2928 291 33 subject subject NOUN iajs-2928 291 34 to to ADP iajs-2928 291 35 π‘₯ π‘₯ DET iajs-2928 291 36 ∈ ∈ PROPN iajs-2928 291 37 𝐴 𝐴 PROPN iajs-2928 291 38 , , PUNCT iajs-2928 291 39 where where SCONJ iajs-2928 291 40 𝐴 𝐴 PROPN iajs-2928 291 41 is be AUX iajs-2928 291 42 ( ( PUNCT iajs-2928 291 43 𝑝 𝑝 NOUN iajs-2928 291 44 , , PUNCT iajs-2928 291 45 𝐸)-convex 𝐸)-convex X iajs-2928 291 46 . . PUNCT iajs-2928 292 1 the the DET iajs-2928 292 2 set set NOUN iajs-2928 292 3 of of ADP iajs-2928 292 4 all all DET iajs-2928 292 5 optimal optimal ADJ iajs-2928 292 6 solutions solution NOUN iajs-2928 292 7 ( ( PUNCT iajs-2928 292 8 or or CCONJ iajs-2928 292 9 global global ADJ iajs-2928 292 10 minimum minimum NOUN iajs-2928 292 11 ) ) PUNCT iajs-2928 292 12 of of ADP iajs-2928 292 13 problem problem NOUN iajs-2928 292 14 ( ( PUNCT iajs-2928 292 15 𝑁𝐿𝑃 𝑁𝐿𝑃 PROPN iajs-2928 292 16 ) ) PUNCT iajs-2928 292 17 is be AUX iajs-2928 292 18 defined define VERB iajs-2928 292 19 as as ADP iajs-2928 292 20 π‘Žπ‘Ÿπ‘”π‘šπ‘–π‘›π΄ π‘Žπ‘Ÿπ‘”π‘šπ‘–π‘›π΄ PROPN iajs-2928 292 21 𝑓= 𝑓= NOUN iajs-2928 292 22 { { PUNCT iajs-2928 292 23 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2928 292 24 ∈ ∈ PROPN iajs-2928 292 25 𝐴 𝐴 PROPN iajs-2928 292 26 : : PUNCT iajs-2928 292 27 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 292 28 ) ) PUNCT iajs-2928 292 29 ≀ ≀ NUM iajs-2928 292 30 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 292 31 ) ) PUNCT iajs-2928 292 32 for for ADP iajs-2928 292 33 all all DET iajs-2928 292 34 ∈ ∈ PROPN iajs-2928 292 35 𝐴 𝐴 PROPN iajs-2928 292 36 } } PUNCT iajs-2928 292 37 . . PUNCT iajs-2928 293 1 a a DET iajs-2928 293 2 point point NOUN iajs-2928 293 3 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2928 293 4 is be AUX iajs-2928 293 5 called call VERB iajs-2928 293 6 a a DET iajs-2928 293 7 local local ADJ iajs-2928 293 8 minimizer minimizer NOUN iajs-2928 293 9 for for ADP iajs-2928 293 10 problem problem NOUN iajs-2928 293 11 ( ( PUNCT iajs-2928 293 12 𝑁𝐿𝑃 𝑁𝐿𝑃 PROPN iajs-2928 293 13 ) ) PUNCT iajs-2928 293 14 if if SCONJ iajs-2928 293 15 there there PRON iajs-2928 293 16 exists exist VERB iajs-2928 293 17 𝛿 𝛿 PROPN iajs-2928 293 18 > > X iajs-2928 293 19 0 0 NUM iajs-2928 293 20 such such ADJ iajs-2928 293 21 that that SCONJ iajs-2928 293 22 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 293 23 ) ) PUNCT iajs-2928 293 24 ≀ ≀ NUM iajs-2928 293 25 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 293 26 ) ) PUNCT iajs-2928 293 27 for for ADP iajs-2928 293 28 all all DET iajs-2928 293 29 π‘₯ π‘₯ PRON iajs-2928 293 30 ∈ ∈ PROPN iajs-2928 293 31 𝐡(π‘₯βˆ— 𝐡(π‘₯βˆ— PROPN iajs-2928 293 32 , , PUNCT iajs-2928 293 33 𝛿 𝛿 ADJ iajs-2928 293 34 ) ) PUNCT iajs-2928 293 35 ∩ ∩ ADJ iajs-2928 293 36 𝐴 𝐴 NOUN iajs-2928 293 37 where where SCONJ iajs-2928 293 38 𝐡 𝐡 PROPN iajs-2928 293 39 𝛿(π‘₯βˆ— 𝛿(π‘₯βˆ— NOUN iajs-2928 293 40 ) ) PUNCT iajs-2928 293 41 is be AUX iajs-2928 293 42 an an DET iajs-2928 293 43 open open ADJ iajs-2928 293 44 ball ball NOUN iajs-2928 293 45 . . PUNCT iajs-2928 294 1 the the DET iajs-2928 294 2 following follow VERB iajs-2928 294 3 optimality optimality NOUN iajs-2928 294 4 properties property NOUN iajs-2928 294 5 are be AUX iajs-2928 294 6 satisfied satisfied ADJ iajs-2928 294 7 under under ADP iajs-2928 294 8 different different ADJ iajs-2928 294 9 conditions condition NOUN iajs-2928 294 10 for for ADP iajs-2928 294 11 the the DET iajs-2928 294 12 objective objective ADJ iajs-2928 294 13 function function NOUN iajs-2928 294 14 𝑓 𝑓 PRON iajs-2928 294 15 and and CCONJ iajs-2928 294 16 the the DET iajs-2928 294 17 mapping mapping NOUN iajs-2928 294 18 𝐸. 𝐸. PROPN iajs-2928 294 19 proposition proposition NOUN iajs-2928 294 20 4.1 4.1 NUM iajs-2928 294 21 . . PUNCT iajs-2928 295 1 let let VERB iajs-2928 295 2 𝑓 𝑓 PRON iajs-2928 295 3 is be AUX iajs-2928 295 4 pseudo pseudo NOUN iajs-2928 295 5 semi semi ADJ iajs-2928 295 6 ( ( PUNCT iajs-2928 295 7 𝑝 𝑝 NOUN iajs-2928 295 8 , , PUNCT iajs-2928 295 9 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 295 10 function function NOUN iajs-2928 295 11 . . PUNCT iajs-2928 296 1 then then ADV iajs-2928 296 2 , , PUNCT iajs-2928 296 3 every every DET iajs-2928 296 4 local local ADJ iajs-2928 296 5 minimum minimum NOUN iajs-2928 296 6 π‘₯βˆ— π‘₯βˆ— NOUN iajs-2928 296 7 = = SYM iajs-2928 296 8 𝐸(π‘₯βˆ— 𝐸(π‘₯βˆ— PROPN iajs-2928 296 9 ) ) PUNCT iajs-2928 296 10 ∈ ∈ PROPN iajs-2928 296 11 𝐸(𝐴 𝐸(𝐴 NOUN iajs-2928 296 12 ) ) PUNCT iajs-2928 296 13 of of ADP iajs-2928 296 14 problem problem NOUN iajs-2928 296 15 ( ( PUNCT iajs-2928 296 16 𝑁𝐿𝑃 𝑁𝐿𝑃 PROPN iajs-2928 296 17 ) ) PUNCT iajs-2928 296 18 is be AUX iajs-2928 296 19 a a DET iajs-2928 296 20 global global ADJ iajs-2928 296 21 minimum minimum NOUN iajs-2928 296 22 . . PUNCT iajs-2928 297 1 proof proof NOUN iajs-2928 297 2 . . PUNCT iajs-2928 298 1 suppose suppose VERB iajs-2928 298 2 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2928 298 3 = = SYM iajs-2928 298 4 𝐸(π‘₯βˆ— 𝐸(π‘₯βˆ— PROPN iajs-2928 298 5 ) ) PUNCT iajs-2928 298 6 is be AUX iajs-2928 298 7 not not PART iajs-2928 298 8 global global ADJ iajs-2928 298 9 minimum minimum NOUN iajs-2928 298 10 , , PUNCT iajs-2928 298 11 then then ADV iajs-2928 298 12 there there PRON iajs-2928 298 13 exists exist VERB iajs-2928 298 14 𝑒 𝑒 PROPN iajs-2928 298 15 ∈ ∈ PROPN iajs-2928 298 16 𝐴 𝐴 PROPN iajs-2928 298 17 with with ADP iajs-2928 298 18 𝑓(𝑒 𝑓(𝑒 PROPN iajs-2928 298 19 ) ) PUNCT iajs-2928 298 20 < < X iajs-2928 298 21 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— X iajs-2928 298 22 ) ) PUNCT iajs-2928 298 23 = = SYM iajs-2928 298 24 𝑓(𝐸π‘₯βˆ— 𝑓(𝐸π‘₯βˆ— NOUN iajs-2928 298 25 ) ) PUNCT iajs-2928 298 26 . . PUNCT iajs-2928 299 1 from from ADP iajs-2928 299 2 the the DET iajs-2928 299 3 assumptions assumption NOUN iajs-2928 299 4 on on ADP iajs-2928 299 5 𝑓 𝑓 PRON iajs-2928 299 6 , , PUNCT iajs-2928 299 7 we we PRON iajs-2928 299 8 have have VERB iajs-2928 299 9 for for ADP iajs-2928 299 10 all all DET iajs-2928 299 11 π‘Ÿ π‘Ÿ NOUN iajs-2928 299 12 , , PUNCT iajs-2928 299 13 𝑠 𝑠 PROPN iajs-2928 299 14 ∈ ∈ PROPN iajs-2928 300 1 [ [ X iajs-2928 300 2 0,1 0,1 NUM iajs-2928 300 3 ] ] PUNCT iajs-2928 300 4 with with ADP iajs-2928 300 5 π‘Ÿπ‘ π‘Ÿπ‘ PRON iajs-2928 301 1 + + NOUN iajs-2928 301 2 𝑠𝑝 𝑠𝑝 NOUN iajs-2928 301 3 = = SYM iajs-2928 301 4 1 1 NUM iajs-2928 302 1 ( ( PUNCT iajs-2928 302 2 𝑖. 𝑖. ADJ iajs-2928 302 3 𝑒. 𝑒. ADJ iajs-2928 302 4 , , PUNCT iajs-2928 302 5 𝑠 𝑠 PROPN iajs-2928 302 6 = = PUNCT iajs-2928 302 7 ( ( PUNCT iajs-2928 302 8 1 1 NUM iajs-2928 302 9 βˆ’ βˆ’ PROPN iajs-2928 302 10 π‘Ÿπ‘ π‘Ÿπ‘ NOUN iajs-2928 302 11 ) ) PUNCT iajs-2928 302 12 1 1 NUM iajs-2928 302 13 𝑝 𝑝 NOUN iajs-2928 302 14 ) ) PUNCT iajs-2928 302 15 , , PUNCT iajs-2928 302 16 if if SCONJ iajs-2928 302 17 𝑓(𝑒 𝑓(𝑒 ADV iajs-2928 302 18 ) ) PUNCT iajs-2928 302 19 < < X iajs-2928 302 20 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— PROPN iajs-2928 302 21 ) ) PUNCT iajs-2928 302 22 . . PUNCT iajs-2928 303 1 then then ADV iajs-2928 303 2 , , PUNCT iajs-2928 303 3 we we PRON iajs-2928 303 4 have have VERB iajs-2928 303 5 𝑓(π‘ŸπΈπ‘’ 𝑓(π‘ŸπΈπ‘’ NOUN iajs-2928 303 6 + + CCONJ iajs-2928 303 7 𝑠𝐸π‘₯βˆ— 𝑠𝐸π‘₯βˆ— NOUN iajs-2928 303 8 ) ) PUNCT iajs-2928 303 9 ≀ ≀ NOUN iajs-2928 303 10 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 303 11 ) ) PUNCT iajs-2928 304 1 + + CCONJ iajs-2928 304 2 ( ( PUNCT iajs-2928 304 3 βˆ’π‘Ÿπ‘ )𝑏(π‘₯βˆ— βˆ’π‘Ÿπ‘ )𝑏(π‘₯βˆ— PROPN iajs-2928 304 4 , , PUNCT iajs-2928 304 5 𝑒 𝑒 NOUN iajs-2928 304 6 ) ) PUNCT iajs-2928 304 7 . . PUNCT iajs-2928 305 1 now now ADV iajs-2928 305 2 , , PUNCT iajs-2928 305 3 if if SCONJ iajs-2928 305 4 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 305 5 ) ) PUNCT iajs-2928 305 6 ≀ ≀ NOUN iajs-2928 305 7 0 0 NUM iajs-2928 305 8 or or CCONJ iajs-2928 305 9 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 305 10 ) ) PUNCT iajs-2928 305 11 β‰₯ β‰₯ NOUN iajs-2928 305 12 0 0 NUM iajs-2928 305 13 . . PUNCT iajs-2928 306 1 then then ADV iajs-2928 306 2 we we PRON iajs-2928 306 3 have have VERB iajs-2928 306 4 𝑓(π‘ŸπΈπ‘’ 𝑓(π‘ŸπΈπ‘’ NOUN iajs-2928 306 5 + + CCONJ iajs-2928 306 6 𝑠𝐸π‘₯βˆ— 𝑠𝐸π‘₯βˆ— NOUN iajs-2928 306 7 ) ) PUNCT iajs-2928 306 8 ≀ ≀ NOUN iajs-2928 306 9 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 306 10 ) ) PUNCT iajs-2928 307 1 + + CCONJ iajs-2928 307 2 ( ( PUNCT iajs-2928 307 3 βˆ’π‘Ÿπ‘ )𝑏(π‘₯βˆ— βˆ’π‘Ÿπ‘ )𝑏(π‘₯βˆ— PROPN iajs-2928 307 4 , , PUNCT iajs-2928 307 5 𝑒 𝑒 NOUN iajs-2928 307 6 ) ) PUNCT iajs-2928 307 7 ≀ ≀ NOUN iajs-2928 307 8 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 307 9 ) ) PUNCT iajs-2928 307 10 . . PUNCT iajs-2928 308 1 ( ( PUNCT iajs-2928 308 2 9 9 X iajs-2928 308 3 ) ) PUNCT iajs-2928 308 4 now now ADV iajs-2928 308 5 , , PUNCT iajs-2928 308 6 for for ADP iajs-2928 308 7 sufficiently sufficiently ADV iajs-2928 308 8 small small ADJ iajs-2928 308 9 , , PUNCT iajs-2928 308 10 π‘Ÿ π‘Ÿ X iajs-2928 308 11 ∈ ∈ PROPN iajs-2928 308 12 ( ( PUNCT iajs-2928 308 13 0,1 0,1 NOUN iajs-2928 308 14 ] ] PUNCT iajs-2928 308 15 ) ) PUNCT iajs-2928 308 16 then then ADV iajs-2928 308 17 π‘ŸπΈπ‘’ π‘ŸπΈπ‘’ NUM iajs-2928 308 18 + + CCONJ iajs-2928 308 19 ( ( PUNCT iajs-2928 308 20 1 1 NUM iajs-2928 308 21 βˆ’ βˆ’ PROPN iajs-2928 308 22 π‘Ÿπ‘ π‘Ÿπ‘ NOUN iajs-2928 308 23 ) ) PUNCT iajs-2928 308 24 1 1 NUM iajs-2928 308 25 𝑝π‘₯βˆ— 𝑝π‘₯βˆ— NOUN iajs-2928 308 26 will will AUX iajs-2928 308 27 be be AUX iajs-2928 308 28 close close ADJ iajs-2928 308 29 enough enough ADV iajs-2928 308 30 to to PART iajs-2928 308 31 π‘₯βˆ—. π‘₯βˆ—. VERB iajs-2928 308 32 i.e. i.e. X iajs-2928 308 33 , , PUNCT iajs-2928 308 34 there there PRON iajs-2928 308 35 exists exist VERB iajs-2928 308 36 𝛿 𝛿 PROPN iajs-2928 308 37 > > X iajs-2928 308 38 0 0 NUM iajs-2928 308 39 such such ADJ iajs-2928 308 40 that that SCONJ iajs-2928 308 41 π‘ŸπΈπ‘’ π‘ŸπΈπ‘’ NUM iajs-2928 308 42 + + CCONJ iajs-2928 308 43 ( ( PUNCT iajs-2928 308 44 1 1 NUM iajs-2928 308 45 βˆ’ βˆ’ PROPN iajs-2928 308 46 π‘Ÿπ‘ π‘Ÿπ‘ NOUN iajs-2928 308 47 ) ) PUNCT iajs-2928 308 48 1 1 NUM iajs-2928 308 49 𝑝π‘₯βˆ— 𝑝π‘₯βˆ— NOUN iajs-2928 308 50 ∈ ∈ PROPN iajs-2928 308 51 𝐡 𝐡 PROPN iajs-2928 308 52 𝛿(π‘₯βˆ— 𝛿(π‘₯βˆ— NOUN iajs-2928 308 53 ) ) PUNCT iajs-2928 308 54 ∩ ∩ NOUN iajs-2928 308 55 𝐴. 𝐴. PROPN iajs-2928 308 56 from from ADP iajs-2928 308 57 the the DET iajs-2928 308 58 local local ADJ iajs-2928 308 59 minimality minimality NOUN iajs-2928 308 60 of of ADP iajs-2928 308 61 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2928 308 62 , , PUNCT iajs-2928 308 63 one one PRON iajs-2928 308 64 obtains obtain VERB iajs-2928 308 65 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NOUN iajs-2928 308 66 ) ) PUNCT iajs-2928 308 67 ≀ ≀ NOUN iajs-2928 308 68 𝑓(π‘ŸπΈπ‘’ 𝑓(π‘ŸπΈπ‘’ PROPN iajs-2928 309 1 + + CCONJ iajs-2928 309 2 ( ( PUNCT iajs-2928 309 3 1 1 NUM iajs-2928 309 4 βˆ’ βˆ’ PROPN iajs-2928 309 5 π‘Ÿπ‘ π‘Ÿπ‘ NOUN iajs-2928 309 6 ) ) PUNCT iajs-2928 309 7 1 1 NUM iajs-2928 309 8 𝑝π‘₯βˆ— 𝑝π‘₯βˆ— NOUN iajs-2928 309 9 ) ) PUNCT iajs-2928 309 10 which which PRON iajs-2928 309 11 contradicts contradict VERB iajs-2928 309 12 ( ( PUNCT iajs-2928 309 13 9 9 NUM iajs-2928 309 14 ) ) PUNCT iajs-2928 309 15 . . PUNCT iajs-2928 310 1 thus thus ADV iajs-2928 310 2 , , PUNCT iajs-2928 310 3 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2928 310 4 is be AUX iajs-2928 310 5 a a DET iajs-2928 310 6 global global ADJ iajs-2928 310 7 minimum minimum NOUN iajs-2928 310 8 . . PUNCT iajs-2928 311 1 β–  β–  PUNCT iajs-2928 311 2 proposition proposition NOUN iajs-2928 311 3 4.2 4.2 NUM iajs-2928 311 4 . . PUNCT iajs-2928 312 1 let let VERB iajs-2928 312 2 𝑓 𝑓 PRON iajs-2928 312 3 is be AUX iajs-2928 312 4 quasi quasi NOUN iajs-2928 312 5 semi semi ADJ iajs-2928 312 6 ( ( PUNCT iajs-2928 312 7 𝑝 𝑝 NOUN iajs-2928 312 8 , , PUNCT iajs-2928 312 9 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 312 10 function function NOUN iajs-2928 312 11 . . PUNCT iajs-2928 313 1 then then ADV iajs-2928 313 2 every every DET iajs-2928 313 3 local local ADJ iajs-2928 313 4 minimum minimum NOUN iajs-2928 313 5 π‘₯βˆ— π‘₯βˆ— NOUN iajs-2928 313 6 = = SYM iajs-2928 313 7 𝐸(π‘₯βˆ— 𝐸(π‘₯βˆ— PROPN iajs-2928 313 8 ) ) PUNCT iajs-2928 313 9 ∈ ∈ PROPN iajs-2928 313 10 𝐸(𝐴 𝐸(𝐴 NOUN iajs-2928 313 11 ) ) PUNCT iajs-2928 313 12 of of ADP iajs-2928 313 13 problem problem NOUN iajs-2928 313 14 ( ( PUNCT iajs-2928 313 15 𝑁𝐿𝑃 𝑁𝐿𝑃 PROPN iajs-2928 313 16 ) ) PUNCT iajs-2928 313 17 is be AUX iajs-2928 313 18 a a DET iajs-2928 313 19 global global ADJ iajs-2928 313 20 minimum minimum NOUN iajs-2928 313 21 . . PUNCT iajs-2928 314 1 ihjpas ihjpas PROPN iajs-2928 314 2 . . PUNCT iajs-2928 315 1 36(1)2023 36(1)2023 NUM iajs-2928 315 2 364 364 NUM iajs-2928 315 3 proof proof NOUN iajs-2928 315 4 . . PUNCT iajs-2928 316 1 suppose suppose VERB iajs-2928 316 2 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2928 316 3 = = SYM iajs-2928 316 4 𝐸(π‘₯βˆ— 𝐸(π‘₯βˆ— PROPN iajs-2928 316 5 ) ) PUNCT iajs-2928 316 6 is be AUX iajs-2928 316 7 not not PART iajs-2928 316 8 global global ADJ iajs-2928 316 9 minimum minimum NOUN iajs-2928 316 10 , , PUNCT iajs-2928 316 11 then then ADV iajs-2928 316 12 there there PRON iajs-2928 316 13 exists exist VERB iajs-2928 316 14 𝑒 𝑒 PROPN iajs-2928 316 15 ∈ ∈ PROPN iajs-2928 316 16 𝐴 𝐴 PROPN iajs-2928 316 17 with with ADP iajs-2928 316 18 𝑓(𝑒 𝑓(𝑒 PROPN iajs-2928 316 19 ) ) PUNCT iajs-2928 316 20 < < X iajs-2928 316 21 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— X iajs-2928 316 22 ) ) PUNCT iajs-2928 316 23 = = SYM iajs-2928 316 24 𝑓(𝐸π‘₯βˆ— 𝑓(𝐸π‘₯βˆ— NOUN iajs-2928 316 25 ) ) PUNCT iajs-2928 316 26 . . PUNCT iajs-2928 317 1 from from ADP iajs-2928 317 2 the the DET iajs-2928 317 3 assumptions assumption NOUN iajs-2928 317 4 on on ADP iajs-2928 317 5 𝑓 𝑓 PRON iajs-2928 317 6 , , PUNCT iajs-2928 317 7 we we PRON iajs-2928 317 8 have have VERB iajs-2928 317 9 for for ADP iajs-2928 317 10 all all DET iajs-2928 317 11 π‘Ÿ π‘Ÿ NOUN iajs-2928 317 12 , , PUNCT iajs-2928 317 13 𝑠 𝑠 PROPN iajs-2928 317 14 ∈ ∈ PROPN iajs-2928 318 1 [ [ X iajs-2928 318 2 0,1 0,1 NUM iajs-2928 318 3 ] ] PUNCT iajs-2928 318 4 with with ADP iajs-2928 318 5 π‘Ÿπ‘ π‘Ÿπ‘ PRON iajs-2928 319 1 + + NOUN iajs-2928 319 2 𝑠𝑝 𝑠𝑝 NOUN iajs-2928 319 3 = = SYM iajs-2928 319 4 1 1 NUM iajs-2928 319 5 ( ( PUNCT iajs-2928 319 6 𝑖. 𝑖. ADJ iajs-2928 319 7 𝑒. 𝑒. ADJ iajs-2928 319 8 , , PUNCT iajs-2928 319 9 𝑠 𝑠 PROPN iajs-2928 319 10 = = PUNCT iajs-2928 319 11 ( ( PUNCT iajs-2928 319 12 1 1 NUM iajs-2928 319 13 βˆ’ βˆ’ PROPN iajs-2928 319 14 π‘Ÿπ‘ π‘Ÿπ‘ NOUN iajs-2928 319 15 ) ) PUNCT iajs-2928 319 16 1 1 NUM iajs-2928 319 17 𝑝 𝑝 NOUN iajs-2928 319 18 ) ) PUNCT iajs-2928 319 19 . . PUNCT iajs-2928 320 1 since since SCONJ iajs-2928 320 2 𝑓(𝑒 𝑓(𝑒 PROPN iajs-2928 320 3 ) ) PUNCT iajs-2928 320 4 < < X iajs-2928 320 5 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— X iajs-2928 320 6 ) ) PUNCT iajs-2928 320 7 and and CCONJ iajs-2928 320 8 𝑓 𝑓 PRON iajs-2928 320 9 is be AUX iajs-2928 320 10 quasi quasi NOUN iajs-2928 320 11 semi semi ADJ iajs-2928 320 12 ( ( PUNCT iajs-2928 320 13 𝑝 𝑝 NUM iajs-2928 320 14 , , PUNCT iajs-2928 320 15 𝐸)-convex 𝐸)-convex PUNCT iajs-2928 320 16 then then ADV iajs-2928 320 17 we we PRON iajs-2928 320 18 have have VERB iajs-2928 320 19 𝑓(π‘ŸπΈπ‘’ 𝑓(π‘ŸπΈπ‘’ NOUN iajs-2928 320 20 + + CCONJ iajs-2928 320 21 𝑠𝐸π‘₯βˆ— 𝑠𝐸π‘₯βˆ— NOUN iajs-2928 320 22 ) ) PUNCT iajs-2928 320 23 ≀ ≀ NOUN iajs-2928 320 24 π‘šπ‘Žπ‘₯{𝑓(𝑒 π‘šπ‘Žπ‘₯{𝑓(𝑒 ADV iajs-2928 320 25 ) ) PUNCT iajs-2928 320 26 , , PUNCT iajs-2928 320 27 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 320 28 ) ) PUNCT iajs-2928 320 29 } } PUNCT iajs-2928 320 30 = = SYM iajs-2928 320 31 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— X iajs-2928 320 32 ) ) PUNCT iajs-2928 320 33 ( ( PUNCT iajs-2928 320 34 10 10 NUM iajs-2928 320 35 ) ) PUNCT iajs-2928 320 36 now now ADV iajs-2928 320 37 , , PUNCT iajs-2928 320 38 for for ADP iajs-2928 320 39 sufficiently sufficiently ADV iajs-2928 320 40 small small ADJ iajs-2928 320 41 , , PUNCT iajs-2928 320 42 π‘Ÿ π‘Ÿ X iajs-2928 320 43 ∈ ∈ PROPN iajs-2928 320 44 ( ( PUNCT iajs-2928 320 45 0,1 0,1 NOUN iajs-2928 320 46 ] ] PUNCT iajs-2928 320 47 ) ) PUNCT iajs-2928 320 48 then then ADV iajs-2928 320 49 π‘ŸπΈπ‘’ π‘ŸπΈπ‘’ NUM iajs-2928 320 50 + + CCONJ iajs-2928 320 51 ( ( PUNCT iajs-2928 320 52 1 1 NUM iajs-2928 320 53 βˆ’ βˆ’ PROPN iajs-2928 320 54 π‘Ÿπ‘ π‘Ÿπ‘ NOUN iajs-2928 320 55 ) ) PUNCT iajs-2928 320 56 1 1 NUM iajs-2928 320 57 𝑝π‘₯βˆ— 𝑝π‘₯βˆ— NOUN iajs-2928 320 58 will will AUX iajs-2928 320 59 be be AUX iajs-2928 320 60 close close ADJ iajs-2928 320 61 enough enough ADV iajs-2928 320 62 to to PART iajs-2928 320 63 π‘₯βˆ—. π‘₯βˆ—. VERB iajs-2928 320 64 i.e. i.e. X iajs-2928 320 65 , , PUNCT iajs-2928 320 66 there there PRON iajs-2928 320 67 exists exist VERB iajs-2928 320 68 𝛿 𝛿 PROPN iajs-2928 320 69 > > X iajs-2928 320 70 0 0 NUM iajs-2928 320 71 such such ADJ iajs-2928 320 72 that that DET iajs-2928 320 73 π‘Ÿνœ€π‘’ π‘Ÿνœ€π‘’ NOUN iajs-2928 320 74 + + CCONJ iajs-2928 320 75 ( ( PUNCT iajs-2928 320 76 1 1 NUM iajs-2928 320 77 βˆ’ βˆ’ PROPN iajs-2928 320 78 π‘Ÿπ‘ π‘Ÿπ‘ NOUN iajs-2928 320 79 ) ) PUNCT iajs-2928 320 80 1 1 NUM iajs-2928 320 81 𝑝π‘₯βˆ— 𝑝π‘₯βˆ— NOUN iajs-2928 320 82 ∈ ∈ PROPN iajs-2928 320 83 𝐡 𝐡 PROPN iajs-2928 320 84 𝛿(π‘₯βˆ— 𝛿(π‘₯βˆ— NOUN iajs-2928 320 85 ) ) PUNCT iajs-2928 320 86 ∩ ∩ NOUN iajs-2928 320 87 𝐴. 𝐴. PROPN iajs-2928 320 88 from from ADP iajs-2928 320 89 the the DET iajs-2928 320 90 local local ADJ iajs-2928 320 91 minimality minimality NOUN iajs-2928 320 92 of of ADP iajs-2928 320 93 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2928 320 94 , , PUNCT iajs-2928 320 95 we we PRON iajs-2928 320 96 have have VERB iajs-2928 320 97 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 320 98 ) ) PUNCT iajs-2928 320 99 ≀ ≀ NOUN iajs-2928 320 100 𝑓(π‘ŸπΈπ‘’ 𝑓(π‘ŸπΈπ‘’ PROPN iajs-2928 321 1 + + CCONJ iajs-2928 321 2 ( ( PUNCT iajs-2928 321 3 1 1 NUM iajs-2928 321 4 βˆ’ βˆ’ PROPN iajs-2928 321 5 π‘Ÿπ‘ π‘Ÿπ‘ NOUN iajs-2928 321 6 ) ) PUNCT iajs-2928 321 7 1 1 NUM iajs-2928 321 8 𝑝π‘₯βˆ— 𝑝π‘₯βˆ— NOUN iajs-2928 321 9 ) ) PUNCT iajs-2928 321 10 which which PRON iajs-2928 321 11 contradicts contradict VERB iajs-2928 321 12 ( ( PUNCT iajs-2928 321 13 10 10 NUM iajs-2928 321 14 ) ) PUNCT iajs-2928 321 15 . . PUNCT iajs-2928 322 1 thus thus ADV iajs-2928 322 2 , , PUNCT iajs-2928 322 3 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2928 322 4 is be AUX iajs-2928 322 5 a a DET iajs-2928 322 6 global global ADJ iajs-2928 322 7 minimum minimum NOUN iajs-2928 322 8 . . PUNCT iajs-2928 323 1 β–  β–  PUNCT iajs-2928 323 2 remark remark VERB iajs-2928 323 3 4.3 4.3 NUM iajs-2928 323 4 . . PUNCT iajs-2928 324 1 the the DET iajs-2928 324 2 conclusions conclusion NOUN iajs-2928 324 3 of of ADP iajs-2928 324 4 propositions proposition NOUN iajs-2928 324 5 4.1 4.1 NUM iajs-2928 324 6 and and CCONJ iajs-2928 324 7 4.2 4.2 NUM iajs-2928 324 8 do do AUX iajs-2928 324 9 not not PART iajs-2928 324 10 hold hold VERB iajs-2928 324 11 if if SCONJ iajs-2928 324 12 the the DET iajs-2928 324 13 objective objective ADJ iajs-2928 324 14 function function NOUN iajs-2928 324 15 𝑓 𝑓 PRON iajs-2928 324 16 is be AUX iajs-2928 324 17 not not PART iajs-2928 324 18 quasi quasi NOUN iajs-2928 324 19 semi semi ADJ iajs-2928 324 20 ( ( PUNCT iajs-2928 324 21 𝑝 𝑝 NOUN iajs-2928 324 22 , , PUNCT iajs-2928 324 23 𝐸)-convex 𝐸)-convex PROPN iajs-2928 324 24 ( ( PUNCT iajs-2928 324 25 respectively respectively ADV iajs-2928 324 26 , , PUNCT iajs-2928 324 27 not not PART iajs-2928 324 28 pseudo pseudo NOUN iajs-2928 324 29 semi semi ADJ iajs-2928 324 30 ( ( PUNCT iajs-2928 324 31 𝑝 𝑝 NOUN iajs-2928 324 32 , , PUNCT iajs-2928 324 33 𝐸)-convex 𝐸)-convex NUM iajs-2928 324 34 ) ) PUNCT iajs-2928 324 35 function function VERB iajs-2928 324 36 as as SCONJ iajs-2928 324 37 the the DET iajs-2928 324 38 following follow VERB iajs-2928 324 39 example example NOUN iajs-2928 324 40 confirms confirm VERB iajs-2928 324 41 . . PUNCT iajs-2928 325 1 example example NOUN iajs-2928 325 2 4.4 4.4 NUM iajs-2928 325 3 . . PUNCT iajs-2928 326 1 consider consider VERB iajs-2928 326 2 the the DET iajs-2928 326 3 optimization optimization NOUN iajs-2928 326 4 problem problem NOUN iajs-2928 326 5 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 326 6 , , PUNCT iajs-2928 326 7 𝑦 𝑦 NOUN iajs-2928 326 8 ) ) PUNCT iajs-2928 326 9 such such ADJ iajs-2928 326 10 that that SCONJ iajs-2928 326 11 ( ( PUNCT iajs-2928 326 12 π‘₯ π‘₯ NOUN iajs-2928 326 13 , , PUNCT iajs-2928 326 14 𝑦 𝑦 NOUN iajs-2928 326 15 ) ) PUNCT iajs-2928 326 16 ∈ ∈ PROPN iajs-2928 326 17 𝐴 𝐴 PROPN iajs-2928 326 18 , , PUNCT iajs-2928 326 19 where where SCONJ iajs-2928 326 20 𝐴 𝐴 PROPN iajs-2928 326 21 = = SYM iajs-2928 326 22 { { PUNCT iajs-2928 326 23 ( ( PUNCT iajs-2928 326 24 π‘₯ π‘₯ PROPN iajs-2928 326 25 , , PUNCT iajs-2928 326 26 𝑦 𝑦 NOUN iajs-2928 326 27 ) ) PUNCT iajs-2928 326 28 ∈ ∈ NOUN iajs-2928 326 29 𝑅2 𝑅2 VERB iajs-2928 327 1 ∢ ∢ NOUN iajs-2928 327 2 |π‘₯| |π‘₯| ADJ iajs-2928 327 3 1 1 NUM iajs-2928 327 4 2 2 NUM iajs-2928 327 5 + + CCONJ iajs-2928 327 6 |𝑦| |𝑦| NOUN iajs-2928 327 7 1 1 NUM iajs-2928 327 8 2 2 NUM iajs-2928 327 9 ≀ ≀ NUM iajs-2928 327 10 4 4 NUM iajs-2928 327 11 } } PUNCT iajs-2928 327 12 and and CCONJ iajs-2928 327 13 𝑓 𝑓 PRON iajs-2928 327 14 : : PUNCT iajs-2928 327 15 𝑅2 𝑅2 ADP iajs-2928 327 16 β†’ β†’ SYM iajs-2928 327 17 𝑅 𝑅 NOUN iajs-2928 327 18 such such ADJ iajs-2928 327 19 that that SCONJ iajs-2928 327 20 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 327 21 , , PUNCT iajs-2928 327 22 𝑦 𝑦 NOUN iajs-2928 327 23 ) ) PUNCT iajs-2928 327 24 = = SYM iajs-2928 327 25 { { PUNCT iajs-2928 327 26 ( ( PUNCT iajs-2928 327 27 𝑦 𝑦 NOUN iajs-2928 327 28 βˆ’ βˆ’ PROPN iajs-2928 327 29 1)2 1)2 NUM iajs-2928 327 30 βˆ’ βˆ’ NOUN iajs-2928 327 31 2 2 NUM iajs-2928 327 32 ≀ ≀ NUM iajs-2928 327 33 π‘₯ π‘₯ DET iajs-2928 327 34 ≀ ≀ NUM iajs-2928 327 35 2 2 NUM iajs-2928 327 36 ( ( PUNCT iajs-2928 327 37 𝑦 𝑦 NOUN iajs-2928 327 38 βˆ’ βˆ’ PROPN iajs-2928 327 39 1)2 1)2 NUM iajs-2928 327 40 βˆ’ βˆ’ PROPN iajs-2928 327 41 ( ( PUNCT iajs-2928 327 42 2 2 NUM iajs-2928 327 43 βˆ’ βˆ’ PROPN iajs-2928 327 44 π‘₯)2 π‘₯)2 PROPN iajs-2928 327 45 o. o. PROPN iajs-2928 327 46 w. w. PROPN iajs-2928 327 47 define define VERB iajs-2928 327 48 𝐸 𝐸 PROPN iajs-2928 327 49 : : PUNCT iajs-2928 327 50 𝑅2 𝑅2 ADP iajs-2928 327 51 β†’ β†’ SYM iajs-2928 327 52 𝑅2 𝑅2 NOUN iajs-2928 327 53 as as ADP iajs-2928 327 54 𝐸(π‘₯. 𝐸(π‘₯. PROPN iajs-2928 327 55 𝑦 𝑦 NOUN iajs-2928 327 56 ) ) PUNCT iajs-2928 327 57 = = SYM iajs-2928 327 58 ( ( PUNCT iajs-2928 327 59 0 0 NUM iajs-2928 327 60 , , PUNCT iajs-2928 327 61 𝑦 𝑦 NOUN iajs-2928 327 62 ) ) PUNCT iajs-2928 327 63 . . PUNCT iajs-2928 328 1 then then ADV iajs-2928 328 2 𝐴 𝐴 PROPN iajs-2928 328 3 is be AUX iajs-2928 328 4 ( ( PUNCT iajs-2928 328 5 1 1 NUM iajs-2928 328 6 2 2 NUM iajs-2928 328 7 , , PUNCT iajs-2928 328 8 𝐸)-convex 𝐸)-convex PRON iajs-2928 328 9 set set VERB iajs-2928 328 10 , , PUNCT iajs-2928 328 11 𝑓 𝑓 PRON iajs-2928 328 12 is be AUX iajs-2928 328 13 not not PART iajs-2928 328 14 quasi quasi NOUN iajs-2928 328 15 semi semi ADV iajs-2928 328 16 ( ( PUNCT iajs-2928 328 17 1 1 NUM iajs-2928 328 18 2 2 NUM iajs-2928 328 19 , , PUNCT iajs-2928 328 20 𝐸 𝐸 PROPN iajs-2928 328 21 ) ) PUNCT iajs-2928 328 22 convex convex NOUN iajs-2928 328 23 and and CCONJ iajs-2928 328 24 not not PART iajs-2928 328 25 pseudo pseudo NOUN iajs-2928 328 26 semi semi ADV iajs-2928 328 27 ( ( PUNCT iajs-2928 328 28 1 1 NUM iajs-2928 328 29 2 2 NUM iajs-2928 328 30 , , PUNCT iajs-2928 328 31 𝐸)convex 𝐸)convex NOUN iajs-2928 328 32 on on ADP iajs-2928 328 33 𝐴. 𝐴. PROPN iajs-2928 328 34 to to PART iajs-2928 328 35 show show VERB iajs-2928 328 36 𝐴 𝐴 PROPN iajs-2928 328 37 is be AUX iajs-2928 328 38 ( ( PUNCT iajs-2928 328 39 1 1 NUM iajs-2928 328 40 2 2 NUM iajs-2928 328 41 , , PUNCT iajs-2928 328 42 𝐸)convex 𝐸)convex PROPN iajs-2928 328 43 set set PROPN iajs-2928 328 44 , , PUNCT iajs-2928 328 45 let let VERB iajs-2928 328 46 ( ( PUNCT iajs-2928 328 47 π‘₯1 π‘₯1 NOUN iajs-2928 328 48 , , PUNCT iajs-2928 328 49 𝑦1),(π‘₯2 𝑦1),(π‘₯2 PROPN iajs-2928 328 50 , , PUNCT iajs-2928 328 51 𝑦2 𝑦2 PROPN iajs-2928 328 52 ) ) PUNCT iajs-2928 329 1 ∈ ∈ PROPN iajs-2928 329 2 𝐴. 𝐴. PROPN iajs-2928 329 3 then then ADV iajs-2928 329 4 , , PUNCT iajs-2928 329 5 |π‘₯1| |π‘₯1| NOUN iajs-2928 329 6 1 1 NUM iajs-2928 329 7 2 2 NUM iajs-2928 329 8 + + NUM iajs-2928 329 9 |𝑦1| |𝑦1| NOUN iajs-2928 329 10 1 1 NUM iajs-2928 329 11 2 2 NUM iajs-2928 329 12 ≀ ≀ NUM iajs-2928 329 13 4 4 NUM iajs-2928 329 14 and and CCONJ iajs-2928 329 15 |π‘₯2| |π‘₯2| NOUN iajs-2928 329 16 1 1 NUM iajs-2928 329 17 2 2 NUM iajs-2928 329 18 + + CCONJ iajs-2928 329 19 |𝑦2| |𝑦2| ADJ iajs-2928 329 20 1 1 NUM iajs-2928 329 21 2 2 NUM iajs-2928 329 22 ≀ ≀ NUM iajs-2928 329 23 4 4 NUM iajs-2928 329 24 . . PUNCT iajs-2928 330 1 now now ADV iajs-2928 330 2 , , PUNCT iajs-2928 330 3 π‘ŸπΈ(π‘₯1 π‘ŸπΈ(π‘₯1 PROPN iajs-2928 330 4 , , PUNCT iajs-2928 330 5 𝑦1 𝑦1 PROPN iajs-2928 330 6 ) ) PUNCT iajs-2928 331 1 + + CCONJ iajs-2928 331 2 𝑠𝐸(π‘₯2 𝑠𝐸(π‘₯2 PROPN iajs-2928 331 3 , , PUNCT iajs-2928 331 4 𝑦2 𝑦2 PROPN iajs-2928 331 5 ) ) PUNCT iajs-2928 331 6 = = PUNCT iajs-2928 331 7 ( ( PUNCT iajs-2928 331 8 0 0 NUM iajs-2928 331 9 , , PUNCT iajs-2928 331 10 π‘Ÿπ‘¦2 π‘Ÿπ‘¦2 NOUN iajs-2928 331 11 + + CCONJ iajs-2928 331 12 𝑠𝑦2 𝑠𝑦2 PROPN iajs-2928 331 13 ) ) PUNCT iajs-2928 331 14 . . PUNCT iajs-2928 332 1 note note VERB iajs-2928 332 2 that that SCONJ iajs-2928 332 3 |0| |0| NOUN iajs-2928 332 4 1 1 NUM iajs-2928 332 5 2 2 NUM iajs-2928 332 6 + + CCONJ iajs-2928 332 7 |π‘Ÿπ‘¦1 |π‘Ÿπ‘¦1 ADJ iajs-2928 333 1 + + NOUN iajs-2928 333 2 𝑠𝑦2| 𝑠𝑦2| NOUN iajs-2928 333 3 1 1 NUM iajs-2928 333 4 2 2 NUM iajs-2928 333 5 ≀ ≀ NOUN iajs-2928 333 6 π‘Ÿ|𝑦1| π‘Ÿ|𝑦1| ADV iajs-2928 333 7 1 1 NUM iajs-2928 333 8 2 2 NUM iajs-2928 333 9 + + CCONJ iajs-2928 333 10 s|𝑦2| s|𝑦2| ADV iajs-2928 333 11 1 1 NUM iajs-2928 333 12 2 2 NUM iajs-2928 333 13 ≀ ≀ NOUN iajs-2928 333 14 4(π‘Ÿ 4(π‘Ÿ NUM iajs-2928 333 15 + + CCONJ iajs-2928 333 16 𝑠 𝑠 X iajs-2928 333 17 ) ) PUNCT iajs-2928 333 18 ≀ ≀ NOUN iajs-2928 333 19 4 4 NUM iajs-2928 333 20 . . PUNCT iajs-2928 333 21 hence hence ADV iajs-2928 333 22 , , PUNCT iajs-2928 333 23 𝐴 𝐴 PROPN iajs-2928 333 24 is be AUX iajs-2928 333 25 ( ( PUNCT iajs-2928 333 26 1 1 NUM iajs-2928 333 27 2 2 NUM iajs-2928 333 28 , , PUNCT iajs-2928 333 29 𝐸)-convex 𝐸)-convex PRON iajs-2928 333 30 set set VERB iajs-2928 333 31 . . PUNCT iajs-2928 334 1 next next ADV iajs-2928 334 2 , , PUNCT iajs-2928 334 3 we we PRON iajs-2928 334 4 show show VERB iajs-2928 334 5 that that SCONJ iajs-2928 334 6 𝑓 𝑓 PRON iajs-2928 334 7 is be AUX iajs-2928 334 8 not not PART iajs-2928 334 9 quasi quasi NOUN iajs-2928 334 10 semi semi ADV iajs-2928 334 11 ( ( PUNCT iajs-2928 334 12 1 1 NUM iajs-2928 334 13 2 2 NUM iajs-2928 334 14 , , PUNCT iajs-2928 334 15 𝐸)convex 𝐸)convex NOUN iajs-2928 334 16 on on ADP iajs-2928 334 17 𝐴. 𝐴. PROPN iajs-2928 334 18 let let VERB iajs-2928 334 19 ( ( PUNCT iajs-2928 334 20 2,1 2,1 NUM iajs-2928 334 21 ) ) PUNCT iajs-2928 334 22 , , PUNCT iajs-2928 334 23 ( ( PUNCT iajs-2928 334 24 βˆ’2,1 βˆ’2,1 NOUN iajs-2928 334 25 ) ) PUNCT iajs-2928 334 26 ∈ ∈ PROPN iajs-2928 334 27 𝐴 𝐴 PROPN iajs-2928 334 28 , , PUNCT iajs-2928 334 29 and and CCONJ iajs-2928 334 30 π‘Ÿ π‘Ÿ X iajs-2928 334 31 = = SYM iajs-2928 334 32 𝑠 𝑠 PROPN iajs-2928 334 33 = = SYM iajs-2928 334 34 1 1 NUM iajs-2928 334 35 4 4 NUM iajs-2928 334 36 , , PUNCT iajs-2928 334 37 𝑝 𝑝 NOUN iajs-2928 334 38 = = SYM iajs-2928 334 39 1 1 NUM iajs-2928 334 40 2 2 NUM iajs-2928 334 41 . . PUNCT iajs-2928 335 1 then then ADV iajs-2928 335 2 𝑓 𝑓 X iajs-2928 335 3 ( ( PUNCT iajs-2928 335 4 1 1 NUM iajs-2928 335 5 4 4 NUM iajs-2928 335 6 𝐸(2,1 𝐸(2,1 NOUN iajs-2928 335 7 ) ) PUNCT iajs-2928 335 8 + + CCONJ iajs-2928 335 9 1 1 NUM iajs-2928 335 10 4 4 NUM iajs-2928 335 11 𝐸(βˆ’2,1 𝐸(βˆ’2,1 NOUN iajs-2928 335 12 ) ) PUNCT iajs-2928 335 13 ) ) PUNCT iajs-2928 336 1 = = SYM iajs-2928 336 2 𝑓 𝑓 X iajs-2928 336 3 ( ( PUNCT iajs-2928 336 4 0 0 NUM iajs-2928 336 5 , , PUNCT iajs-2928 336 6 1 1 NUM iajs-2928 336 7 2 2 NUM iajs-2928 336 8 ) ) PUNCT iajs-2928 336 9 = = SYM iajs-2928 336 10 1 1 NUM iajs-2928 336 11 4 4 NUM iajs-2928 336 12 > > SYM iajs-2928 336 13 π‘šπ‘Žπ‘₯{𝑓(2,1 π‘šπ‘Žπ‘₯{𝑓(2,1 PROPN iajs-2928 336 14 ) ) PUNCT iajs-2928 336 15 , , PUNCT iajs-2928 336 16 𝑓(βˆ’2,1 𝑓(βˆ’2,1 PROPN iajs-2928 336 17 ) ) PUNCT iajs-2928 336 18 } } PUNCT iajs-2928 337 1 = = SYM iajs-2928 337 2 0 0 X iajs-2928 337 3 . . PUNCT iajs-2928 338 1 also also ADV iajs-2928 338 2 , , PUNCT iajs-2928 338 3 𝑓 𝑓 PRON iajs-2928 338 4 is be AUX iajs-2928 338 5 not not PART iajs-2928 338 6 pseudo pseudo NOUN iajs-2928 338 7 semi semi ADV iajs-2928 338 8 ( ( PUNCT iajs-2928 338 9 1 1 NUM iajs-2928 338 10 2 2 NUM iajs-2928 338 11 , , PUNCT iajs-2928 338 12 𝐸)convex 𝐸)convex NOUN iajs-2928 338 13 on on ADP iajs-2928 338 14 𝐴. 𝐴. PROPN iajs-2928 338 15 to to ADP iajs-2928 338 16 this this DET iajs-2928 338 17 end end NOUN iajs-2928 338 18 , , PUNCT iajs-2928 338 19 let let VERB iajs-2928 338 20 π‘₯ π‘₯ VERB iajs-2928 338 21 = = SYM iajs-2928 338 22 ( ( PUNCT iajs-2928 338 23 2 2 NUM iajs-2928 338 24 , , PUNCT iajs-2928 338 25 1 1 NUM iajs-2928 338 26 2 2 NUM iajs-2928 338 27 ) ) PUNCT iajs-2928 338 28 , , PUNCT iajs-2928 338 29 𝑦 𝑦 NOUN iajs-2928 338 30 = = SYM iajs-2928 338 31 ( ( PUNCT iajs-2928 338 32 2,1 2,1 NUM iajs-2928 338 33 ) ) PUNCT iajs-2928 338 34 ∈ ∈ PROPN iajs-2928 338 35 𝐴 𝐴 PROPN iajs-2928 338 36 and and CCONJ iajs-2928 338 37 π‘Ÿ π‘Ÿ NOUN iajs-2928 338 38 = = SYM iajs-2928 338 39 𝑠 𝑠 PROPN iajs-2928 338 40 = = SYM iajs-2928 338 41 1 1 NUM iajs-2928 338 42 4 4 NUM iajs-2928 338 43 , , PUNCT iajs-2928 338 44 𝑝 𝑝 NOUN iajs-2928 338 45 = = SYM iajs-2928 338 46 1 1 NUM iajs-2928 338 47 2 2 NUM iajs-2928 338 48 such such ADJ iajs-2928 338 49 that that SCONJ iajs-2928 338 50 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 338 51 ) ) PUNCT iajs-2928 338 52 = = SYM iajs-2928 338 53 1 1 NUM iajs-2928 338 54 4 4 NUM iajs-2928 338 55 < < X iajs-2928 338 56 𝑓(𝑦 𝑓(𝑦 PROPN iajs-2928 338 57 ) ) PUNCT iajs-2928 338 58 = = SYM iajs-2928 339 1 0 0 X iajs-2928 339 2 . . PUNCT iajs-2928 340 1 then then ADV iajs-2928 340 2 , , PUNCT iajs-2928 340 3 𝑓 𝑓 PRON iajs-2928 340 4 ( ( PUNCT iajs-2928 340 5 1 1 NUM iajs-2928 340 6 4 4 NUM iajs-2928 340 7 𝐸 𝐸 PROPN iajs-2928 340 8 ( ( PUNCT iajs-2928 340 9 2 2 NUM iajs-2928 340 10 , , PUNCT iajs-2928 340 11 1 1 NUM iajs-2928 340 12 2 2 NUM iajs-2928 340 13 ) ) PUNCT iajs-2928 340 14 + + CCONJ iajs-2928 340 15 1 1 NUM iajs-2928 340 16 4 4 NUM iajs-2928 340 17 𝐸(2,1 𝐸(2,1 NOUN iajs-2928 340 18 ) ) PUNCT iajs-2928 340 19 ) ) PUNCT iajs-2928 341 1 = = PUNCT iajs-2928 341 2 ( ( PUNCT iajs-2928 341 3 1 1 NUM iajs-2928 341 4 8 8 NUM iajs-2928 341 5 + + NUM iajs-2928 341 6 1 1 NUM iajs-2928 341 7 4 4 NUM iajs-2928 341 8 βˆ’ βˆ’ NUM iajs-2928 341 9 1 1 NUM iajs-2928 341 10 ) ) SYM iajs-2928 341 11 2 2 NUM iajs-2928 341 12 = = SYM iajs-2928 341 13 25 25 NUM iajs-2928 341 14 64 64 NUM iajs-2928 341 15 > > SYM iajs-2928 341 16 𝑓(2,1 𝑓(2,1 NOUN iajs-2928 341 17 ) ) PUNCT iajs-2928 341 18 βˆ’ βˆ’ PROPN iajs-2928 342 1 1 1 NUM iajs-2928 342 2 16 16 NUM iajs-2928 342 3 𝑏(π‘₯ 𝑏(π‘₯ PROPN iajs-2928 342 4 , , PUNCT iajs-2928 342 5 𝑦 𝑦 NOUN iajs-2928 342 6 ) ) PUNCT iajs-2928 342 7 = = SYM iajs-2928 343 1 βˆ’ βˆ’ PROPN iajs-2928 343 2 1 1 NUM iajs-2928 343 3 16 16 NUM iajs-2928 343 4 𝑏(π‘₯ 𝑏(π‘₯ PROPN iajs-2928 343 5 , , PUNCT iajs-2928 343 6 𝑦 𝑦 NOUN iajs-2928 343 7 ) ) PUNCT iajs-2928 343 8 , , PUNCT iajs-2928 343 9 for for ADP iajs-2928 343 10 any any DET iajs-2928 343 11 strictly strictly ADV iajs-2928 343 12 positive positive ADJ iajs-2928 343 13 function function NOUN iajs-2928 343 14 𝑏(π‘₯ 𝑏(π‘₯ NOUN iajs-2928 343 15 , , PUNCT iajs-2928 343 16 𝑦 𝑦 NOUN iajs-2928 343 17 ) ) PUNCT iajs-2928 343 18 . . PUNCT iajs-2928 344 1 thus thus ADV iajs-2928 344 2 , , PUNCT iajs-2928 344 3 𝑓 𝑓 PRON iajs-2928 344 4 is be AUX iajs-2928 344 5 not not PART iajs-2928 344 6 pseudo pseudo NOUN iajs-2928 344 7 semi semi ADV iajs-2928 344 8 ( ( PUNCT iajs-2928 344 9 1 1 NUM iajs-2928 344 10 2 2 NUM iajs-2928 344 11 , , PUNCT iajs-2928 344 12 𝐸)convex 𝐸)convex NOUN iajs-2928 344 13 on on ADP iajs-2928 344 14 𝐴 𝐴 PROPN iajs-2928 344 15 as as SCONJ iajs-2928 344 16 claimed claim VERB iajs-2928 344 17 . . PUNCT iajs-2928 345 1 now now ADV iajs-2928 345 2 , , PUNCT iajs-2928 345 3 take take VERB iajs-2928 345 4 π‘₯0 π‘₯0 NOUN iajs-2928 345 5 = = SYM iajs-2928 345 6 ( ( PUNCT iajs-2928 345 7 0,1 0,1 NUM iajs-2928 345 8 ) ) PUNCT iajs-2928 345 9 ∈ ∈ PROPN iajs-2928 345 10 𝐴 𝐴 PROPN iajs-2928 345 11 such such ADJ iajs-2928 345 12 that that DET iajs-2928 345 13 𝐸(0,1 𝐸(0,1 NOUN iajs-2928 345 14 ) ) PUNCT iajs-2928 345 15 = = SYM iajs-2928 345 16 ( ( PUNCT iajs-2928 345 17 0,1 0,1 NUM iajs-2928 345 18 ) ) PUNCT iajs-2928 345 19 and and CCONJ iajs-2928 345 20 𝑓(0,1 𝑓(0,1 NOUN iajs-2928 345 21 ) ) PUNCT iajs-2928 346 1 = = SYM iajs-2928 346 2 0 0 NUM iajs-2928 347 1 ≀ ≀ NUM iajs-2928 347 2 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 347 3 , , PUNCT iajs-2928 347 4 𝑦 𝑦 NOUN iajs-2928 347 5 ) ) PUNCT iajs-2928 347 6 for for ADP iajs-2928 347 7 all all DET iajs-2928 347 8 ihjpas ihjpa NOUN iajs-2928 347 9 . . PUNCT iajs-2928 348 1 36(1)2023 36(1)2023 NUM iajs-2928 348 2 365 365 NUM iajs-2928 348 3 ( ( PUNCT iajs-2928 348 4 π‘₯ π‘₯ NOUN iajs-2928 348 5 , , PUNCT iajs-2928 348 6 𝑦 𝑦 NOUN iajs-2928 348 7 ) ) PUNCT iajs-2928 348 8 ∈ ∈ PROPN iajs-2928 348 9 𝐴 𝐴 PROPN iajs-2928 348 10 ∩ ∩ ADJ iajs-2928 348 11 [ [ X iajs-2928 348 12 βˆ’2,2 βˆ’2,2 X iajs-2928 348 13 ] ] X iajs-2928 348 14 Γ— Γ— NOUN iajs-2928 348 15 𝑅. 𝑅. ADV iajs-2928 348 16 hence hence ADV iajs-2928 348 17 , , PUNCT iajs-2928 348 18 π‘₯0 π‘₯0 NOUN iajs-2928 348 19 = = SYM iajs-2928 348 20 ( ( PUNCT iajs-2928 348 21 0,1 0,1 NUM iajs-2928 348 22 ) ) PUNCT iajs-2928 348 23 is be AUX iajs-2928 348 24 a a DET iajs-2928 348 25 local local ADJ iajs-2928 348 26 minimum minimum NOUN iajs-2928 348 27 . . PUNCT iajs-2928 349 1 however however ADV iajs-2928 349 2 , , PUNCT iajs-2928 349 3 the the DET iajs-2928 349 4 conclusion conclusion NOUN iajs-2928 349 5 of of ADP iajs-2928 349 6 propositions proposition NOUN iajs-2928 349 7 4.1 4.1 NUM iajs-2928 349 8 and and CCONJ iajs-2928 349 9 4.2 4.2 NUM iajs-2928 349 10 does do AUX iajs-2928 349 11 not not PART iajs-2928 349 12 satisfy satisfy VERB iajs-2928 349 13 , , PUNCT iajs-2928 349 14 i.e. i.e. X iajs-2928 349 15 , , PUNCT iajs-2928 349 16 π‘₯0 π‘₯0 NOUN iajs-2928 349 17 is be AUX iajs-2928 349 18 not not PART iajs-2928 349 19 a a DET iajs-2928 349 20 global global ADJ iajs-2928 349 21 minimum minimum NOUN iajs-2928 349 22 . . PUNCT iajs-2928 350 1 indeed indeed ADV iajs-2928 350 2 , , PUNCT iajs-2928 350 3 take take VERB iajs-2928 350 4 = = PUNCT iajs-2928 350 5 ( ( PUNCT iajs-2928 350 6 3,1 3,1 NUM iajs-2928 350 7 ) ) PUNCT iajs-2928 350 8 ∈ ∈ PROPN iajs-2928 350 9 𝐴 𝐴 PROPN iajs-2928 350 10 . . PUNCT iajs-2928 351 1 then then ADV iajs-2928 351 2 𝑓(3,1 𝑓(3,1 NUM iajs-2928 351 3 ) ) PUNCT iajs-2928 351 4 = = PUNCT iajs-2928 352 1 ( ( PUNCT iajs-2928 352 2 1 1 NUM iajs-2928 352 3 βˆ’ βˆ’ PROPN iajs-2928 352 4 1)2 1)2 NUM iajs-2928 352 5 βˆ’ βˆ’ NOUN iajs-2928 352 6 ( ( PUNCT iajs-2928 352 7 2 2 NUM iajs-2928 352 8 βˆ’ βˆ’ NUM iajs-2928 352 9 3)2 3)2 NUM iajs-2928 352 10 = = SYM iajs-2928 352 11 βˆ’1 βˆ’1 NOUN iajs-2928 352 12 < < X iajs-2928 352 13 𝑓(π‘₯0 𝑓(π‘₯0 PROPN iajs-2928 352 14 ) ) PUNCT iajs-2928 352 15 = = SYM iajs-2928 353 1 0 0 X iajs-2928 353 2 . . PUNCT iajs-2928 353 3 proposition proposition NOUN iajs-2928 353 4 4.5 4.5 NUM iajs-2928 353 5 . . PUNCT iajs-2928 354 1 let let VERB iajs-2928 354 2 𝑓 𝑓 PRON iajs-2928 354 3 is be AUX iajs-2928 354 4 strictly strictly ADV iajs-2928 354 5 quasi quasi ADJ iajs-2928 354 6 semi semi ADJ iajs-2928 354 7 ( ( PUNCT iajs-2928 354 8 𝑝 𝑝 PROPN iajs-2928 354 9 , , PUNCT iajs-2928 354 10 𝐸)convex 𝐸)convex NOUN iajs-2928 354 11 on on ADP iajs-2928 354 12 𝐴. 𝐴. PROPN iajs-2928 354 13 then then ADV iajs-2928 354 14 , , PUNCT iajs-2928 354 15 the the DET iajs-2928 354 16 global global ADJ iajs-2928 354 17 minimum minimum NOUN iajs-2928 354 18 of of ADP iajs-2928 354 19 problem problem NOUN iajs-2928 354 20 ( ( PUNCT iajs-2928 354 21 𝑁𝐿𝑃 𝑁𝐿𝑃 PROPN iajs-2928 354 22 ) ) PUNCT iajs-2928 354 23 is be AUX iajs-2928 354 24 singleton singleton NOUN iajs-2928 354 25 . . PUNCT iajs-2928 355 1 proof proof NOUN iajs-2928 355 2 . . PUNCT iajs-2928 356 1 let let VERB iajs-2928 356 2 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2928 356 3 , , PUNCT iajs-2928 356 4 π‘¦βˆ— π‘¦βˆ— PROPN iajs-2928 356 5 be be VERB iajs-2928 356 6 two two NUM iajs-2928 356 7 different different ADJ iajs-2928 356 8 global global ADJ iajs-2928 356 9 minima minima NOUN iajs-2928 356 10 of of ADP iajs-2928 356 11 ( ( PUNCT iajs-2928 356 12 𝑁𝐿𝑃 𝑁𝐿𝑃 PROPN iajs-2928 356 13 ) ) PUNCT iajs-2928 356 14 then then ADV iajs-2928 356 15 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 356 16 ) ) PUNCT iajs-2928 356 17 = = SYM iajs-2928 356 18 𝑓(π‘¦βˆ— 𝑓(π‘¦βˆ— X iajs-2928 356 19 ) ) PUNCT iajs-2928 356 20 ≀ ≀ NUM iajs-2928 356 21 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 356 22 ) ) PUNCT iajs-2928 356 23 for for ADP iajs-2928 356 24 any any DET iajs-2928 356 25 π‘₯ π‘₯ PROPN iajs-2928 356 26 ∈ ∈ PROPN iajs-2928 356 27 𝐴. 𝐴. NOUN iajs-2928 356 28 from from ADP iajs-2928 356 29 the the DET iajs-2928 356 30 assumptions assumption NOUN iajs-2928 356 31 on on ADP iajs-2928 356 32 𝑓 𝑓 PRON iajs-2928 356 33 and and CCONJ iajs-2928 356 34 𝐴 𝐴 PROPN iajs-2928 356 35 , , PUNCT iajs-2928 356 36 we we PRON iajs-2928 356 37 have have VERB iajs-2928 356 38 π‘ŸπΈπ‘₯βˆ— π‘ŸπΈπ‘₯βˆ— NUM iajs-2928 356 39 + + CCONJ iajs-2928 356 40 π‘ πΈπ‘¦βˆ— π‘ πΈπ‘¦βˆ— X iajs-2928 356 41 ∈ ∈ PROPN iajs-2928 356 42 𝐴 𝐴 PROPN iajs-2928 356 43 and and CCONJ iajs-2928 356 44 𝑓(π‘ŸπΈπ‘₯βˆ— 𝑓(π‘ŸπΈπ‘₯βˆ— VERB iajs-2928 356 45 + + CCONJ iajs-2928 356 46 π‘ πΈπ‘¦βˆ— π‘ πΈπ‘¦βˆ— X iajs-2928 356 47 ) ) PUNCT iajs-2928 356 48 < < X iajs-2928 356 49 π‘šπ‘Žπ‘₯{𝑓(π‘₯βˆ— π‘šπ‘Žπ‘₯{𝑓(π‘₯βˆ— PROPN iajs-2928 356 50 ) ) PUNCT iajs-2928 356 51 , , PUNCT iajs-2928 356 52 𝑓(π‘¦βˆ— 𝑓(π‘¦βˆ— NUM iajs-2928 356 53 ) ) PUNCT iajs-2928 356 54 } } PUNCT iajs-2928 356 55 = = SYM iajs-2928 356 56 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 356 57 ) ) PUNCT iajs-2928 356 58 . . PUNCT iajs-2928 357 1 the the DET iajs-2928 357 2 above above ADJ iajs-2928 357 3 inequality inequality NOUN iajs-2928 357 4 yields yield NOUN iajs-2928 357 5 that that PRON iajs-2928 357 6 π‘ŸπΈπ‘₯βˆ— π‘ŸπΈπ‘₯βˆ— VERB iajs-2928 357 7 + + CCONJ iajs-2928 357 8 π‘ πΈπ‘¦βˆ— π‘ πΈπ‘¦βˆ— X iajs-2928 357 9 is be AUX iajs-2928 357 10 a a DET iajs-2928 357 11 global global ADJ iajs-2928 357 12 minimum minimum NOUN iajs-2928 357 13 which which PRON iajs-2928 357 14 is be AUX iajs-2928 357 15 a a DET iajs-2928 357 16 contradiction contradiction NOUN iajs-2928 357 17 . . PUNCT iajs-2928 358 1 hence hence ADV iajs-2928 358 2 , , PUNCT iajs-2928 358 3 there there PRON iajs-2928 358 4 is be VERB iajs-2928 358 5 a a DET iajs-2928 358 6 unique unique ADJ iajs-2928 358 7 global global ADJ iajs-2928 358 8 minimum minimum NOUN iajs-2928 358 9 . . PUNCT iajs-2928 359 1 β–  β–  PUNCT iajs-2928 359 2 proposition proposition NOUN iajs-2928 359 3 4.6 4.6 NUM iajs-2928 359 4 . . PUNCT iajs-2928 360 1 let let VERB iajs-2928 360 2 𝑓 𝑓 PRON iajs-2928 360 3 is be AUX iajs-2928 360 4 quasi quasi NOUN iajs-2928 360 5 semi semi ADJ iajs-2928 360 6 ( ( PUNCT iajs-2928 360 7 𝑝 𝑝 PROPN iajs-2928 360 8 , , PUNCT iajs-2928 360 9 𝐸)convex 𝐸)convex NOUN iajs-2928 360 10 on on ADP iajs-2928 360 11 𝐴. 𝐴. PROPN iajs-2928 360 12 then then ADV iajs-2928 360 13 , , PUNCT iajs-2928 360 14 the the DET iajs-2928 360 15 set set NOUN iajs-2928 360 16 of of ADP iajs-2928 360 17 global global ADJ iajs-2928 360 18 minima minima NOUN iajs-2928 360 19 of of ADP iajs-2928 360 20 problem problem NOUN iajs-2928 360 21 ( ( PUNCT iajs-2928 360 22 𝑁𝐿𝑃 𝑁𝐿𝑃 PROPN iajs-2928 360 23 ) ) PUNCT iajs-2928 360 24 is be AUX iajs-2928 360 25 ( ( PUNCT iajs-2928 360 26 𝑝 𝑝 PROPN iajs-2928 360 27 , , PUNCT iajs-2928 360 28 𝐸)convex 𝐸)convex NOUN iajs-2928 360 29 . . PUNCT iajs-2928 361 1 proof proof NOUN iajs-2928 361 2 . . PUNCT iajs-2928 362 1 let let VERB iajs-2928 362 2 π‘₯1 π‘₯1 NOUN iajs-2928 362 3 βˆ— βˆ— NOUN iajs-2928 362 4 , , PUNCT iajs-2928 362 5 π‘₯2 π‘₯2 NOUN iajs-2928 362 6 βˆ— βˆ— NOUN iajs-2928 362 7 ∈ ∈ PROPN iajs-2928 362 8 π‘Žπ‘Ÿπ‘”π‘šπ‘–π‘›π΄π‘“ π‘Žπ‘Ÿπ‘”π‘šπ‘–π‘›π΄π‘“ PROPN iajs-2928 362 9 = = PRON iajs-2928 362 10 { { PUNCT iajs-2928 362 11 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2928 362 12 ∈ ∈ PROPN iajs-2928 362 13 𝐴 𝐴 PROPN iajs-2928 362 14 : : PUNCT iajs-2928 362 15 𝑓(π‘₯βˆ— 𝑓(π‘₯βˆ— NUM iajs-2928 362 16 ) ) PUNCT iajs-2928 362 17 ≀ ≀ NUM iajs-2928 362 18 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 362 19 ) ) PUNCT iajs-2928 362 20 βˆ€π‘₯ βˆ€π‘₯ PROPN iajs-2928 362 21 ∈ ∈ PROPN iajs-2928 362 22 𝐴 𝐴 PROPN iajs-2928 362 23 } } PUNCT iajs-2928 362 24 the the DET iajs-2928 362 25 set set NOUN iajs-2928 362 26 of of ADP iajs-2928 362 27 global global ADJ iajs-2928 362 28 minima minima NOUN iajs-2928 362 29 of of ADP iajs-2928 362 30 problem problem NOUN iajs-2928 362 31 ( ( PUNCT iajs-2928 362 32 𝑁𝐿𝑃 𝑁𝐿𝑃 PROPN iajs-2928 362 33 ) ) PUNCT iajs-2928 362 34 we we PRON iajs-2928 362 35 must must AUX iajs-2928 362 36 prove prove VERB iajs-2928 362 37 π‘ŸπΈπ‘₯1 π‘ŸπΈπ‘₯1 PROPN iajs-2928 362 38 βˆ— βˆ— NOUN iajs-2928 362 39 + + CCONJ iajs-2928 362 40 𝑠𝐸π‘₯2 𝑠𝐸π‘₯2 PROPN iajs-2928 362 41 βˆ— βˆ— NOUN iajs-2928 362 42 ∈ ∈ PROPN iajs-2928 363 1 π‘Žπ‘Ÿπ‘”π‘šπ‘–π‘›π΄π‘“. π‘Žπ‘Ÿπ‘”π‘šπ‘–π‘›π΄π‘“. PROPN iajs-2928 363 2 since since SCONJ iajs-2928 363 3 𝑓 𝑓 ADV iajs-2928 363 4 is be AUX iajs-2928 363 5 quasi quasi NOUN iajs-2928 363 6 semi semi ADJ iajs-2928 363 7 ( ( PUNCT iajs-2928 363 8 𝑝 𝑝 NOUN iajs-2928 363 9 , , PUNCT iajs-2928 363 10 𝐸)-convex 𝐸)-convex PROPN iajs-2928 363 11 on on ADP iajs-2928 363 12 the the DET iajs-2928 363 13 ( ( PUNCT iajs-2928 363 14 𝑝 𝑝 PROPN iajs-2928 363 15 , , PUNCT iajs-2928 363 16 𝐸)-convex 𝐸)-convex PRON iajs-2928 363 17 set set VERB iajs-2928 363 18 𝐴 𝐴 PROPN iajs-2928 363 19 , , PUNCT iajs-2928 363 20 we we PRON iajs-2928 363 21 have have VERB iajs-2928 363 22 π‘ŸπΈπ‘₯1 π‘ŸπΈπ‘₯1 PROPN iajs-2928 363 23 βˆ— βˆ— NOUN iajs-2928 363 24 + + CCONJ iajs-2928 363 25 𝑠𝐸π‘₯2 𝑠𝐸π‘₯2 PROPN iajs-2928 363 26 βˆ— βˆ— NOUN iajs-2928 363 27 ∈ ∈ PROPN iajs-2928 363 28 𝐴 𝐴 PROPN iajs-2928 363 29 and and CCONJ iajs-2928 363 30 for for ADP iajs-2928 363 31 each each DET iajs-2928 363 32 π‘₯ π‘₯ PRON iajs-2928 363 33 ∈ ∈ PROPN iajs-2928 363 34 𝐴 𝐴 PROPN iajs-2928 363 35 , , PUNCT iajs-2928 363 36 𝑓(π‘ŸπΈπ‘₯1 𝑓(π‘ŸπΈπ‘₯1 PROPN iajs-2928 363 37 βˆ— βˆ— NOUN iajs-2928 363 38 + + CCONJ iajs-2928 363 39 𝑠𝐸π‘₯2 𝑠𝐸π‘₯2 PROPN iajs-2928 363 40 βˆ— βˆ— NOUN iajs-2928 363 41 ) ) PUNCT iajs-2928 363 42 ≀ ≀ NUM iajs-2928 363 43 π‘šπ‘Žπ‘₯{𝑓(π‘₯1 π‘šπ‘Žπ‘₯{𝑓(π‘₯1 ADJ iajs-2928 363 44 βˆ— βˆ— NOUN iajs-2928 363 45 ) ) PUNCT iajs-2928 363 46 , , PUNCT iajs-2928 363 47 𝑓(π‘₯2 𝑓(π‘₯2 NOUN iajs-2928 363 48 βˆ— βˆ— NOUN iajs-2928 363 49 ) ) PUNCT iajs-2928 363 50 } } PUNCT iajs-2928 363 51 ≀ ≀ NUM iajs-2928 363 52 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2928 363 53 ) ) PUNCT iajs-2928 363 54 . . PUNCT iajs-2928 364 1 therefore therefore ADV iajs-2928 364 2 , , PUNCT iajs-2928 364 3 π‘ŸπΈπ‘₯1 π‘ŸπΈπ‘₯1 PROPN iajs-2928 364 4 βˆ— βˆ— NOUN iajs-2928 364 5 + + CCONJ iajs-2928 364 6 𝑠𝐸π‘₯2 𝑠𝐸π‘₯2 PROPN iajs-2928 364 7 βˆ— βˆ— PROPN iajs-2928 364 8 ∈ ∈ PROPN iajs-2928 364 9 π‘Žπ‘Ÿπ‘”π‘šπ‘–π‘›π΄π‘“ π‘Žπ‘Ÿπ‘”π‘šπ‘–π‘›π΄π‘“ PROPN iajs-2928 364 10 as as SCONJ iajs-2928 364 11 required require VERB iajs-2928 364 12 . . PUNCT iajs-2928 365 1 β–  β–  PUNCT iajs-2928 365 2 5.conclusion 5.conclusion NOUN iajs-2928 365 3 in in ADP iajs-2928 365 4 this this DET iajs-2928 365 5 paper paper NOUN iajs-2928 365 6 , , PUNCT iajs-2928 365 7 new new ADJ iajs-2928 365 8 generalized generalize VERB iajs-2928 365 9 convex convex NOUN iajs-2928 365 10 functions function NOUN iajs-2928 365 11 ( ( PUNCT iajs-2928 365 12 quasi quasi X iajs-2928 365 13 semi semi ADJ iajs-2928 365 14 ( ( PUNCT iajs-2928 365 15 𝑝 𝑝 NOUN iajs-2928 365 16 , , PUNCT iajs-2928 365 17 𝐸)-convex 𝐸)-convex X iajs-2928 365 18 , , PUNCT iajs-2928 365 19 and and CCONJ iajs-2928 365 20 pseudo pseudo NOUN iajs-2928 365 21 semi semi ADJ iajs-2928 365 22 ( ( PUNCT iajs-2928 365 23 𝑝 𝑝 NOUN iajs-2928 365 24 , , PUNCT iajs-2928 365 25 𝐸)-convex 𝐸)-convex NOUN iajs-2928 365 26 functions function NOUN iajs-2928 365 27 ) ) PUNCT iajs-2928 365 28 are be AUX iajs-2928 365 29 defined define VERB iajs-2928 365 30 , , PUNCT iajs-2928 365 31 and and CCONJ iajs-2928 365 32 their their PRON iajs-2928 365 33 various various ADJ iajs-2928 365 34 general general ADJ iajs-2928 365 35 and and CCONJ iajs-2928 365 36 optimality optimality NOUN iajs-2928 365 37 properties property NOUN iajs-2928 365 38 are be AUX iajs-2928 365 39 studied study VERB iajs-2928 365 40 . . PUNCT iajs-2928 366 1 these these DET iajs-2928 366 2 functions function NOUN iajs-2928 366 3 are be AUX iajs-2928 366 4 a a DET iajs-2928 366 5 combination combination NOUN iajs-2928 366 6 of of ADP iajs-2928 366 7 𝑝-convex 𝑝-convex NOUN iajs-2928 366 8 and and CCONJ iajs-2928 366 9 𝐸-convex 𝐸-convex PROPN iajs-2928 366 10 functions function NOUN iajs-2928 366 11 introduced introduce VERB iajs-2928 366 12 in in ADP iajs-2928 366 13 the the DET iajs-2928 366 14 literature literature NOUN iajs-2928 366 15 . . PUNCT iajs-2928 367 1 different different ADJ iajs-2928 367 2 examples example NOUN iajs-2928 367 3 are be AUX iajs-2928 367 4 established establish VERB iajs-2928 367 5 to to PART iajs-2928 367 6 illustrate illustrate VERB iajs-2928 367 7 these these DET iajs-2928 367 8 functions function NOUN iajs-2928 367 9 and and CCONJ iajs-2928 367 10 to to PART iajs-2928 367 11 confirm confirm VERB iajs-2928 367 12 some some DET iajs-2928 367 13 properties property NOUN iajs-2928 367 14 proved prove VERB iajs-2928 367 15 throughout throughout ADP iajs-2928 367 16 the the DET iajs-2928 367 17 work work NOUN iajs-2928 367 18 . . PUNCT iajs-2928 368 1 references reference NOUN iajs-2928 368 2 1 1 NUM iajs-2928 368 3 . . PUNCT iajs-2928 368 4 youness youness NOUN iajs-2928 368 5 , , PUNCT iajs-2928 368 6 e. e. PROPN iajs-2928 368 7 a. a. PROPN iajs-2928 368 8 e e PROPN iajs-2928 368 9 - - ADJ iajs-2928 368 10 convex convex ADJ iajs-2928 368 11 sets set NOUN iajs-2928 368 12 , , PUNCT iajs-2928 368 13 𝐸-convex 𝐸-convex PROPN iajs-2928 368 14 functions function NOUN iajs-2928 368 15 , , PUNCT iajs-2928 368 16 and and CCONJ iajs-2928 368 17 e e X iajs-2928 368 18 - - ADJ iajs-2928 368 19 convex convex ADJ iajs-2928 368 20 programming programming NOUN iajs-2928 368 21 , , PUNCT iajs-2928 368 22 journal journal NOUN iajs-2928 368 23 of of ADP iajs-2928 368 24 optimization optimization NOUN iajs-2928 368 25 theory theory NOUN iajs-2928 368 26 and and CCONJ iajs-2928 368 27 applications application NOUN iajs-2928 368 28 1999 1999 NUM iajs-2928 368 29 , , PUNCT iajs-2928 368 30 102 102 NUM iajs-2928 368 31 , , PUNCT iajs-2928 368 32 439 439 NUM iajs-2928 368 33 - - SYM iajs-2928 368 34 450 450 NUM iajs-2928 368 35 . . NOUN iajs-2928 369 1 2 2 NUM iajs-2928 369 2 . . X iajs-2928 369 3 chen chen PROPN iajs-2928 369 4 , , PUNCT iajs-2928 369 5 x. x. VERB iajs-2928 369 6 some some DET iajs-2928 369 7 properties property NOUN iajs-2928 369 8 of of ADP iajs-2928 369 9 semi semi ADJ iajs-2928 369 10 𝐸-convex 𝐸-convex PROPN iajs-2928 369 11 functions function NOUN iajs-2928 369 12 . . PUNCT iajs-2928 370 1 journal journal NOUN iajs-2928 370 2 of of ADP iajs-2928 370 3 mathematical mathematical ADJ iajs-2928 370 4 analysis analysis NOUN iajs-2928 370 5 and and CCONJ iajs-2928 370 6 applications application NOUN iajs-2928 370 7 2002 2002 NUM iajs-2928 370 8 , , PUNCT iajs-2928 370 9 275 275 NUM iajs-2928 370 10 , , PUNCT iajs-2928 370 11 251 251 NUM iajs-2928 370 12 - - SYM iajs-2928 370 13 262 262 NUM iajs-2928 370 14 . . PUNCT iajs-2928 371 1 3.chen 3.chen NUM iajs-2928 371 2 , , PUNCT iajs-2928 371 3 x. x. NOUN iajs-2928 371 4 some some DET iajs-2928 371 5 properties property NOUN iajs-2928 371 6 of of ADP iajs-2928 371 7 semi semi ADJ iajs-2928 371 8 𝐸-convex 𝐸-convex PROPN iajs-2928 371 9 functions function NOUN iajs-2928 371 10 and and CCONJ iajs-2928 371 11 semi-𝐸-convex semi-𝐸-convex PROPN iajs-2928 371 12 programming programming NOUN iajs-2928 371 13 . . PUNCT iajs-2928 372 1 the the DET iajs-2928 372 2 eighth eighth ADJ iajs-2928 372 3 international international ADJ iajs-2928 372 4 symposium symposium NOUN iajs-2928 372 5 on on ADP iajs-2928 372 6 operations operation NOUN iajs-2928 372 7 research research NOUN iajs-2928 372 8 and and CCONJ iajs-2928 372 9 its its PRON iajs-2928 372 10 applications application NOUN iajs-2928 372 11 ( ( PUNCT iajs-2928 372 12 isora'09 isora'09 NOUN iajs-2928 372 13 ) ) PUNCT iajs-2928 372 14 2009 2009 NUM iajs-2928 372 15 , , PUNCT iajs-2928 372 16 20 20 NUM iajs-2928 372 17 - - SYM iajs-2928 372 18 22 22 NUM iajs-2928 372 19 . . PUNCT iajs-2928 373 1 4 4 NUM iajs-2928 373 2 . . X iajs-2928 373 3 jian jian PROPN iajs-2928 373 4 , , PUNCT iajs-2928 373 5 j. j. PROPN iajs-2928 373 6 b. b. PROPN iajs-2928 373 7 on on ADP iajs-2928 373 8 ( ( PUNCT iajs-2928 373 9 𝐸 𝐸 PROPN iajs-2928 373 10 , , PUNCT iajs-2928 373 11 𝐹 𝐹 PROPN iajs-2928 373 12 ) ) PUNCT iajs-2928 373 13 generalized generalized ADJ iajs-2928 373 14 convexity convexity NOUN iajs-2928 373 15 . . PUNCT iajs-2928 374 1 international international ADJ iajs-2928 374 2 journal journal PROPN iajs-2928 374 3 of of ADP iajs-2928 374 4 mathematical mathematical ADJ iajs-2928 374 5 sciences science NOUN iajs-2928 374 6 2003 2003 NUM iajs-2928 374 7 , , PUNCT iajs-2928 374 8 2 2 NUM iajs-2928 374 9 ( ( PUNCT iajs-2928 374 10 1 1 NUM iajs-2928 374 11 ) ) PUNCT iajs-2928 374 12 , , PUNCT iajs-2928 374 13 121 121 NUM iajs-2928 374 14 - - SYM iajs-2928 374 15 132 132 NUM iajs-2928 374 16 . . PUNCT iajs-2928 375 1 5.abdulmaged 5.abdulmaged NUM iajs-2928 375 2 , , PUNCT iajs-2928 375 3 m. m. NOUN iajs-2928 375 4 i. i. NOUN iajs-2928 375 5 on on ADP iajs-2928 375 6 some some DET iajs-2928 375 7 generalization generalization NOUN iajs-2928 375 8 of of ADP iajs-2928 375 9 convex convex NOUN iajs-2928 375 10 sets set NOUN iajs-2928 375 11 , , PUNCT iajs-2928 375 12 convex convex NOUN iajs-2928 375 13 functions function NOUN iajs-2928 375 14 , , PUNCT iajs-2928 375 15 and and CCONJ iajs-2928 375 16 convex convex VERB iajs-2928 375 17 optimization optimization NOUN iajs-2928 375 18 problems problem NOUN iajs-2928 375 19 . . PUNCT iajs-2928 376 1 ms.c ms.c PROPN iajs-2928 376 2 . . PUNCT iajs-2928 377 1 thesis thesis NOUN iajs-2928 377 2 , , PUNCT iajs-2928 377 3 department department NOUN iajs-2928 377 4 of of ADP iajs-2928 377 5 mathematics mathematics PROPN iajs-2928 377 6 , , PUNCT iajs-2928 377 7 college college NOUN iajs-2928 377 8 of of ADP iajs-2928 377 9 education education PROPN iajs-2928 377 10 ibn ibn PROPN iajs-2928 377 11 alhaitham alhaitham NOUN iajs-2928 377 12 , , PUNCT iajs-2928 377 13 university university NOUN iajs-2928 377 14 of of ADP iajs-2928 377 15 baghdad baghdad PROPN iajs-2928 377 16 , , PUNCT iajs-2928 377 17 2018 2018 NUM iajs-2928 377 18 . . PUNCT iajs-2928 378 1 6 6 NUM iajs-2928 378 2 . . X iajs-2928 378 3 syau syau PROPN iajs-2928 378 4 , , PUNCT iajs-2928 378 5 y y PROPN iajs-2928 378 6 - - PUNCT iajs-2928 378 7 r. r. PROPN iajs-2928 378 8 ; ; PUNCT iajs-2928 378 9 lee lee PROPN iajs-2928 378 10 . . PUNCT iajs-2928 379 1 e.s e.s PROPN iajs-2928 379 2 . . PUNCT iajs-2928 380 1 some some DET iajs-2928 380 2 properties property NOUN iajs-2928 380 3 of of ADP iajs-2928 380 4 𝐸-convex 𝐸-convex PROPN iajs-2928 380 5 functions function NOUN iajs-2928 380 6 . . PUNCT iajs-2928 381 1 applied apply VERB iajs-2928 381 2 mathematics mathematics NOUN iajs-2928 381 3 letters letter NOUN iajs-2928 381 4 2005 2005 NUM iajs-2928 381 5 , , PUNCT iajs-2928 381 6 18 18 NUM iajs-2928 381 7 , , PUNCT iajs-2928 381 8 1074 1074 NUM iajs-2928 381 9 - - SYM iajs-2928 381 10 1080 1080 NUM iajs-2928 381 11 . . PUNCT iajs-2928 382 1 7.bayoumi 7.bayoumi NUM iajs-2928 382 2 , , PUNCT iajs-2928 382 3 a. a. NOUN iajs-2928 382 4 ; ; PUNCT iajs-2928 382 5 fathy fathy PROPN iajs-2928 382 6 , , PUNCT iajs-2928 382 7 a. a. NOUN iajs-2928 382 8 𝑝 𝑝 NOUN iajs-2928 382 9 -convex -convex NOUN iajs-2928 382 10 function function NOUN iajs-2928 382 11 in in ADP iajs-2928 382 12 discrete discrete ADJ iajs-2928 382 13 sets set NOUN iajs-2928 382 14 . . 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