id	sid	tid	token	lemma	pos
ejpam-2541	1	1	compile	compile	NOUN
ejpam-2541	1	2	/	/	SYM
ejpam-2541	1	3	output.dvi	output.dvi	NOUN
ejpam-2541	1	4	european	european	ADJ
ejpam-2541	1	5	journal	journal	NOUN
ejpam-2541	1	6	of	of	ADP
ejpam-2541	1	7	pure	pure	ADJ
ejpam-2541	1	8	and	and	CCONJ
ejpam-2541	1	9	applied	apply	VERB
ejpam-2541	1	10	mathematics	mathematic	NOUN
ejpam-2541	1	11	vol	vol	NOUN
ejpam-2541	1	12	.	.	PROPN
ejpam-2541	2	1	9	9	NUM
ejpam-2541	2	2	,	,	PUNCT
ejpam-2541	2	3	no	no	INTJ
ejpam-2541	2	4	.	.	NOUN
ejpam-2541	2	5	1	1	NUM
ejpam-2541	2	6	,	,	PUNCT
ejpam-2541	2	7	2016	2016	NUM
ejpam-2541	2	8	,	,	PUNCT
ejpam-2541	2	9	57	57	NUM
ejpam-2541	2	10	-	-	SYM
ejpam-2541	2	11	63	63	NUM
ejpam-2541	2	12	issn	issn	PROPN
ejpam-2541	2	13	1307	1307	NUM
ejpam-2541	2	14	-	-	SYM
ejpam-2541	2	15	5543	5543	NUM
ejpam-2541	2	16	–	–	PUNCT
ejpam-2541	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2541	2	18	on	on	ADP
ejpam-2541	2	19	convexity	convexity	NOUN
ejpam-2541	2	20	in	in	ADP
ejpam-2541	2	21	product	product	NOUN
ejpam-2541	2	22	of	of	ADP
ejpam-2541	2	23	riemannian	riemannian	PROPN
ejpam-2541	2	24	manifolds	manifold	VERB
ejpam-2541	2	25	wedad	wedad	PROPN
ejpam-2541	2	26	saleh	saleh	PROPN
ejpam-2541	2	27	,	,	PUNCT
ejpam-2541	2	28	adem	adem	PROPN
ejpam-2541	2	29	kiliçman	kiliçman	PROPN
ejpam-2541	2	30	∗	∗	PROPN
ejpam-2541	2	31	department	department	PROPN
ejpam-2541	2	32	of	of	ADP
ejpam-2541	2	33	mathematics	mathematics	PROPN
ejpam-2541	2	34	and	and	CCONJ
ejpam-2541	2	35	institute	institute	PROPN
ejpam-2541	2	36	for	for	ADP
ejpam-2541	2	37	mathematical	mathematical	ADJ
ejpam-2541	2	38	research	research	NOUN
ejpam-2541	2	39	,	,	PUNCT
ejpam-2541	2	40	university	university	NOUN
ejpam-2541	2	41	putra	putra	PROPN
ejpam-2541	2	42	malaysia	malaysia	PROPN
ejpam-2541	2	43	,	,	PUNCT
ejpam-2541	2	44	43400	43400	NUM
ejpam-2541	2	45	upm	upm	PROPN
ejpam-2541	2	46	,	,	PUNCT
ejpam-2541	2	47	serdang	serdang	PROPN
ejpam-2541	2	48	,	,	PUNCT
ejpam-2541	2	49	selangor	selangor	PROPN
ejpam-2541	2	50	,	,	PUNCT
ejpam-2541	2	51	malaysia	malaysia	PROPN
ejpam-2541	2	52	abstract	abstract	NOUN
ejpam-2541	2	53	.	.	PUNCT
ejpam-2541	3	1	in	in	ADP
ejpam-2541	3	2	this	this	DET
ejpam-2541	3	3	paper	paper	NOUN
ejpam-2541	3	4	,	,	PUNCT
ejpam-2541	3	5	the	the	DET
ejpam-2541	3	6	concept	concept	NOUN
ejpam-2541	3	7	of	of	ADP
ejpam-2541	3	8	convexity	convexity	NOUN
ejpam-2541	3	9	and	and	CCONJ
ejpam-2541	3	10	starshapedness	starshapedness	NOUN
ejpam-2541	3	11	in	in	ADP
ejpam-2541	3	12	the	the	DET
ejpam-2541	3	13	cartesian	cartesian	ADJ
ejpam-2541	3	14	product	product	NOUN
ejpam-2541	3	15	of	of	ADP
ejpam-2541	3	16	two	two	NUM
ejpam-2541	3	17	complete	complete	ADJ
ejpam-2541	3	18	,	,	PUNCT
ejpam-2541	3	19	simple	simple	ADJ
ejpam-2541	3	20	connected	connect	VERB
ejpam-2541	3	21	smooth	smooth	ADJ
ejpam-2541	3	22	riemannian	riemannian	ADJ
ejpam-2541	3	23	manifolds	manifold	NOUN
ejpam-2541	3	24	without	without	ADP
ejpam-2541	3	25	conjugate	conjugate	ADJ
ejpam-2541	3	26	points	point	NOUN
ejpam-2541	3	27	are	be	AUX
ejpam-2541	3	28	studied	study	VERB
ejpam-2541	3	29	in	in	ADP
ejpam-2541	3	30	terms	term	NOUN
ejpam-2541	3	31	of	of	ADP
ejpam-2541	3	32	the	the	DET
ejpam-2541	3	33	same	same	ADJ
ejpam-2541	3	34	concepts	concept	NOUN
ejpam-2541	3	35	in	in	ADP
ejpam-2541	3	36	the	the	DET
ejpam-2541	3	37	components	component	NOUN
ejpam-2541	3	38	of	of	ADP
ejpam-2541	3	39	product	product	NOUN
ejpam-2541	3	40	.	.	PUNCT
ejpam-2541	4	1	we	we	PRON
ejpam-2541	4	2	also	also	ADV
ejpam-2541	4	3	discuss	discuss	VERB
ejpam-2541	4	4	some	some	PRON
ejpam-2541	4	5	of	of	ADP
ejpam-2541	4	6	their	their	PRON
ejpam-2541	4	7	properties	property	NOUN
ejpam-2541	4	8	in	in	ADP
ejpam-2541	4	9	the	the	DET
ejpam-2541	4	10	cartesian	cartesian	ADJ
ejpam-2541	4	11	product	product	NOUN
ejpam-2541	4	12	of	of	ADP
ejpam-2541	4	13	riemannian	riemannian	ADJ
ejpam-2541	4	14	manifolds	manifold	NOUN
ejpam-2541	4	15	without	without	ADP
ejpam-2541	4	16	conjugate	conjugate	ADJ
ejpam-2541	4	17	points	point	NOUN
ejpam-2541	4	18	.	.	PUNCT
ejpam-2541	5	1	results	result	NOUN
ejpam-2541	5	2	obtained	obtain	VERB
ejpam-2541	5	3	in	in	ADP
ejpam-2541	5	4	this	this	DET
ejpam-2541	5	5	paper	paper	NOUN
ejpam-2541	5	6	may	may	AUX
ejpam-2541	5	7	inspire	inspire	VERB
ejpam-2541	5	8	future	future	ADJ
ejpam-2541	5	9	research	research	NOUN
ejpam-2541	5	10	in	in	ADP
ejpam-2541	5	11	convex	convex	ADJ
ejpam-2541	5	12	analysis	analysis	NOUN
ejpam-2541	5	13	and	and	CCONJ
ejpam-2541	5	14	related	relate	VERB
ejpam-2541	5	15	optimization	optimization	NOUN
ejpam-2541	5	16	fields	field	NOUN
ejpam-2541	5	17	.	.	PUNCT
ejpam-2541	6	1	2010	2010	NUM
ejpam-2541	6	2	mathematics	mathematic	NOUN
ejpam-2541	6	3	subject	subject	NOUN
ejpam-2541	6	4	classifications	classification	NOUN
ejpam-2541	6	5	:	:	PUNCT
ejpam-2541	6	6	52a20	52a20	NUM
ejpam-2541	6	7	,	,	PUNCT
ejpam-2541	6	8	52a30,53b20	52a30,53b20	NUM
ejpam-2541	6	9	key	key	ADJ
ejpam-2541	6	10	words	word	NOUN
ejpam-2541	6	11	and	and	CCONJ
ejpam-2541	6	12	phrases	phrase	NOUN
ejpam-2541	6	13	:	:	PUNCT
ejpam-2541	6	14	convex	convex	NOUN
ejpam-2541	6	15	sets	set	NOUN
ejpam-2541	6	16	,	,	PUNCT
ejpam-2541	6	17	conjugate	conjugate	ADJ
ejpam-2541	6	18	points	point	NOUN
ejpam-2541	6	19	,	,	PUNCT
ejpam-2541	6	20	kernel	kernel	PROPN
ejpam-2541	6	21	,	,	PUNCT
ejpam-2541	6	22	starshaped	starshape	VERB
ejpam-2541	6	23	sets	set	VERB
ejpam-2541	6	24	1	1	NUM
ejpam-2541	6	25	.	.	PUNCT
ejpam-2541	6	26	introduction	introduction	NOUN
ejpam-2541	6	27	convexity	convexity	NOUN
ejpam-2541	6	28	and	and	CCONJ
ejpam-2541	6	29	starshapedness	starshapedness	NOUN
ejpam-2541	6	30	play	play	VERB
ejpam-2541	6	31	an	an	DET
ejpam-2541	6	32	important	important	ADJ
ejpam-2541	6	33	role	role	NOUN
ejpam-2541	6	34	in	in	ADP
ejpam-2541	6	35	optimization	optimization	NOUN
ejpam-2541	6	36	theory	theory	NOUN
ejpam-2541	6	37	,	,	PUNCT
ejpam-2541	6	38	convex	convex	VERB
ejpam-2541	6	39	analysis	analysis	NOUN
ejpam-2541	6	40	,	,	PUNCT
ejpam-2541	6	41	minkowski	minkowski	ADJ
ejpam-2541	6	42	space	space	NOUN
ejpam-2541	6	43	and	and	CCONJ
ejpam-2541	6	44	fractal	fractal	ADJ
ejpam-2541	6	45	mathematics	mathematic	NOUN
ejpam-2541	6	46	[	[	X
ejpam-2541	6	47	4	4	NUM
ejpam-2541	6	48	,	,	PUNCT
ejpam-2541	6	49	7–9	7–9	NUM
ejpam-2541	6	50	,	,	PUNCT
ejpam-2541	6	51	11–14	11–14	NUM
ejpam-2541	6	52	,	,	PUNCT
ejpam-2541	6	53	16	16	NUM
ejpam-2541	6	54	]	]	PUNCT
ejpam-2541	6	55	.	.	PUNCT
ejpam-2541	7	1	in	in	ADP
ejpam-2541	7	2	[	[	X
ejpam-2541	7	3	15	15	NUM
ejpam-2541	7	4	]	]	PUNCT
ejpam-2541	7	5	,	,	PUNCT
ejpam-2541	7	6	pandey	pandey	PROPN
ejpam-2541	7	7	introduced	introduce	VERB
ejpam-2541	7	8	an	an	DET
ejpam-2541	7	9	interesting	interesting	ADJ
ejpam-2541	7	10	form	form	NOUN
ejpam-2541	7	11	of	of	ADP
ejpam-2541	7	12	a	a	DET
ejpam-2541	7	13	riemannian	riemannian	ADJ
ejpam-2541	7	14	metric	metric	ADJ
ejpam-2541	7	15	g	g	NOUN
ejpam-2541	7	16	and	and	CCONJ
ejpam-2541	7	17	connection	connection	NOUN
ejpam-2541	7	18	▽	▽	NOUN
ejpam-2541	7	19	on	on	ADP
ejpam-2541	7	20	the	the	DET
ejpam-2541	7	21	cartesian	cartesian	ADJ
ejpam-2541	7	22	product	product	NOUN
ejpam-2541	7	23	m1	m1	PROPN
ejpam-2541	7	24	×	×	PROPN
ejpam-2541	7	25	m2	m2	PROPN
ejpam-2541	7	26	of	of	ADP
ejpam-2541	7	27	two	two	NUM
ejpam-2541	7	28	c∞	c∞	ADJ
ejpam-2541	7	29	riemannian	riemannian	NOUN
ejpam-2541	7	30	manifolds	manifold	NOUN
ejpam-2541	7	31	m1	m1	PROPN
ejpam-2541	7	32	and	and	CCONJ
ejpam-2541	7	33	m2	m2	PROPN
ejpam-2541	7	34	.	.	PUNCT
ejpam-2541	8	1	the	the	DET
ejpam-2541	8	2	main	main	ADJ
ejpam-2541	8	3	result	result	NOUN
ejpam-2541	8	4	in	in	ADP
ejpam-2541	8	5	[	[	X
ejpam-2541	8	6	2	2	NUM
ejpam-2541	8	7	]	]	PUNCT
ejpam-2541	8	8	is	be	AUX
ejpam-2541	8	9	that	that	SCONJ
ejpam-2541	8	10	the	the	DET
ejpam-2541	8	11	product	product	NOUN
ejpam-2541	8	12	m1×m2	m1×m2	NOUN
ejpam-2541	8	13	of	of	ADP
ejpam-2541	8	14	two	two	NUM
ejpam-2541	8	15	riemannian	riemannian	ADJ
ejpam-2541	8	16	manifolds	manifold	NOUN
ejpam-2541	8	17	is	be	AUX
ejpam-2541	8	18	free	free	ADJ
ejpam-2541	8	19	from	from	ADP
ejpam-2541	8	20	conjugate	conjugate	ADJ
ejpam-2541	8	21	(	(	PUNCT
ejpam-2541	8	22	rep.focal	rep.focal	ADJ
ejpam-2541	8	23	)	)	PUNCT
ejpam-2541	8	24	points	point	NOUN
ejpam-2541	8	25	under	under	ADP
ejpam-2541	8	26	the	the	DET
ejpam-2541	8	27	metric	metric	ADJ
ejpam-2541	8	28	and	and	CCONJ
ejpam-2541	8	29	connection	connection	NOUN
ejpam-2541	8	30	given	give	VERB
ejpam-2541	8	31	in	in	ADP
ejpam-2541	8	32	[	[	X
ejpam-2541	8	33	15	15	NUM
ejpam-2541	8	34	]	]	X
ejpam-2541	8	35	if	if	SCONJ
ejpam-2541	8	36	and	and	CCONJ
ejpam-2541	8	37	only	only	ADV
ejpam-2541	8	38	if	if	SCONJ
ejpam-2541	8	39	both	both	DET
ejpam-2541	8	40	m1	m1	PROPN
ejpam-2541	8	41	and	and	CCONJ
ejpam-2541	8	42	m2	m2	PROPN
ejpam-2541	8	43	are	be	AUX
ejpam-2541	8	44	free	free	ADJ
ejpam-2541	8	45	from	from	ADP
ejpam-2541	8	46	conjugate	conjugate	ADJ
ejpam-2541	8	47	(	(	PUNCT
ejpam-2541	8	48	resp	resp	NOUN
ejpam-2541	8	49	.	.	PUNCT
ejpam-2541	9	1	focal	focal	ADJ
ejpam-2541	9	2	)	)	PUNCT
ejpam-2541	9	3	points	point	NOUN
ejpam-2541	9	4	under	under	ADP
ejpam-2541	9	5	their	their	PRON
ejpam-2541	9	6	own	own	ADJ
ejpam-2541	9	7	metrics	metric	NOUN
ejpam-2541	9	8	and	and	CCONJ
ejpam-2541	9	9	connections	connection	NOUN
ejpam-2541	9	10	.	.	PUNCT
ejpam-2541	10	1	in	in	ADP
ejpam-2541	10	2	[	[	X
ejpam-2541	10	3	3	3	NUM
ejpam-2541	10	4	]	]	PUNCT
ejpam-2541	10	5	,	,	PUNCT
ejpam-2541	10	6	there	there	PRON
ejpam-2541	10	7	are	be	VERB
ejpam-2541	10	8	some	some	DET
ejpam-2541	10	9	interesting	interesting	ADJ
ejpam-2541	10	10	results	result	NOUN
ejpam-2541	10	11	in	in	ADP
ejpam-2541	10	12	product	product	NOUN
ejpam-2541	10	13	of	of	ADP
ejpam-2541	10	14	two	two	NUM
ejpam-2541	10	15	c∞	c∞	ADJ
ejpam-2541	10	16	complete	complete	ADJ
ejpam-2541	10	17	,	,	PUNCT
ejpam-2541	10	18	simple	simple	ADJ
ejpam-2541	10	19	connected	connect	VERB
ejpam-2541	10	20	smooth	smooth	ADJ
ejpam-2541	10	21	riemannian	riemannian	ADJ
ejpam-2541	10	22	manifolds	manifold	NOUN
ejpam-2541	10	23	without	without	ADP
ejpam-2541	10	24	conjugate	conjugate	ADJ
ejpam-2541	10	25	points	point	NOUN
ejpam-2541	10	26	.	.	PUNCT
ejpam-2541	11	1	the	the	DET
ejpam-2541	11	2	main	main	ADJ
ejpam-2541	11	3	aim	aim	NOUN
ejpam-2541	11	4	of	of	ADP
ejpam-2541	11	5	this	this	DET
ejpam-2541	11	6	paper	paper	NOUN
ejpam-2541	11	7	is	be	AUX
ejpam-2541	11	8	studying	study	VERB
ejpam-2541	11	9	the	the	DET
ejpam-2541	11	10	convexity	convexity	NOUN
ejpam-2541	11	11	and	and	CCONJ
ejpam-2541	11	12	starshapedness	starshapedness	NOUN
ejpam-2541	11	13	in	in	ADP
ejpam-2541	11	14	the	the	DET
ejpam-2541	11	15	cartesian	cartesian	ADJ
ejpam-2541	11	16	product	product	NOUN
ejpam-2541	11	17	of	of	ADP
ejpam-2541	11	18	two	two	NUM
ejpam-2541	11	19	complete	complete	ADJ
ejpam-2541	11	20	,	,	PUNCT
ejpam-2541	11	21	simple	simple	ADJ
ejpam-2541	11	22	connected	connect	VERB
ejpam-2541	11	23	smooth	smooth	ADJ
ejpam-2541	11	24	riemannian	riemannian	ADJ
ejpam-2541	11	25	manifolds	manifold	NOUN
ejpam-2541	11	26	without	without	ADP
ejpam-2541	11	27	conjugate	conjugate	ADJ
ejpam-2541	11	28	points	point	NOUN
ejpam-2541	11	29	.	.	PUNCT
ejpam-2541	12	1	2	2	X
ejpam-2541	12	2	.	.	X
ejpam-2541	12	3	preliminaries	preliminary	NOUN
ejpam-2541	12	4	in	in	ADP
ejpam-2541	12	5	this	this	DET
ejpam-2541	12	6	section	section	NOUN
ejpam-2541	12	7	,	,	PUNCT
ejpam-2541	12	8	we	we	PRON
ejpam-2541	12	9	recall	recall	VERB
ejpam-2541	12	10	some	some	DET
ejpam-2541	12	11	definitions	definition	NOUN
ejpam-2541	12	12	and	and	CCONJ
ejpam-2541	12	13	properties	property	NOUN
ejpam-2541	12	14	,	,	PUNCT
ejpam-2541	12	15	which	which	PRON
ejpam-2541	12	16	are	be	AUX
ejpam-2541	12	17	used	use	VERB
ejpam-2541	12	18	further	far	ADV
ejpam-2541	12	19	in	in	ADP
ejpam-2541	12	20	this	this	DET
ejpam-2541	12	21	paper	paper	NOUN
ejpam-2541	12	22	.	.	PUNCT
ejpam-2541	13	1	we	we	PRON
ejpam-2541	13	2	refer	refer	VERB
ejpam-2541	13	3	to	to	ADP
ejpam-2541	13	4	[	[	X
ejpam-2541	13	5	18	18	NUM
ejpam-2541	13	6	]	]	PUNCT
ejpam-2541	13	7	for	for	ADP
ejpam-2541	13	8	the	the	DET
ejpam-2541	13	9	standard	standard	ADJ
ejpam-2541	13	10	material	material	NOUN
ejpam-2541	13	11	on	on	ADP
ejpam-2541	13	12	differential	differential	ADJ
ejpam-2541	13	13	geometry	geometry	NOUN
ejpam-2541	13	14	.	.	PUNCT
ejpam-2541	14	1	∗corresponding	∗corresponde	VERB
ejpam-2541	14	2	author	author	NOUN
ejpam-2541	14	3	.	.	PUNCT
ejpam-2541	15	1	email	email	NOUN
ejpam-2541	15	2	addresses	address	NOUN
ejpam-2541	15	3	:	:	PUNCT
ejpam-2541	15	4	wed_10_777@hotmail.com	wed_10_777@hotmail.com	X
ejpam-2541	15	5	(	(	PUNCT
ejpam-2541	15	6	w.	w.	PROPN
ejpam-2541	15	7	saleh	saleh	PROPN
ejpam-2541	15	8	)	)	PUNCT
ejpam-2541	15	9	,	,	PUNCT
ejpam-2541	15	10	akilic@upm.edu.my	akilic@upm.edu.my	PROPN
ejpam-2541	15	11	(	(	PUNCT
ejpam-2541	15	12	a.	a.	NOUN
ejpam-2541	15	13	kiliçman	kiliçman	PROPN
ejpam-2541	15	14	)	)	PUNCT
ejpam-2541	15	15	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2541	16	1	57	57	NUM
ejpam-2541	16	2	c	c	X
ejpam-2541	16	3	©	©	PROPN
ejpam-2541	16	4	2016	2016	NUM
ejpam-2541	16	5	ejpam	ejpam	VERB
ejpam-2541	16	6	all	all	DET
ejpam-2541	16	7	rights	right	NOUN
ejpam-2541	16	8	reserved	reserve	VERB
ejpam-2541	16	9	.	.	PUNCT
ejpam-2541	17	1	w.	w.	PROPN
ejpam-2541	17	2	saleh	saleh	PROPN
ejpam-2541	17	3	,	,	PUNCT
ejpam-2541	17	4	a.	a.	NOUN
ejpam-2541	17	5	kiliccman	kiliccman	PROPN
ejpam-2541	17	6	/	/	SYM
ejpam-2541	17	7	eur	eur	PROPN
ejpam-2541	17	8	.	.	PUNCT
ejpam-2541	18	1	j.	j.	PROPN
ejpam-2541	18	2	pure	pure	PROPN
ejpam-2541	18	3	appl	appl	PROPN
ejpam-2541	18	4	.	.	PROPN
ejpam-2541	18	5	math	math	PROPN
ejpam-2541	18	6	,	,	PUNCT
ejpam-2541	18	7	9	9	NUM
ejpam-2541	18	8	(	(	PUNCT
ejpam-2541	18	9	2016	2016	NUM
ejpam-2541	18	10	)	)	PUNCT
ejpam-2541	18	11	,	,	PUNCT
ejpam-2541	18	12	57	57	NUM
ejpam-2541	18	13	-	-	SYM
ejpam-2541	18	14	63	63	NUM
ejpam-2541	18	15	58	58	NUM
ejpam-2541	18	16	let	let	VERB
ejpam-2541	18	17	n	n	PRON
ejpam-2541	18	18	be	be	AUX
ejpam-2541	18	19	a	a	DET
ejpam-2541	18	20	c∞	c∞	ADJ
ejpam-2541	18	21	n	n	CCONJ
ejpam-2541	18	22	-	-	PUNCT
ejpam-2541	18	23	dimensional	dimensional	ADJ
ejpam-2541	18	24	riemannian	riemannian	ADJ
ejpam-2541	18	25	manifold	manifold	NOUN
ejpam-2541	18	26	,	,	PUNCT
ejpam-2541	18	27	and	and	CCONJ
ejpam-2541	18	28	tzn	tzn	PROPN
ejpam-2541	18	29	be	be	VERB
ejpam-2541	18	30	the	the	DET
ejpam-2541	18	31	tangent	tangent	ADJ
ejpam-2541	18	32	space	space	NOUN
ejpam-2541	18	33	to	to	ADP
ejpam-2541	18	34	n	n	PROPN
ejpam-2541	18	35	at	at	ADP
ejpam-2541	18	36	z.	z.	PROPN
ejpam-2541	18	37	also	also	ADV
ejpam-2541	18	38	,	,	PUNCT
ejpam-2541	18	39	assume	assume	VERB
ejpam-2541	18	40	that	that	SCONJ
ejpam-2541	18	41	µz(x1	µz(x1	NOUN
ejpam-2541	18	42	,	,	PUNCT
ejpam-2541	18	43	x2	x2	PROPN
ejpam-2541	18	44	)	)	PUNCT
ejpam-2541	18	45	is	be	AUX
ejpam-2541	18	46	a	a	DET
ejpam-2541	18	47	positive	positive	ADJ
ejpam-2541	18	48	inner	inner	ADJ
ejpam-2541	18	49	product	product	NOUN
ejpam-2541	18	50	on	on	ADP
ejpam-2541	18	51	the	the	DET
ejpam-2541	18	52	tangent	tangent	ADJ
ejpam-2541	18	53	space	space	NOUN
ejpam-2541	18	54	tzn	tzn	NOUN
ejpam-2541	18	55	(	(	PUNCT
ejpam-2541	18	56	x1	x1	PROPN
ejpam-2541	18	57	,	,	PUNCT
ejpam-2541	18	58	x2	x2	PROPN
ejpam-2541	18	59	∈	∈	PROPN
ejpam-2541	18	60	tzn	tzn	PROPN
ejpam-2541	18	61	)	)	PUNCT
ejpam-2541	18	62	,	,	PUNCT
ejpam-2541	18	63	which	which	PRON
ejpam-2541	18	64	is	be	AUX
ejpam-2541	18	65	given	give	VERB
ejpam-2541	18	66	for	for	ADP
ejpam-2541	18	67	each	each	DET
ejpam-2541	18	68	point	point	NOUN
ejpam-2541	18	69	of	of	ADP
ejpam-2541	18	70	n	n	PROPN
ejpam-2541	18	71	.	.	PUNCT
ejpam-2541	19	1	then	then	ADV
ejpam-2541	19	2	,	,	PUNCT
ejpam-2541	19	3	a	a	DET
ejpam-2541	19	4	c∞	c∞	PROPN
ejpam-2541	19	5	map	map	NOUN
ejpam-2541	19	6	µ	µ	NOUN
ejpam-2541	19	7	:	:	PUNCT
ejpam-2541	19	8	z	z	NOUN
ejpam-2541	19	9	−→	−→	PROPN
ejpam-2541	19	10	µz	µz	PROPN
ejpam-2541	19	11	,	,	PUNCT
ejpam-2541	19	12	which	which	PRON
ejpam-2541	19	13	assigns	assign	VERB
ejpam-2541	19	14	a	a	DET
ejpam-2541	19	15	positive	positive	ADJ
ejpam-2541	19	16	inner	inner	ADJ
ejpam-2541	19	17	product	product	NOUN
ejpam-2541	19	18	µz	µz	NOUN
ejpam-2541	19	19	to	to	ADP
ejpam-2541	19	20	tzn	tzn	PROPN
ejpam-2541	19	21	for	for	ADP
ejpam-2541	19	22	each	each	DET
ejpam-2541	19	23	point	point	NOUN
ejpam-2541	19	24	z	z	NOUN
ejpam-2541	19	25	of	of	ADP
ejpam-2541	19	26	n	n	PROPN
ejpam-2541	19	27	is	be	AUX
ejpam-2541	19	28	called	call	VERB
ejpam-2541	19	29	a	a	DET
ejpam-2541	19	30	riemannian	riemannian	ADJ
ejpam-2541	19	31	metric	metric	NOUN
ejpam-2541	19	32	.	.	PUNCT
ejpam-2541	20	1	the	the	DET
ejpam-2541	20	2	length	length	NOUN
ejpam-2541	20	3	of	of	ADP
ejpam-2541	20	4	a	a	DET
ejpam-2541	20	5	piecewise	piecewise	NOUN
ejpam-2541	20	6	c1	c1	NOUN
ejpam-2541	20	7	curve	curve	NOUN
ejpam-2541	20	8	η	η	PROPN
ejpam-2541	20	9	:	:	PUNCT
ejpam-2541	21	1	[	[	X
ejpam-2541	21	2	a1	a1	NOUN
ejpam-2541	21	3	,	,	PUNCT
ejpam-2541	21	4	a2	a2	PROPN
ejpam-2541	21	5	]	]	PUNCT
ejpam-2541	21	6	−→	−→	NOUN
ejpam-2541	21	7	n	n	CCONJ
ejpam-2541	21	8	which	which	PRON
ejpam-2541	21	9	is	be	AUX
ejpam-2541	21	10	defined	define	VERB
ejpam-2541	21	11	as	as	SCONJ
ejpam-2541	21	12	follows	follow	VERB
ejpam-2541	21	13	:	:	PUNCT
ejpam-2541	21	14	l(η	l(η	NOUN
ejpam-2541	21	15	)	)	PUNCT
ejpam-2541	22	1	=	=	SYM
ejpam-2541	22	2	∫	∫	PROPN
ejpam-2541	22	3	a2	a2	PROPN
ejpam-2541	22	4	a1	a1	PROPN
ejpam-2541	22	5	‖ή(x)‖d	‖ή(x)‖d	PUNCT
ejpam-2541	22	6	x	x	X
ejpam-2541	22	7	.	.	PUNCT
ejpam-2541	23	1	we	we	PRON
ejpam-2541	23	2	define	define	VERB
ejpam-2541	23	3	d(z1	d(z1	NOUN
ejpam-2541	23	4	,	,	PUNCT
ejpam-2541	23	5	z2	z2	NOUN
ejpam-2541	23	6	)	)	PUNCT
ejpam-2541	23	7	=	=	PUNCT
ejpam-2541	23	8	in	in	ADP
ejpam-2541	23	9	f	f	PROPN
ejpam-2541	23	10	�	�	PROPN
ejpam-2541	23	11	l(η	l(η	PROPN
ejpam-2541	23	12	):	):	PUNCT
ejpam-2541	23	13	η	η	PROPN
ejpam-2541	23	14	is	be	AUX
ejpam-2541	23	15	a	a	DET
ejpam-2541	23	16	piecewise	piecewise	NOUN
ejpam-2541	23	17	c1	c1	NOUN
ejpam-2541	23	18	curve	curve	NOUN
ejpam-2541	23	19	joining	join	VERB
ejpam-2541	23	20	z1	z1	PROPN
ejpam-2541	23	21	to	to	ADP
ejpam-2541	23	22	z2	z2	PROPN
ejpam-2541	23	23	for	for	ADP
ejpam-2541	23	24	any	any	DET
ejpam-2541	23	25	points	point	NOUN
ejpam-2541	23	26	z1	z1	VERB
ejpam-2541	23	27	,	,	PUNCT
ejpam-2541	23	28	z2	z2	PROPN
ejpam-2541	23	29	∈	∈	PROPN
ejpam-2541	23	30	n	n	X
ejpam-2541	23	31	.	.	PUNCT
ejpam-2541	24	1	▽	▽	ADJ
ejpam-2541	24	2	x	x	PUNCT
ejpam-2541	24	3	y	y	PROPN
ejpam-2541	24	4	,	,	PUNCT
ejpam-2541	24	5	x	x	INTJ
ejpam-2541	24	6	,	,	PUNCT
ejpam-2541	24	7	y	y	PROPN
ejpam-2541	24	8	∈	∈	PROPN
ejpam-2541	24	9	n	n	PRON
ejpam-2541	24	10	is	be	AUX
ejpam-2541	24	11	a	a	DET
ejpam-2541	24	12	unique	unique	ADJ
ejpam-2541	24	13	determined	determine	VERB
ejpam-2541	24	14	riemannian	riemannian	ADJ
ejpam-2541	24	15	connection	connection	NOUN
ejpam-2541	24	16	which	which	PRON
ejpam-2541	24	17	called	call	VERB
ejpam-2541	24	18	levicivita	levicivita	NOUN
ejpam-2541	24	19	connection	connection	NOUN
ejpam-2541	24	20	on	on	ADP
ejpam-2541	24	21	every	every	DET
ejpam-2541	24	22	riemannian	riemannian	ADJ
ejpam-2541	24	23	manifolds	manifold	NOUN
ejpam-2541	24	24	.	.	PUNCT
ejpam-2541	25	1	furthermore	furthermore	ADV
ejpam-2541	25	2	,	,	PUNCT
ejpam-2541	25	3	a	a	DET
ejpam-2541	25	4	smooth	smooth	ADJ
ejpam-2541	25	5	path	path	NOUN
ejpam-2541	25	6	η	η	PROPN
ejpam-2541	25	7	is	be	AUX
ejpam-2541	25	8	a	a	DET
ejpam-2541	25	9	geodesic	geodesic	NOUN
ejpam-2541	25	10	if	if	SCONJ
ejpam-2541	25	11	and	and	CCONJ
ejpam-2541	25	12	only	only	ADV
ejpam-2541	25	13	if	if	SCONJ
ejpam-2541	25	14	its	its	PRON
ejpam-2541	25	15	tangent	tangent	NOUN
ejpam-2541	25	16	vector	vector	NOUN
ejpam-2541	25	17	is	be	AUX
ejpam-2541	25	18	a	a	DET
ejpam-2541	25	19	parallel	parallel	ADJ
ejpam-2541	25	20	vector	vector	NOUN
ejpam-2541	25	21	field	field	NOUN
ejpam-2541	25	22	along	along	ADP
ejpam-2541	25	23	the	the	DET
ejpam-2541	25	24	path	path	NOUN
ejpam-2541	25	25	η	η	PROPN
ejpam-2541	25	26	,	,	PUNCT
ejpam-2541	25	27	i.e	i.e	PRON
ejpam-2541	25	28	,	,	PUNCT
ejpam-2541	25	29	η	η	PROPN
ejpam-2541	25	30	satisfies	satisfy	VERB
ejpam-2541	25	31	the	the	DET
ejpam-2541	25	32	equation	equation	NOUN
ejpam-2541	25	33	▽	▽	NOUN
ejpam-2541	25	34	ή(t)ή(t	ή(t)ή(t	NOUN
ejpam-2541	25	35	)	)	PUNCT
ejpam-2541	25	36	=	=	SYM
ejpam-2541	26	1	0	0	X
ejpam-2541	26	2	.	.	PUNCT
ejpam-2541	27	1	every	every	DET
ejpam-2541	27	2	path	path	NOUN
ejpam-2541	27	3	η	η	PROPN
ejpam-2541	27	4	is	be	AUX
ejpam-2541	27	5	joining	join	VERB
ejpam-2541	27	6	z1	z1	NUM
ejpam-2541	27	7	,	,	PUNCT
ejpam-2541	27	8	z2	z2	PROPN
ejpam-2541	27	9	∈	∈	PROPN
ejpam-2541	27	10	n	n	CCONJ
ejpam-2541	27	11	where	where	SCONJ
ejpam-2541	27	12	l(η	l(η	NOUN
ejpam-2541	27	13	)	)	PUNCT
ejpam-2541	27	14	=	=	SYM
ejpam-2541	27	15	d(z1	d(z1	NOUN
ejpam-2541	27	16	,	,	PUNCT
ejpam-2541	27	17	z2	z2	PROPN
ejpam-2541	27	18	)	)	PUNCT
ejpam-2541	27	19	is	be	AUX
ejpam-2541	27	20	a	a	DET
ejpam-2541	27	21	minimal	minimal	ADJ
ejpam-2541	27	22	geodesic	geodesic	NOUN
ejpam-2541	27	23	.	.	PUNCT
ejpam-2541	28	1	finally	finally	ADV
ejpam-2541	28	2	,	,	PUNCT
ejpam-2541	28	3	assume	assume	VERB
ejpam-2541	28	4	that	that	SCONJ
ejpam-2541	28	5	(	(	PUNCT
ejpam-2541	28	6	n	n	X
ejpam-2541	28	7	,	,	PUNCT
ejpam-2541	28	8	η	η	NOUN
ejpam-2541	28	9	)	)	PUNCT
ejpam-2541	28	10	is	be	AUX
ejpam-2541	28	11	a	a	DET
ejpam-2541	28	12	complete	complete	ADJ
ejpam-2541	28	13	n	n	CCONJ
ejpam-2541	28	14	-	-	PUNCT
ejpam-2541	28	15	dimensional	dimensional	ADJ
ejpam-2541	28	16	riemannian	riemannian	NOUN
ejpam-2541	28	17	manifold	manifold	NOUN
ejpam-2541	28	18	with	with	ADP
ejpam-2541	28	19	riemannian	riemannian	ADJ
ejpam-2541	28	20	connection	connection	NOUN
ejpam-2541	28	21	▽	▽	PROPN
ejpam-2541	28	22	.	.	PUNCT
ejpam-2541	29	1	let	let	VERB
ejpam-2541	29	2	x1	x1	NUM
ejpam-2541	29	3	,	,	PUNCT
ejpam-2541	29	4	x2	x2	PROPN
ejpam-2541	29	5	∈	∈	PROPN
ejpam-2541	29	6	n	n	PROPN
ejpam-2541	29	7	and	and	CCONJ
ejpam-2541	29	8	η	η	PROPN
ejpam-2541	29	9	:	:	PUNCT
ejpam-2541	30	1	[	[	X
ejpam-2541	30	2	0,1	0,1	NUM
ejpam-2541	30	3	]	]	X
ejpam-2541	30	4	−→	−→	NOUN
ejpam-2541	30	5	n	n	AUX
ejpam-2541	30	6	be	be	AUX
ejpam-2541	30	7	a	a	DET
ejpam-2541	30	8	geodesic	geodesic	NOUN
ejpam-2541	30	9	joining	join	VERB
ejpam-2541	30	10	the	the	DET
ejpam-2541	30	11	points	point	NOUN
ejpam-2541	30	12	x1	x1	PROPN
ejpam-2541	30	13	and	and	CCONJ
ejpam-2541	30	14	x2	x2	PROPN
ejpam-2541	30	15	,	,	PUNCT
ejpam-2541	30	16	which	which	PRON
ejpam-2541	30	17	means	mean	VERB
ejpam-2541	30	18	that	that	SCONJ
ejpam-2541	30	19	ηx1,x2	ηx1,x2	ADJ
ejpam-2541	30	20	(	(	PUNCT
ejpam-2541	30	21	0	0	NUM
ejpam-2541	30	22	)	)	PUNCT
ejpam-2541	31	1	=	=	SYM
ejpam-2541	31	2	x2	x2	PROPN
ejpam-2541	31	3	and	and	CCONJ
ejpam-2541	31	4	ηx1,x2	ηx1,x2	ADJ
ejpam-2541	31	5	(	(	PUNCT
ejpam-2541	31	6	1	1	NUM
ejpam-2541	31	7	)	)	PUNCT
ejpam-2541	31	8	=	=	SYM
ejpam-2541	32	1	x1	x1	PROPN
ejpam-2541	32	2	.	.	PUNCT
ejpam-2541	33	1	definition	definition	NOUN
ejpam-2541	33	2	1	1	NUM
ejpam-2541	33	3	(	(	PUNCT
ejpam-2541	33	4	see[10	see[10	NUM
ejpam-2541	33	5	]	]	PUNCT
ejpam-2541	33	6	)	)	PUNCT
ejpam-2541	33	7	.	.	PUNCT
ejpam-2541	34	1	a	a	DET
ejpam-2541	34	2	subset	subset	NOUN
ejpam-2541	34	3	b	b	NOUN
ejpam-2541	34	4	in	in	ADP
ejpam-2541	34	5	a	a	DET
ejpam-2541	34	6	riemannian	riemannian	ADJ
ejpam-2541	34	7	manifold	manifold	NOUN
ejpam-2541	34	8	n	n	NOUN
ejpam-2541	34	9	is	be	AUX
ejpam-2541	34	10	convex	convex	ADJ
ejpam-2541	34	11	if	if	SCONJ
ejpam-2541	34	12	for	for	SCONJ
ejpam-2541	34	13	each	each	DET
ejpam-2541	34	14	pair	pair	NOUN
ejpam-2541	34	15	points	point	VERB
ejpam-2541	34	16	p	p	PRON
ejpam-2541	34	17	,	,	PUNCT
ejpam-2541	34	18	q	q	PROPN
ejpam-2541	34	19	∈	∈	PROPN
ejpam-2541	34	20	n	n	CCONJ
ejpam-2541	34	21	,	,	PUNCT
ejpam-2541	34	22	there	there	PRON
ejpam-2541	34	23	is	be	VERB
ejpam-2541	34	24	a	a	DET
ejpam-2541	34	25	unique	unique	ADJ
ejpam-2541	34	26	minimal	minimal	ADJ
ejpam-2541	34	27	geodesic	geodesic	ADJ
ejpam-2541	34	28	segment	segment	NOUN
ejpam-2541	34	29	from	from	ADP
ejpam-2541	34	30	p	p	NOUN
ejpam-2541	34	31	to	to	ADP
ejpam-2541	34	32	q	q	PROPN
ejpam-2541	35	1	and	and	CCONJ
ejpam-2541	35	2	this	this	DET
ejpam-2541	35	3	segment	segment	NOUN
ejpam-2541	35	4	is	be	AUX
ejpam-2541	35	5	in	in	ADP
ejpam-2541	35	6	b.	b.	PROPN
ejpam-2541	35	7	when	when	SCONJ
ejpam-2541	35	8	dealing	deal	VERB
ejpam-2541	35	9	with	with	ADP
ejpam-2541	35	10	a	a	DET
ejpam-2541	35	11	subset	subset	NOUN
ejpam-2541	35	12	b	b	X
ejpam-2541	35	13	⊂	⊂	PROPN
ejpam-2541	35	14	w	w	PROPN
ejpam-2541	35	15	,	,	PUNCT
ejpam-2541	35	16	where	where	SCONJ
ejpam-2541	35	17	w	w	NOUN
ejpam-2541	35	18	is	be	AUX
ejpam-2541	35	19	a	a	DET
ejpam-2541	35	20	c∞	c∞	PROPN
ejpam-2541	35	21	complete	complete	ADJ
ejpam-2541	35	22	,	,	PUNCT
ejpam-2541	35	23	simply	simply	ADV
ejpam-2541	35	24	connected	connected	ADJ
ejpam-2541	35	25	ndimensional	ndimensional	ADJ
ejpam-2541	35	26	riemannian	riemannian	NOUN
ejpam-2541	35	27	manifold	manifold	ADJ
ejpam-2541	35	28	without	without	ADP
ejpam-2541	35	29	conjugate	conjugate	ADJ
ejpam-2541	35	30	points	point	NOUN
ejpam-2541	35	31	,	,	PUNCT
ejpam-2541	35	32	the	the	DET
ejpam-2541	35	33	word	word	NOUN
ejpam-2541	35	34	"	"	PUNCT
ejpam-2541	35	35	a	a	DET
ejpam-2541	35	36	unique	unique	ADJ
ejpam-2541	35	37	minimal	minimal	ADJ
ejpam-2541	35	38	geodesic	geodesic	ADJ
ejpam-2541	35	39	segment	segment	NOUN
ejpam-2541	35	40	"	"	PUNCT
ejpam-2541	35	41	should	should	AUX
ejpam-2541	35	42	be	be	AUX
ejpam-2541	35	43	replaced	replace	VERB
ejpam-2541	35	44	by	by	ADP
ejpam-2541	35	45	"	"	PUNCT
ejpam-2541	35	46	the	the	DET
ejpam-2541	35	47	geodesic	geodesic	ADJ
ejpam-2541	35	48	segment	segment	NOUN
ejpam-2541	35	49	"	"	PUNCT
ejpam-2541	35	50	.	.	PUNCT
ejpam-2541	36	1	the	the	DET
ejpam-2541	36	2	following	follow	VERB
ejpam-2541	36	3	theorem	theorem	NOUN
ejpam-2541	36	4	was	be	AUX
ejpam-2541	36	5	proved	prove	VERB
ejpam-2541	36	6	in	in	ADP
ejpam-2541	36	7	[	[	X
ejpam-2541	36	8	1	1	NUM
ejpam-2541	36	9	]	]	PUNCT
ejpam-2541	36	10	:	:	PUNCT
ejpam-2541	36	11	theorem	theorem	NOUN
ejpam-2541	36	12	1	1	X
ejpam-2541	36	13	.	.	PUNCT
ejpam-2541	37	1	let	let	VERB
ejpam-2541	37	2	a⊂w	a⊂w	NOUN
ejpam-2541	37	3	be	be	AUX
ejpam-2541	37	4	an	an	DET
ejpam-2541	37	5	open	open	ADJ
ejpam-2541	37	6	convex	convex	NOUN
ejpam-2541	37	7	subset	subset	NOUN
ejpam-2541	37	8	.	.	PUNCT
ejpam-2541	38	1	then	then	ADV
ejpam-2541	38	2	,	,	PUNCT
ejpam-2541	38	3	(	(	PUNCT
ejpam-2541	38	4	i	i	NOUN
ejpam-2541	38	5	)	)	PUNCT
ejpam-2541	38	6	the	the	DET
ejpam-2541	38	7	closure	closure	NOUN
ejpam-2541	38	8	of	of	ADP
ejpam-2541	38	9	a	a	DET
ejpam-2541	38	10	(	(	PUNCT
ejpam-2541	38	11	ā	ā	NOUN
ejpam-2541	38	12	)	)	PUNCT
ejpam-2541	38	13	is	be	AUX
ejpam-2541	38	14	also	also	ADV
ejpam-2541	38	15	convex	convex	ADJ
ejpam-2541	38	16	.	.	PUNCT
ejpam-2541	39	1	(	(	PUNCT
ejpam-2541	39	2	ii	ii	NOUN
ejpam-2541	39	3	)	)	PUNCT
ejpam-2541	39	4	the	the	DET
ejpam-2541	39	5	interior	interior	NOUN
ejpam-2541	39	6	of	of	ADP
ejpam-2541	39	7	a	a	DET
ejpam-2541	39	8	(	(	PUNCT
ejpam-2541	39	9	int(a	int(a	NOUN
ejpam-2541	39	10	)	)	PUNCT
ejpam-2541	39	11	)	)	PUNCT
ejpam-2541	39	12	is	be	AUX
ejpam-2541	39	13	also	also	ADV
ejpam-2541	39	14	convex	convex	ADJ
ejpam-2541	39	15	.	.	PUNCT
ejpam-2541	40	1	the	the	DET
ejpam-2541	40	2	following	follow	VERB
ejpam-2541	40	3	theorem	theorem	NOUN
ejpam-2541	40	4	gives	give	VERB
ejpam-2541	40	5	the	the	DET
ejpam-2541	40	6	relationship	relationship	NOUN
ejpam-2541	40	7	between	between	ADP
ejpam-2541	40	8	global	global	ADJ
ejpam-2541	40	9	supporting	supporting	NOUN
ejpam-2541	40	10	and	and	CCONJ
ejpam-2541	40	11	convexity	convexity	NOUN
ejpam-2541	40	12	:	:	PUNCT
ejpam-2541	40	13	theorem	theorem	NOUN
ejpam-2541	40	14	2	2	NUM
ejpam-2541	40	15	(	(	PUNCT
ejpam-2541	40	16	see	see	VERB
ejpam-2541	40	17	[	[	X
ejpam-2541	40	18	5	5	NUM
ejpam-2541	40	19	]	]	PUNCT
ejpam-2541	40	20	)	)	PUNCT
ejpam-2541	40	21	.	.	PUNCT
ejpam-2541	41	1	let	let	VERB
ejpam-2541	41	2	a⊂w	a⊂w	NOUN
ejpam-2541	41	3	be	be	AUX
ejpam-2541	41	4	an	an	DET
ejpam-2541	41	5	open	open	ADJ
ejpam-2541	41	6	subset	subset	NOUN
ejpam-2541	41	7	whose	whose	DET
ejpam-2541	41	8	boundary	boundary	NOUN
ejpam-2541	41	9	a	a	PRON
ejpam-2541	41	10	is	be	AUX
ejpam-2541	41	11	a	a	DET
ejpam-2541	41	12	smooth	smooth	ADJ
ejpam-2541	41	13	hypersurface	hypersurface	NOUN
ejpam-2541	41	14	of	of	ADP
ejpam-2541	41	15	w.	w.	PROPN
ejpam-2541	41	16	then	then	ADV
ejpam-2541	41	17	,	,	PUNCT
ejpam-2541	41	18	a	a	PRON
ejpam-2541	41	19	is	be	AUX
ejpam-2541	41	20	convex	convex	ADJ
ejpam-2541	41	21	if	if	SCONJ
ejpam-2541	41	22	and	and	CCONJ
ejpam-2541	41	23	only	only	ADV
ejpam-2541	41	24	if	if	SCONJ
ejpam-2541	41	25	a	a	PRON
ejpam-2541	41	26	is	be	AUX
ejpam-2541	41	27	globally	globally	ADV
ejpam-2541	41	28	supported	support	VERB
ejpam-2541	41	29	at	at	ADP
ejpam-2541	41	30	each	each	DET
ejpam-2541	41	31	boundary	boundary	ADJ
ejpam-2541	41	32	point	point	NOUN
ejpam-2541	41	33	.	.	PUNCT
ejpam-2541	42	1	definition	definition	NOUN
ejpam-2541	42	2	2	2	NUM
ejpam-2541	42	3	(	(	PUNCT
ejpam-2541	42	4	see	see	VERB
ejpam-2541	42	5	[	[	X
ejpam-2541	42	6	17	17	NUM
ejpam-2541	42	7	]	]	NUM
ejpam-2541	42	8	)	)	PUNCT
ejpam-2541	42	9	.	.	PUNCT
ejpam-2541	43	1	a	a	DET
ejpam-2541	43	2	subset	subset	NOUN
ejpam-2541	43	3	s	s	X
ejpam-2541	43	4	in	in	ADP
ejpam-2541	43	5	a	a	DET
ejpam-2541	43	6	riemannian	riemannian	ADJ
ejpam-2541	43	7	manifold	manifold	NOUN
ejpam-2541	43	8	n	n	NUM
ejpam-2541	43	9	is	be	AUX
ejpam-2541	43	10	starshaped	starshape	VERB
ejpam-2541	43	11	if	if	SCONJ
ejpam-2541	43	12	there	there	PRON
ejpam-2541	43	13	is	be	VERB
ejpam-2541	43	14	a	a	DET
ejpam-2541	43	15	point	point	NOUN
ejpam-2541	43	16	p	p	X
ejpam-2541	43	17	∈	∈	PROPN
ejpam-2541	43	18	s	s	VERB
ejpam-2541	43	19	such	such	ADJ
ejpam-2541	43	20	that	that	PRON
ejpam-2541	43	21	for	for	ADP
ejpam-2541	43	22	all	all	DET
ejpam-2541	43	23	q	q	PROPN
ejpam-2541	43	24	∈	∈	PROPN
ejpam-2541	43	25	s	s	VERB
ejpam-2541	43	26	there	there	PRON
ejpam-2541	43	27	is	be	VERB
ejpam-2541	43	28	a	a	DET
ejpam-2541	43	29	unique	unique	ADJ
ejpam-2541	43	30	minimal	minimal	ADJ
ejpam-2541	43	31	geodesic	geodesic	ADJ
ejpam-2541	43	32	segment	segment	NOUN
ejpam-2541	43	33	γpq	γpq	ADV
ejpam-2541	43	34	from	from	ADP
ejpam-2541	43	35	p	p	NOUN
ejpam-2541	43	36	to	to	ADP
ejpam-2541	43	37	q	q	PROPN
ejpam-2541	44	1	and	and	CCONJ
ejpam-2541	44	2	this	this	DET
ejpam-2541	44	3	segment	segment	NOUN
ejpam-2541	44	4	is	be	AUX
ejpam-2541	44	5	in	in	ADP
ejpam-2541	44	6	s.	s.	PROPN
ejpam-2541	44	7	in	in	ADP
ejpam-2541	44	8	such	such	DET
ejpam-2541	44	9	a	a	DET
ejpam-2541	44	10	case	case	NOUN
ejpam-2541	44	11	,	,	PUNCT
ejpam-2541	44	12	the	the	DET
ejpam-2541	44	13	set	set	NOUN
ejpam-2541	44	14	s	s	PART
ejpam-2541	44	15	is	be	AUX
ejpam-2541	44	16	starshaped	starshape	VERB
ejpam-2541	44	17	with	with	ADP
ejpam-2541	44	18	respect	respect	NOUN
ejpam-2541	44	19	to	to	ADP
ejpam-2541	44	20	p	p	NOUN
ejpam-2541	44	21	or	or	CCONJ
ejpam-2541	44	22	p	p	NOUN
ejpam-2541	44	23	sees	see	VERB
ejpam-2541	44	24	s	s	PRON
ejpam-2541	44	25	via	via	ADP
ejpam-2541	44	26	s.	s.	PROPN
ejpam-2541	44	27	remark	remark	PROPN
ejpam-2541	44	28	1	1	NUM
ejpam-2541	44	29	.	.	PUNCT
ejpam-2541	45	1	(	(	PUNCT
ejpam-2541	45	2	i	i	NOUN
ejpam-2541	45	3	)	)	PUNCT
ejpam-2541	45	4	the	the	DET
ejpam-2541	45	5	subset	subset	NOUN
ejpam-2541	45	6	of	of	ADP
ejpam-2541	45	7	s	s	PROPN
ejpam-2541	45	8	consisting	consist	VERB
ejpam-2541	45	9	of	of	ADP
ejpam-2541	45	10	all	all	DET
ejpam-2541	45	11	points	point	NOUN
ejpam-2541	45	12	like	like	SCONJ
ejpam-2541	45	13	p	p	NOUN
ejpam-2541	45	14	is	be	AUX
ejpam-2541	45	15	called	call	VERB
ejpam-2541	45	16	the	the	DET
ejpam-2541	45	17	kernel	kernel	NOUN
ejpam-2541	45	18	of	of	ADP
ejpam-2541	45	19	s	s	PROPN
ejpam-2541	45	20	(	(	PUNCT
ejpam-2541	45	21	kers	ker	NOUN
ejpam-2541	45	22	)	)	PUNCT
ejpam-2541	45	23	.	.	PUNCT
ejpam-2541	46	1	(	(	PUNCT
ejpam-2541	46	2	ii	ii	NOUN
ejpam-2541	46	3	)	)	PUNCT
ejpam-2541	46	4	in	in	ADP
ejpam-2541	46	5	w	w	PROPN
ejpam-2541	46	6	,	,	PUNCT
ejpam-2541	46	7	a	a	DET
ejpam-2541	46	8	subset	subset	NOUN
ejpam-2541	46	9	s	s	X
ejpam-2541	46	10	is	be	AUX
ejpam-2541	46	11	starshaped	starshape	VERB
ejpam-2541	46	12	if	if	SCONJ
ejpam-2541	46	13	there	there	PRON
ejpam-2541	46	14	is	be	VERB
ejpam-2541	46	15	a	a	DET
ejpam-2541	46	16	point	point	NOUN
ejpam-2541	46	17	p	p	X
ejpam-2541	46	18	∈	∈	PROPN
ejpam-2541	46	19	ssuch	ssuch	NOUN
ejpam-2541	46	20	that	that	SCONJ
ejpam-2541	46	21	for	for	ADP
ejpam-2541	46	22	all	all	DET
ejpam-2541	46	23	q	q	PROPN
ejpam-2541	46	24	∈	∈	PROPN
ejpam-2541	46	25	s	s	PROPN
ejpam-2541	46	26	,	,	PUNCT
ejpam-2541	46	27	the	the	DET
ejpam-2541	46	28	geodesic	geodesic	ADJ
ejpam-2541	46	29	segment	segment	NOUN
ejpam-2541	46	30	γpq	γpq	ADV
ejpam-2541	46	31	joining	join	VERB
ejpam-2541	46	32	p	p	NOUN
ejpam-2541	46	33	and	and	CCONJ
ejpam-2541	46	34	q	q	NOUN
ejpam-2541	46	35	is	be	AUX
ejpam-2541	46	36	contained	contain	VERB
ejpam-2541	46	37	in	in	ADP
ejpam-2541	46	38	s.	s.	PROPN
ejpam-2541	46	39	w.	w.	PROPN
ejpam-2541	46	40	saleh	saleh	PROPN
ejpam-2541	46	41	,	,	PUNCT
ejpam-2541	46	42	a.	a.	NOUN
ejpam-2541	46	43	kiliccman	kiliccman	PROPN
ejpam-2541	46	44	/	/	SYM
ejpam-2541	46	45	eur	eur	PROPN
ejpam-2541	46	46	.	.	PUNCT
ejpam-2541	47	1	j.	j.	PROPN
ejpam-2541	47	2	pure	pure	PROPN
ejpam-2541	47	3	appl	appl	PROPN
ejpam-2541	47	4	.	.	PROPN
ejpam-2541	47	5	math	math	PROPN
ejpam-2541	47	6	,	,	PUNCT
ejpam-2541	47	7	9	9	NUM
ejpam-2541	47	8	(	(	PUNCT
ejpam-2541	47	9	2016	2016	NUM
ejpam-2541	47	10	)	)	PUNCT
ejpam-2541	47	11	,	,	PUNCT
ejpam-2541	47	12	57	57	NUM
ejpam-2541	47	13	-	-	SYM
ejpam-2541	47	14	63	63	NUM
ejpam-2541	47	15	59	59	NUM
ejpam-2541	47	16	theorem	theorem	NOUN
ejpam-2541	47	17	3	3	NUM
ejpam-2541	47	18	(	(	PUNCT
ejpam-2541	47	19	see	see	VERB
ejpam-2541	47	20	[	[	X
ejpam-2541	47	21	6	6	NUM
ejpam-2541	47	22	]	]	NUM
ejpam-2541	47	23	)	)	PUNCT
ejpam-2541	47	24	.	.	PUNCT
ejpam-2541	48	1	let	let	VERB
ejpam-2541	48	2	s	s	PRON
ejpam-2541	48	3	⊂w	⊂w	PROPN
ejpam-2541	48	4	be	be	AUX
ejpam-2541	48	5	an	an	DET
ejpam-2541	48	6	open	open	ADJ
ejpam-2541	48	7	starshaped	starshape	VERB
ejpam-2541	48	8	subset	subset	VERB
ejpam-2541	48	9	with	with	ADP
ejpam-2541	48	10	respect	respect	NOUN
ejpam-2541	48	11	to	to	ADP
ejpam-2541	48	12	some	some	DET
ejpam-2541	48	13	point	point	NOUN
ejpam-2541	48	14	p	p	X
ejpam-2541	48	15	∈	∈	PROPN
ejpam-2541	48	16	s	s	NOUN
ejpam-2541	48	17	,	,	PUNCT
ejpam-2541	48	18	then	then	ADV
ejpam-2541	48	19	the	the	DET
ejpam-2541	48	20	closure	closure	NOUN
ejpam-2541	48	21	s̄	s̄	NOUN
ejpam-2541	48	22	is	be	AUX
ejpam-2541	48	23	also	also	ADV
ejpam-2541	48	24	starshaped	starshape	VERB
ejpam-2541	48	25	with	with	ADP
ejpam-2541	48	26	respect	respect	NOUN
ejpam-2541	48	27	to	to	ADP
ejpam-2541	48	28	the	the	DET
ejpam-2541	48	29	same	same	ADJ
ejpam-2541	48	30	point	point	NOUN
ejpam-2541	48	31	p	p	NOUN
ejpam-2541	48	32	.	.	PUNCT
ejpam-2541	49	1	definition	definition	NOUN
ejpam-2541	49	2	3	3	NUM
ejpam-2541	49	3	(	(	PUNCT
ejpam-2541	49	4	see	see	VERB
ejpam-2541	49	5	[	[	X
ejpam-2541	49	6	5	5	NUM
ejpam-2541	49	7	]	]	PUNCT
ejpam-2541	49	8	)	)	PUNCT
ejpam-2541	49	9	.	.	PUNCT
ejpam-2541	50	1	let	let	VERB
ejpam-2541	50	2	a	a	PRON
ejpam-2541	50	3	be	be	AUX
ejpam-2541	50	4	an	an	DET
ejpam-2541	50	5	open	open	ADJ
ejpam-2541	50	6	subset	subset	NOUN
ejpam-2541	50	7	of	of	ADP
ejpam-2541	50	8	w	w	ADP
ejpam-2541	50	9	whose	whose	DET
ejpam-2541	50	10	boundary	boundary	NOUN
ejpam-2541	50	11	∂	∂	NOUN
ejpam-2541	50	12	a	a	PRON
ejpam-2541	50	13	is	be	AUX
ejpam-2541	50	14	a	a	DET
ejpam-2541	50	15	smooth	smooth	ADJ
ejpam-2541	50	16	hypersurface	hypersurface	NOUN
ejpam-2541	50	17	of	of	ADP
ejpam-2541	50	18	w.	w.	PROPN
ejpam-2541	50	19	a	a	PROPN
ejpam-2541	50	20	is	be	AUX
ejpam-2541	50	21	called	call	VERB
ejpam-2541	50	22	globally	globally	ADV
ejpam-2541	50	23	supported	support	VERB
ejpam-2541	50	24	at	at	ADP
ejpam-2541	50	25	p	p	PROPN
ejpam-2541	50	26	∈	∈	PROPN
ejpam-2541	50	27	∂	∂	NOUN
ejpam-2541	50	28	a	a	PRON
ejpam-2541	50	29	if	if	SCONJ
ejpam-2541	50	30	a	a	PRON
ejpam-2541	50	31	is	be	AUX
ejpam-2541	50	32	contained	contain	VERB
ejpam-2541	50	33	in	in	ADP
ejpam-2541	50	34	one	one	NUM
ejpam-2541	50	35	side	side	NOUN
ejpam-2541	50	36	of	of	ADP
ejpam-2541	50	37	the	the	DET
ejpam-2541	50	38	tangent	tangent	ADJ
ejpam-2541	50	39	geodesic	geodesic	NOUN
ejpam-2541	50	40	hypersurface	hypersurface	NOUN
ejpam-2541	50	41	sp	sp	ADP
ejpam-2541	50	42	at	at	ADP
ejpam-2541	50	43	p	p	PROPN
ejpam-2541	50	44	∈	∈	PROPN
ejpam-2541	50	45	∂	∂	NOUN
ejpam-2541	50	46	a.	a.	NOUN
ejpam-2541	50	47	let	let	VERB
ejpam-2541	50	48	n1	n1	NOUN
ejpam-2541	50	49	and	and	CCONJ
ejpam-2541	50	50	n2	n2	ADJ
ejpam-2541	50	51	be	be	AUX
ejpam-2541	50	52	two	two	NUM
ejpam-2541	50	53	complete	complete	ADJ
ejpam-2541	50	54	riemannian	riemannian	NOUN
ejpam-2541	50	55	manifolds	manifold	NOUN
ejpam-2541	50	56	with	with	ADP
ejpam-2541	50	57	riemannian	riemannian	ADJ
ejpam-2541	50	58	metrics	metric	NOUN
ejpam-2541	50	59	g1	g1	NOUN
ejpam-2541	50	60	and	and	CCONJ
ejpam-2541	50	61	g2	g2	PROPN
ejpam-2541	50	62	and	and	CCONJ
ejpam-2541	50	63	riemannian	riemannian	ADJ
ejpam-2541	50	64	connections	connection	NOUN
ejpam-2541	50	65	▽	▽	ADJ
ejpam-2541	50	66	1	1	NUM
ejpam-2541	50	67	and	and	CCONJ
ejpam-2541	50	68	▽	▽	ADJ
ejpam-2541	50	69	2	2	NUM
ejpam-2541	50	70	,	,	PUNCT
ejpam-2541	50	71	respectively	respectively	ADV
ejpam-2541	50	72	.	.	PUNCT
ejpam-2541	51	1	a	a	DET
ejpam-2541	51	2	riemannian	riemannian	ADJ
ejpam-2541	51	3	metric	metric	ADJ
ejpam-2541	51	4	g	g	NOUN
ejpam-2541	51	5	on	on	ADP
ejpam-2541	51	6	n1×n2	n1×n2	NOUN
ejpam-2541	51	7	was	be	AUX
ejpam-2541	51	8	defined	define	VERB
ejpam-2541	51	9	as	as	ADP
ejpam-2541	51	10	follows	follow	VERB
ejpam-2541	51	11	(	(	PUNCT
ejpam-2541	51	12	see[19	see[19	PRON
ejpam-2541	51	13	]	]	X
ejpam-2541	51	14	)	)	PUNCT
ejpam-2541	52	1	g(x	g(x	NOUN
ejpam-2541	52	2	,	,	PUNCT
ejpam-2541	52	3	y	y	PROPN
ejpam-2541	52	4	)	)	PUNCT
ejpam-2541	52	5	=	=	SYM
ejpam-2541	52	6	g((x1	g((x1	NOUN
ejpam-2541	52	7	,	,	PUNCT
ejpam-2541	52	8	x2	x2	PROPN
ejpam-2541	52	9	)	)	PUNCT
ejpam-2541	52	10	,	,	PUNCT
ejpam-2541	52	11	(	(	PUNCT
ejpam-2541	52	12	y1	y1	INTJ
ejpam-2541	52	13	,	,	PUNCT
ejpam-2541	52	14	y2	y2	PROPN
ejpam-2541	52	15	)	)	PUNCT
ejpam-2541	52	16	)	)	PUNCT
ejpam-2541	53	1	=	=	PUNCT
ejpam-2541	53	2	g1(x1	g1(x1	NOUN
ejpam-2541	53	3	,	,	PUNCT
ejpam-2541	53	4	y1	y1	NOUN
ejpam-2541	53	5	)	)	PUNCT
ejpam-2541	53	6	+	+	NUM
ejpam-2541	53	7	g2(x2	g2(x2	NOUN
ejpam-2541	53	8	,	,	PUNCT
ejpam-2541	53	9	y2	y2	PROPN
ejpam-2541	53	10	)	)	PUNCT
ejpam-2541	53	11	where	where	SCONJ
ejpam-2541	53	12	x	x	X
ejpam-2541	53	13	i	i	PRON
ejpam-2541	53	14	,	,	PUNCT
ejpam-2541	53	15	yi	yi	PROPN
ejpam-2541	53	16	∈	∈	PROPN
ejpam-2541	53	17	ℑ(ni	ℑ(ni	NOUN
ejpam-2541	53	18	)	)	PUNCT
ejpam-2541	53	19	and	and	CCONJ
ejpam-2541	53	20	ℑ	ℑ	PROPN
ejpam-2541	53	21	denotes	denote	VERB
ejpam-2541	53	22	the	the	DET
ejpam-2541	53	23	set	set	NOUN
ejpam-2541	53	24	of	of	ADP
ejpam-2541	53	25	all	all	DET
ejpam-2541	53	26	vector	vector	NOUN
ejpam-2541	53	27	fields	field	NOUN
ejpam-2541	53	28	on	on	ADP
ejpam-2541	53	29	ni	ni	PROPN
ejpam-2541	53	30	,	,	PUNCT
ejpam-2541	53	31	i	i	NOUN
ejpam-2541	53	32	=	=	SYM
ejpam-2541	53	33	1,2	1,2	NUM
ejpam-2541	53	34	.	.	PUNCT
ejpam-2541	54	1	similarly	similarly	ADV
ejpam-2541	54	2	,	,	PUNCT
ejpam-2541	54	3	a	a	DET
ejpam-2541	54	4	riemannian	riemannian	ADJ
ejpam-2541	54	5	connection	connection	NOUN
ejpam-2541	54	6	▽	▽	NOUN
ejpam-2541	54	7	on	on	ADP
ejpam-2541	54	8	n1	n1	PROPN
ejpam-2541	54	9	×	×	PROPN
ejpam-2541	54	10	n2	n2	NOUN
ejpam-2541	54	11	wwas	wwas	AUX
ejpam-2541	54	12	given	give	VERB
ejpam-2541	54	13	by	by	ADP
ejpam-2541	54	14	[	[	X
ejpam-2541	54	15	19	19	NUM
ejpam-2541	54	16	]	]	PUNCT
ejpam-2541	54	17	▽	▽	PROPN
ejpam-2541	54	18	x	x	SYM
ejpam-2541	54	19	y	y	NOUN
ejpam-2541	54	20	=	=	NOUN
ejpam-2541	54	21	▽	▽	X
ejpam-2541	54	22	(	(	PUNCT
ejpam-2541	54	23	x1,x2	x1,x2	PROPN
ejpam-2541	54	24	)	)	PUNCT
ejpam-2541	54	25	(	(	PUNCT
ejpam-2541	54	26	y1	y1	INTJ
ejpam-2541	54	27	,	,	PUNCT
ejpam-2541	54	28	y2	y2	NOUN
ejpam-2541	54	29	)	)	PUNCT
ejpam-2541	55	1	=	=	SYM
ejpam-2541	55	2	(	(	PUNCT
ejpam-2541	55	3	▽	▽	NOUN
ejpam-2541	55	4	1	1	NUM
ejpam-2541	55	5	x1	x1	NUM
ejpam-2541	55	6	y1,	y1,	ADJ
ejpam-2541	55	7	▽	▽	ADJ
ejpam-2541	55	8	2	2	NUM
ejpam-2541	55	9	x2	x2	NOUN
ejpam-2541	55	10	y2	y2	NOUN
ejpam-2541	55	11	)	)	PUNCT
ejpam-2541	55	12	.	.	PUNCT
ejpam-2541	56	1	if	if	SCONJ
ejpam-2541	56	2	γ	γ	X
ejpam-2541	56	3	:	:	PUNCT
ejpam-2541	56	4	[	[	X
ejpam-2541	56	5	0,λ]→	0,λ]→	X
ejpam-2541	56	6	n1	n1	PROPN
ejpam-2541	56	7	×	×	PROPN
ejpam-2541	56	8	n2	n2	NOUN
ejpam-2541	56	9	is	be	AUX
ejpam-2541	56	10	a	a	DET
ejpam-2541	56	11	smooth	smooth	ADJ
ejpam-2541	56	12	curve	curve	NOUN
ejpam-2541	56	13	in	in	ADP
ejpam-2541	56	14	n1	n1	PROPN
ejpam-2541	56	15	×	×	PROPN
ejpam-2541	56	16	n2	n2	NOUN
ejpam-2541	56	17	,	,	PUNCT
ejpam-2541	56	18	then	then	ADV
ejpam-2541	56	19	the	the	DET
ejpam-2541	56	20	natural	natural	ADJ
ejpam-2541	56	21	projections	projection	NOUN
ejpam-2541	56	22	γ1	γ1	NOUN
ejpam-2541	56	23	:	:	PUNCT
ejpam-2541	57	1	[	[	X
ejpam-2541	57	2	0,λ]→	0,λ]→	X
ejpam-2541	57	3	n1	n1	PROPN
ejpam-2541	57	4	and	and	CCONJ
ejpam-2541	57	5	γ2	γ2	ADJ
ejpam-2541	57	6	:	:	PUNCT
ejpam-2541	58	1	[	[	X
ejpam-2541	58	2	0,λ]→	0,λ]→	X
ejpam-2541	58	3	n2	n2	NOUN
ejpam-2541	58	4	of	of	ADP
ejpam-2541	58	5	γ	γ	NOUN
ejpam-2541	58	6	on	on	ADP
ejpam-2541	58	7	both	both	CCONJ
ejpam-2541	58	8	n1	n1	NOUN
ejpam-2541	58	9	and	and	CCONJ
ejpam-2541	58	10	n2	n2	ADJ
ejpam-2541	58	11	,	,	PUNCT
ejpam-2541	58	12	respectively	respectively	ADV
ejpam-2541	58	13	,	,	PUNCT
ejpam-2541	58	14	are	be	AUX
ejpam-2541	58	15	smooth	smooth	ADJ
ejpam-2541	58	16	curves	curve	NOUN
ejpam-2541	58	17	.	.	PUNCT
ejpam-2541	59	1	moreover	moreover	ADV
ejpam-2541	59	2	,	,	PUNCT
ejpam-2541	59	3	γ	γ	PROPN
ejpam-2541	59	4	is	be	AUX
ejpam-2541	59	5	a	a	DET
ejpam-2541	59	6	geodesic	geodesic	NOUN
ejpam-2541	59	7	in	in	ADP
ejpam-2541	59	8	n1	n1	ADJ
ejpam-2541	59	9	×	×	PROPN
ejpam-2541	59	10	n2	n2	NOUN
ejpam-2541	59	11	if	if	SCONJ
ejpam-2541	59	12	and	and	CCONJ
ejpam-2541	59	13	only	only	ADV
ejpam-2541	59	14	if	if	SCONJ
ejpam-2541	59	15	both	both	DET
ejpam-2541	59	16	γ1	γ1	NOUN
ejpam-2541	59	17	and	and	CCONJ
ejpam-2541	59	18	γ2	γ2	PROPN
ejpam-2541	59	19	are	be	AUX
ejpam-2541	59	20	geodesics	geodesic	NOUN
ejpam-2541	59	21	in	in	ADP
ejpam-2541	59	22	n1	n1	NOUN
ejpam-2541	59	23	and	and	CCONJ
ejpam-2541	59	24	n2	n2	ADJ
ejpam-2541	59	25	,	,	PUNCT
ejpam-2541	59	26	respectively	respectively	ADV
ejpam-2541	59	27	.	.	PUNCT
ejpam-2541	60	1	which	which	PRON
ejpam-2541	60	2	means	mean	VERB
ejpam-2541	60	3	▽	▽	ADJ
ejpam-2541	60	4	γ̇γ	γ̇γ	NOUN
ejpam-2541	60	5	=	=	PRON
ejpam-2541	60	6	▽	▽	X
ejpam-2541	60	7	(	(	PUNCT
ejpam-2541	60	8	γ̇1,γ̇2	γ̇1,γ̇2	NOUN
ejpam-2541	60	9	)	)	PUNCT
ejpam-2541	60	10	(	(	PUNCT
ejpam-2541	60	11	γ̇1	γ̇1	PROPN
ejpam-2541	60	12	,	,	PUNCT
ejpam-2541	60	13	γ̇2	γ̇2	PROPN
ejpam-2541	60	14	)	)	PUNCT
ejpam-2541	61	1	=	=	SYM
ejpam-2541	61	2	(	(	PUNCT
ejpam-2541	61	3	▽	▽	NOUN
ejpam-2541	61	4	1	1	NUM
ejpam-2541	61	5	γ̇1	γ̇1	PROPN
ejpam-2541	61	6	γ̇1,	γ̇1,	NUM
ejpam-2541	61	7	▽	▽	X
ejpam-2541	61	8	2	2	NUM
ejpam-2541	61	9	γ̇2	γ̇2	PROPN
ejpam-2541	61	10	γ̇2	γ̇2	PROPN
ejpam-2541	61	11	)	)	PUNCT
ejpam-2541	61	12	,	,	PUNCT
ejpam-2541	61	13	where	where	SCONJ
ejpam-2541	61	14	γ̇	γ̇	PROPN
ejpam-2541	61	15	is	be	AUX
ejpam-2541	61	16	the	the	DET
ejpam-2541	61	17	velocity	velocity	NOUN
ejpam-2541	61	18	vector	vector	NOUN
ejpam-2541	61	19	field	field	NOUN
ejpam-2541	61	20	along	along	ADP
ejpam-2541	61	21	the	the	DET
ejpam-2541	61	22	curve	curve	NOUN
ejpam-2541	61	23	γ	γ	PROPN
ejpam-2541	61	24	.	.	PUNCT
ejpam-2541	61	25	consequently	consequently	ADV
ejpam-2541	61	26	,	,	PUNCT
ejpam-2541	61	27	▽	▽	NOUN
ejpam-2541	61	28	γ̇γ̇=	γ̇γ̇=	X
ejpam-2541	61	29	0	0	NUM
ejpam-2541	61	30	if	if	SCONJ
ejpam-2541	61	31	and	and	CCONJ
ejpam-2541	61	32	only	only	ADV
ejpam-2541	61	33	if	if	SCONJ
ejpam-2541	61	34	▽	▽	NOUN
ejpam-2541	61	35	i	i	PRON
ejpam-2541	61	36	γ̇i	γ̇i	ADJ
ejpam-2541	61	37	γ̇i	γ̇i	NOUN
ejpam-2541	61	38	=	=	SYM
ejpam-2541	61	39	0	0	NUM
ejpam-2541	62	1	for	for	ADP
ejpam-2541	62	2	i	i	PRON
ejpam-2541	62	3	=	=	SYM
ejpam-2541	62	4	1,2	1,2	NUM
ejpam-2541	62	5	,	,	PUNCT
ejpam-2541	62	6	see	see	VERB
ejpam-2541	62	7	[	[	X
ejpam-2541	62	8	3	3	NUM
ejpam-2541	62	9	]	]	PUNCT
ejpam-2541	62	10	.	.	PUNCT
ejpam-2541	63	1	let	let	VERB
ejpam-2541	63	2	w1	w1	NOUN
ejpam-2541	63	3	and	and	CCONJ
ejpam-2541	63	4	w2	w2	NOUN
ejpam-2541	63	5	be	be	AUX
ejpam-2541	63	6	c∞	c∞	PROPN
ejpam-2541	63	7	complete	complete	ADJ
ejpam-2541	63	8	,	,	PUNCT
ejpam-2541	63	9	simply	simply	ADV
ejpam-2541	63	10	connected	connected	ADJ
ejpam-2541	63	11	riemannian	riemannian	ADJ
ejpam-2541	63	12	manifolds	manifold	NOUN
ejpam-2541	63	13	without	without	ADP
ejpam-2541	63	14	conjugate	conjugate	ADJ
ejpam-2541	63	15	points	point	NOUN
ejpam-2541	63	16	,	,	PUNCT
ejpam-2541	63	17	then	then	ADV
ejpam-2541	63	18	w1	w1	NOUN
ejpam-2541	63	19	×w2	×w2	NOUN
ejpam-2541	63	20	is	be	AUX
ejpam-2541	63	21	also	also	ADV
ejpam-2541	63	22	a	a	DET
ejpam-2541	63	23	c∞	c∞	PROPN
ejpam-2541	63	24	complete	complete	ADJ
ejpam-2541	63	25	,	,	PUNCT
ejpam-2541	63	26	simply	simply	ADV
ejpam-2541	63	27	connected	connected	ADJ
ejpam-2541	63	28	riemannian	riemannian	NOUN
ejpam-2541	63	29	manifold	manifold	ADJ
ejpam-2541	63	30	without	without	ADP
ejpam-2541	63	31	conjugate	conjugate	ADJ
ejpam-2541	63	32	points	point	NOUN
ejpam-2541	63	33	.	.	PUNCT
ejpam-2541	64	1	notice	notice	VERB
ejpam-2541	64	2	that	that	SCONJ
ejpam-2541	64	3	dim(w1	dim(w1	PROPN
ejpam-2541	64	4	×w2	×w2	NOUN
ejpam-2541	64	5	)	)	PUNCT
ejpam-2541	64	6	=	=	SYM
ejpam-2541	64	7	dim(w1	dim(w1	X
ejpam-2541	64	8	)	)	PUNCT
ejpam-2541	65	1	+	+	SYM
ejpam-2541	65	2	dim(w2	dim(w2	NOUN
ejpam-2541	65	3	)	)	PUNCT
ejpam-2541	65	4	.	.	PUNCT
ejpam-2541	66	1	consequently	consequently	ADV
ejpam-2541	66	2	,	,	PUNCT
ejpam-2541	66	3	each	each	DET
ejpam-2541	66	4	pair	pair	NOUN
ejpam-2541	66	5	of	of	ADP
ejpam-2541	66	6	different	different	ADJ
ejpam-2541	66	7	points	point	NOUN
ejpam-2541	66	8	(	(	PUNCT
ejpam-2541	66	9	p1	p1	NOUN
ejpam-2541	66	10	,	,	PUNCT
ejpam-2541	66	11	p2	p2	PROPN
ejpam-2541	66	12	)	)	PUNCT
ejpam-2541	66	13	and	and	CCONJ
ejpam-2541	66	14	(	(	PUNCT
ejpam-2541	66	15	q1,q2	q1,q2	PROPN
ejpam-2541	66	16	)	)	PUNCT
ejpam-2541	66	17	in	in	ADP
ejpam-2541	66	18	w1×w2	w1×w2	PROPN
ejpam-2541	66	19	are	be	AUX
ejpam-2541	66	20	joined	join	VERB
ejpam-2541	66	21	by	by	ADP
ejpam-2541	66	22	a	a	DET
ejpam-2541	66	23	unique	unique	ADJ
ejpam-2541	66	24	geodesic	geodesic	NOUN
ejpam-2541	66	25	γ	γ	NOUN
ejpam-2541	66	26	.	.	PUNCT
ejpam-2541	67	1	this	this	DET
ejpam-2541	67	2	segment	segment	NOUN
ejpam-2541	67	3	when	when	SCONJ
ejpam-2541	67	4	naturally	naturally	ADV
ejpam-2541	67	5	projected	project	VERB
ejpam-2541	67	6	on	on	ADP
ejpam-2541	67	7	w1	w1	NOUN
ejpam-2541	67	8	and	and	CCONJ
ejpam-2541	67	9	w2	w2	NOUN
ejpam-2541	67	10	yields	yield	VERB
ejpam-2541	67	11	two	two	NUM
ejpam-2541	67	12	geodesic	geodesic	ADJ
ejpam-2541	67	13	segments	segment	NOUN
ejpam-2541	67	14	γi	γi	ADP
ejpam-2541	67	15	⊂	⊂	PROPN
ejpam-2541	67	16	wi	wi	PROPN
ejpam-2541	67	17	joining	join	VERB
ejpam-2541	67	18	pi	pi	NOUN
ejpam-2541	67	19	and	and	CCONJ
ejpam-2541	67	20	qi	qi	PROPN
ejpam-2541	67	21	,	,	PUNCT
ejpam-2541	67	22	i	i	PRON
ejpam-2541	67	23	=	=	NOUN
ejpam-2541	67	24	1,2	1,2	NUM
ejpam-2541	67	25	each	each	DET
ejpam-2541	67	26	one	one	NOUN
ejpam-2541	67	27	is	be	AUX
ejpam-2541	67	28	unique	unique	ADJ
ejpam-2541	67	29	in	in	ADP
ejpam-2541	67	30	its	its	PRON
ejpam-2541	67	31	own	own	ADJ
ejpam-2541	67	32	manifolds.the	manifolds.the	DET
ejpam-2541	67	33	natural	natural	ADJ
ejpam-2541	67	34	projection	projection	NOUN
ejpam-2541	67	35	will	will	AUX
ejpam-2541	67	36	be	be	AUX
ejpam-2541	67	37	denoted	denote	VERB
ejpam-2541	67	38	by	by	ADP
ejpam-2541	67	39	ηi	ηi	PROPN
ejpam-2541	67	40	:	:	PUNCT
ejpam-2541	67	41	w1	w1	NOUN
ejpam-2541	67	42	×w2→wi	×w2→wi	PROPN
ejpam-2541	67	43	where	where	SCONJ
ejpam-2541	67	44	ηi(p1	ηi(p1	NOUN
ejpam-2541	67	45	,	,	PUNCT
ejpam-2541	67	46	p2	p2	X
ejpam-2541	67	47	)	)	PUNCT
ejpam-2541	67	48	=	=	SYM
ejpam-2541	68	1	pi	pi	NOUN
ejpam-2541	68	2	,	,	PUNCT
ejpam-2541	68	3	i	i	NOUN
ejpam-2541	68	4	=	=	NOUN
ejpam-2541	68	5	1,2	1,2	NUM
ejpam-2541	68	6	see[3	see[3	NUM
ejpam-2541	68	7	]	]	PUNCT
ejpam-2541	68	8	.	.	PUNCT
ejpam-2541	69	1	the	the	DET
ejpam-2541	69	2	following	follow	VERB
ejpam-2541	69	3	propositions	proposition	NOUN
ejpam-2541	69	4	were	be	AUX
ejpam-2541	69	5	proved	prove	VERB
ejpam-2541	69	6	in	in	ADP
ejpam-2541	69	7	[	[	X
ejpam-2541	69	8	3	3	NUM
ejpam-2541	69	9	]	]	PUNCT
ejpam-2541	69	10	:	:	PUNCT
ejpam-2541	69	11	proposition	proposition	NOUN
ejpam-2541	69	12	1	1	X
ejpam-2541	69	13	.	.	PUNCT
ejpam-2541	70	1	let	let	VERB
ejpam-2541	70	2	a1	a1	PROPN
ejpam-2541	70	3	⊂w1	⊂w1	PROPN
ejpam-2541	70	4	and	and	CCONJ
ejpam-2541	70	5	a2	a2	PROPN
ejpam-2541	70	6	⊂w2	⊂w2	PROPN
ejpam-2541	70	7	be	be	AUX
ejpam-2541	70	8	subsets	subset	NOUN
ejpam-2541	70	9	of	of	ADP
ejpam-2541	70	10	w1	w1	NOUN
ejpam-2541	70	11	and	and	CCONJ
ejpam-2541	70	12	w2	w2	NOUN
ejpam-2541	70	13	.	.	PUNCT
ejpam-2541	71	1	then	then	ADV
ejpam-2541	71	2	,	,	PUNCT
ejpam-2541	71	3	a1	a1	PROPN
ejpam-2541	71	4	×a2	×a2	PROPN
ejpam-2541	71	5	⊂w1	⊂w1	NOUN
ejpam-2541	71	6	×w2	×w2	NOUN
ejpam-2541	71	7	is	be	AUX
ejpam-2541	71	8	convex	convex	ADJ
ejpam-2541	71	9	if	if	SCONJ
ejpam-2541	71	10	and	and	CCONJ
ejpam-2541	71	11	only	only	ADV
ejpam-2541	71	12	if	if	SCONJ
ejpam-2541	71	13	both	both	PRON
ejpam-2541	71	14	a1	a1	NOUN
ejpam-2541	71	15	and	and	CCONJ
ejpam-2541	71	16	a2	a2	PROPN
ejpam-2541	71	17	are	be	AUX
ejpam-2541	71	18	convex	convex	ADJ
ejpam-2541	71	19	.	.	PUNCT
ejpam-2541	72	1	proposition	proposition	NOUN
ejpam-2541	72	2	2	2	NUM
ejpam-2541	72	3	.	.	PUNCT
ejpam-2541	73	1	let	let	VERB
ejpam-2541	73	2	a1	a1	PROPN
ejpam-2541	73	3	⊂w1	⊂w1	PROPN
ejpam-2541	73	4	and	and	CCONJ
ejpam-2541	73	5	a2	a2	PROPN
ejpam-2541	73	6	⊂w2	⊂w2	PROPN
ejpam-2541	73	7	be	be	AUX
ejpam-2541	73	8	two	two	NUM
ejpam-2541	73	9	subsets	subset	NOUN
ejpam-2541	73	10	.	.	PUNCT
ejpam-2541	74	1	then	then	ADV
ejpam-2541	74	2	,	,	PUNCT
ejpam-2541	74	3	(	(	PUNCT
ejpam-2541	74	4	i	i	NOUN
ejpam-2541	74	5	)	)	PUNCT
ejpam-2541	74	6	a1	a1	PROPN
ejpam-2541	74	7	×	×	PROPN
ejpam-2541	74	8	a2	a2	PROPN
ejpam-2541	74	9	⊂w1	⊂w1	PROPN
ejpam-2541	74	10	×w2	×w2	PROPN
ejpam-2541	74	11	is	be	AUX
ejpam-2541	74	12	starshaped	starshape	VERB
ejpam-2541	74	13	if	if	SCONJ
ejpam-2541	74	14	and	and	CCONJ
ejpam-2541	74	15	only	only	ADV
ejpam-2541	74	16	if	if	SCONJ
ejpam-2541	74	17	both	both	PRON
ejpam-2541	74	18	a1	a1	NOUN
ejpam-2541	74	19	and	and	CCONJ
ejpam-2541	74	20	a2	a2	NOUN
ejpam-2541	74	21	are	be	AUX
ejpam-2541	74	22	starshaped	starshape	VERB
ejpam-2541	74	23	.	.	PUNCT
ejpam-2541	75	1	(	(	PUNCT
ejpam-2541	75	2	ii	ii	X
ejpam-2541	75	3	)	)	PUNCT
ejpam-2541	75	4	ker(a1	ker(a1	PROPN
ejpam-2541	75	5	×	×	PROPN
ejpam-2541	75	6	a2	a2	PROPN
ejpam-2541	75	7	)	)	PUNCT
ejpam-2541	75	8	=	=	PUNCT
ejpam-2541	75	9	(	(	PUNCT
ejpam-2541	75	10	kera1)×	kera1)×	PROPN
ejpam-2541	75	11	(	(	PUNCT
ejpam-2541	75	12	kera2	kera2	PROPN
ejpam-2541	75	13	)	)	PUNCT
ejpam-2541	75	14	3	3	NUM
ejpam-2541	75	15	.	.	X
ejpam-2541	75	16	convexity	convexity	NOUN
ejpam-2541	75	17	in	in	ADP
ejpam-2541	75	18	riemannian	riemannian	ADJ
ejpam-2541	75	19	manifolds	manifold	NOUN
ejpam-2541	75	20	product	product	NOUN
ejpam-2541	75	21	in	in	ADP
ejpam-2541	75	22	this	this	DET
ejpam-2541	75	23	section	section	NOUN
ejpam-2541	75	24	,	,	PUNCT
ejpam-2541	75	25	we	we	PRON
ejpam-2541	75	26	study	study	VERB
ejpam-2541	75	27	some	some	DET
ejpam-2541	75	28	properties	property	NOUN
ejpam-2541	75	29	of	of	ADP
ejpam-2541	75	30	convexity	convexity	NOUN
ejpam-2541	75	31	in	in	ADP
ejpam-2541	75	32	riemannian	riemannian	ADJ
ejpam-2541	75	33	manifolds	manifold	NOUN
ejpam-2541	75	34	product	product	NOUN
ejpam-2541	75	35	.	.	PUNCT
ejpam-2541	76	1	proposition	proposition	NOUN
ejpam-2541	76	2	3	3	NUM
ejpam-2541	76	3	.	.	PUNCT
ejpam-2541	77	1	the	the	DET
ejpam-2541	77	2	intersection	intersection	NOUN
ejpam-2541	77	3	of	of	ADP
ejpam-2541	77	4	any	any	DET
ejpam-2541	77	5	number	number	NOUN
ejpam-2541	77	6	of	of	ADP
ejpam-2541	77	7	product	product	NOUN
ejpam-2541	77	8	convex	convex	NOUN
ejpam-2541	77	9	subsets	subset	NOUN
ejpam-2541	77	10	is	be	AUX
ejpam-2541	77	11	convex	convex	NOUN
ejpam-2541	77	12	subset	subset	NOUN
ejpam-2541	77	13	.	.	PUNCT
ejpam-2541	78	1	w.	w.	PROPN
ejpam-2541	78	2	saleh	saleh	PROPN
ejpam-2541	78	3	,	,	PUNCT
ejpam-2541	78	4	a.	a.	NOUN
ejpam-2541	78	5	kiliccman	kiliccman	PROPN
ejpam-2541	78	6	/	/	SYM
ejpam-2541	78	7	eur	eur	PROPN
ejpam-2541	78	8	.	.	PUNCT
ejpam-2541	79	1	j.	j.	PROPN
ejpam-2541	79	2	pure	pure	PROPN
ejpam-2541	79	3	appl	appl	PROPN
ejpam-2541	79	4	.	.	PROPN
ejpam-2541	79	5	math	math	PROPN
ejpam-2541	79	6	,	,	PUNCT
ejpam-2541	79	7	9	9	NUM
ejpam-2541	79	8	(	(	PUNCT
ejpam-2541	79	9	2016	2016	NUM
ejpam-2541	79	10	)	)	PUNCT
ejpam-2541	79	11	,	,	PUNCT
ejpam-2541	79	12	57	57	NUM
ejpam-2541	79	13	-	-	SYM
ejpam-2541	79	14	63	63	NUM
ejpam-2541	79	15	60	60	NUM
ejpam-2541	79	16	proof	proof	NOUN
ejpam-2541	79	17	.	.	PUNCT
ejpam-2541	80	1	let	let	VERB
ejpam-2541	80	2	a1	a1	NOUN
ejpam-2541	80	3	,	,	PUNCT
ejpam-2541	80	4	a2	a2	PROPN
ejpam-2541	80	5	,	,	PUNCT
ejpam-2541	80	6	b1	b1	NOUN
ejpam-2541	80	7	,	,	PUNCT
ejpam-2541	80	8	and	and	CCONJ
ejpam-2541	80	9	b2	b2	NOUN
ejpam-2541	80	10	be	be	VERB
ejpam-2541	80	11	convex	convex	ADJ
ejpam-2541	80	12	subsets	subset	NOUN
ejpam-2541	80	13	,	,	PUNCT
ejpam-2541	80	14	then	then	ADV
ejpam-2541	80	15	a1	a1	NOUN
ejpam-2541	80	16	×	×	PROPN
ejpam-2541	80	17	a2	a2	PROPN
ejpam-2541	80	18	,	,	PUNCT
ejpam-2541	80	19	and	and	CCONJ
ejpam-2541	80	20	b1	b1	NOUN
ejpam-2541	80	21	×	×	PROPN
ejpam-2541	80	22	b2	b2	NOUN
ejpam-2541	80	23	are	be	AUX
ejpam-2541	80	24	convex	convex	ADJ
ejpam-2541	80	25	subsets	subset	NOUN
ejpam-2541	80	26	.	.	PUNCT
ejpam-2541	81	1	we	we	PRON
ejpam-2541	81	2	know	know	VERB
ejpam-2541	81	3	that	that	SCONJ
ejpam-2541	81	4	(	(	PUNCT
ejpam-2541	81	5	a1×a2)∩	a1×a2)∩	ADV
ejpam-2541	81	6	(	(	PUNCT
ejpam-2541	81	7	b1×b2	b1×b2	PROPN
ejpam-2541	81	8	)	)	PUNCT
ejpam-2541	81	9	=	=	SYM
ejpam-2541	81	10	(	(	PUNCT
ejpam-2541	81	11	a1∩b1)×	a1∩b1)×	PROPN
ejpam-2541	81	12	(	(	PUNCT
ejpam-2541	81	13	a2∩b2	a2∩b2	NOUN
ejpam-2541	81	14	)	)	PUNCT
ejpam-2541	81	15	.	.	PUNCT
ejpam-2541	82	1	since	since	SCONJ
ejpam-2541	82	2	a1∩b1	a1∩b1	PROPN
ejpam-2541	82	3	,	,	PUNCT
ejpam-2541	82	4	and	and	CCONJ
ejpam-2541	82	5	a2∩b2	a2∩b2	NOUN
ejpam-2541	82	6	are	be	AUX
ejpam-2541	82	7	convex	convex	ADJ
ejpam-2541	82	8	,	,	PUNCT
ejpam-2541	82	9	then	then	ADV
ejpam-2541	82	10	(	(	PUNCT
ejpam-2541	82	11	a1	a1	NOUN
ejpam-2541	82	12	∩	∩	X
ejpam-2541	82	13	b1)×	b1)×	X
ejpam-2541	82	14	(	(	PUNCT
ejpam-2541	82	15	a2	a2	PROPN
ejpam-2541	82	16	∩	∩	ADJ
ejpam-2541	82	17	b2	b2	NOUN
ejpam-2541	82	18	)	)	PUNCT
ejpam-2541	82	19	is	be	AUX
ejpam-2541	82	20	also	also	ADV
ejpam-2541	82	21	convex	convex	ADJ
ejpam-2541	82	22	.	.	PUNCT
ejpam-2541	83	1	therefore	therefore	ADV
ejpam-2541	83	2	,	,	PUNCT
ejpam-2541	83	3	the	the	DET
ejpam-2541	83	4	proof	proof	NOUN
ejpam-2541	83	5	is	be	AUX
ejpam-2541	83	6	complete	complete	ADJ
ejpam-2541	83	7	.	.	PUNCT
ejpam-2541	84	1	remark	remark	NOUN
ejpam-2541	84	2	2	2	NUM
ejpam-2541	84	3	.	.	PUNCT
ejpam-2541	85	1	the	the	DET
ejpam-2541	85	2	above	above	ADJ
ejpam-2541	85	3	proposition	proposition	NOUN
ejpam-2541	85	4	is	be	AUX
ejpam-2541	85	5	not	not	PART
ejpam-2541	85	6	true	true	ADJ
ejpam-2541	85	7	in	in	ADP
ejpam-2541	85	8	general	general	ADJ
ejpam-2541	85	9	for	for	ADP
ejpam-2541	85	10	the	the	DET
ejpam-2541	85	11	union	union	NOUN
ejpam-2541	85	12	of	of	ADP
ejpam-2541	85	13	subsets	subset	NOUN
ejpam-2541	85	14	of	of	ADP
ejpam-2541	85	15	w1	w1	NOUN
ejpam-2541	85	16	×w2	×w2	PROPN
ejpam-2541	85	17	.	.	PUNCT
ejpam-2541	86	1	theorem	theorem	NOUN
ejpam-2541	86	2	4	4	NUM
ejpam-2541	86	3	.	.	PUNCT
ejpam-2541	87	1	let	let	VERB
ejpam-2541	87	2	a1	a1	PROPN
ejpam-2541	87	3	⊂	⊂	PROPN
ejpam-2541	87	4	w1	w1	PROPN
ejpam-2541	87	5	and	and	CCONJ
ejpam-2541	87	6	a2	a2	PROPN
ejpam-2541	87	7	⊂	⊂	PROPN
ejpam-2541	87	8	w2	w2	PROPN
ejpam-2541	87	9	be	be	AUX
ejpam-2541	87	10	an	an	DET
ejpam-2541	87	11	open	open	ADJ
ejpam-2541	87	12	convex	convex	NOUN
ejpam-2541	87	13	subsets	subset	NOUN
ejpam-2541	87	14	,	,	PUNCT
ejpam-2541	87	15	then	then	ADV
ejpam-2541	87	16	the	the	DET
ejpam-2541	87	17	closure	closure	NOUN
ejpam-2541	87	18	a1	a1	NOUN
ejpam-2541	87	19	×	×	PROPN
ejpam-2541	87	20	a2	a2	PROPN
ejpam-2541	87	21	is	be	AUX
ejpam-2541	87	22	also	also	ADV
ejpam-2541	87	23	convex	convex	ADJ
ejpam-2541	87	24	.	.	PUNCT
ejpam-2541	88	1	proof	proof	NOUN
ejpam-2541	88	2	.	.	PUNCT
ejpam-2541	89	1	assume	assume	VERB
ejpam-2541	89	2	that	that	SCONJ
ejpam-2541	89	3	both	both	PRON
ejpam-2541	89	4	a1	a1	PROPN
ejpam-2541	89	5	⊂	⊂	PROPN
ejpam-2541	89	6	w1	w1	PROPN
ejpam-2541	89	7	and	and	CCONJ
ejpam-2541	89	8	a2	a2	PROPN
ejpam-2541	89	9	⊂	⊂	PROPN
ejpam-2541	89	10	w2	w2	PROPN
ejpam-2541	89	11	are	be	AUX
ejpam-2541	89	12	convex	convex	ADJ
ejpam-2541	89	13	subsets	subset	NOUN
ejpam-2541	89	14	.	.	PUNCT
ejpam-2541	90	1	then	then	ADV
ejpam-2541	90	2	,	,	PUNCT
ejpam-2541	90	3	ā1	ā1	PROPN
ejpam-2541	90	4	⊂	⊂	PROPN
ejpam-2541	90	5	w1	w1	PROPN
ejpam-2541	90	6	and	and	CCONJ
ejpam-2541	90	7	ā2	ā2	NUM
ejpam-2541	90	8	⊂w2	⊂w2	NOUN
ejpam-2541	90	9	are	be	AUX
ejpam-2541	90	10	convex	convex	PROPN
ejpam-2541	90	11	,	,	PUNCT
ejpam-2541	90	12	which	which	PRON
ejpam-2541	90	13	means	mean	VERB
ejpam-2541	90	14	that	that	SCONJ
ejpam-2541	90	15	ā1	ā1	ADJ
ejpam-2541	90	16	×	×	NOUN
ejpam-2541	90	17	ā2	ā2	PRON
ejpam-2541	90	18	is	be	AUX
ejpam-2541	90	19	convex	convex	ADJ
ejpam-2541	90	20	.	.	PUNCT
ejpam-2541	91	1	then	then	ADV
ejpam-2541	91	2	,	,	PUNCT
ejpam-2541	91	3	a1	a1	PROPN
ejpam-2541	91	4	×	×	PROPN
ejpam-2541	91	5	a2	a2	PROPN
ejpam-2541	91	6	is	be	AUX
ejpam-2541	91	7	convex	convex	PROPN
ejpam-2541	91	8	.	.	PUNCT
ejpam-2541	92	1	theorem	theorem	ADJ
ejpam-2541	92	2	5	5	NUM
ejpam-2541	92	3	.	.	PUNCT
ejpam-2541	93	1	let	let	VERB
ejpam-2541	93	2	a1	a1	PROPN
ejpam-2541	93	3	⊂w1	⊂w1	PROPN
ejpam-2541	93	4	and	and	CCONJ
ejpam-2541	93	5	a2	a2	PROPN
ejpam-2541	93	6	⊂w2	⊂w2	NOUN
ejpam-2541	93	7	be	be	AUX
ejpam-2541	93	8	convex	convex	ADJ
ejpam-2541	93	9	subsets	subset	NOUN
ejpam-2541	93	10	,	,	PUNCT
ejpam-2541	93	11	then	then	ADV
ejpam-2541	93	12	the	the	DET
ejpam-2541	93	13	interior	interior	NOUN
ejpam-2541	93	14	of	of	ADP
ejpam-2541	93	15	a1	a1	PROPN
ejpam-2541	93	16	×	×	PROPN
ejpam-2541	93	17	a2	a2	PROPN
ejpam-2541	93	18	(	(	PUNCT
ejpam-2541	93	19	int(a1)×	int(a1)×	PROPN
ejpam-2541	93	20	int(a2	int(a2	PROPN
ejpam-2541	93	21	)	)	PUNCT
ejpam-2541	93	22	)	)	PUNCT
ejpam-2541	93	23	is	be	AUX
ejpam-2541	93	24	also	also	ADV
ejpam-2541	93	25	convex	convex	ADJ
ejpam-2541	93	26	.	.	PUNCT
ejpam-2541	94	1	proof	proof	NOUN
ejpam-2541	94	2	.	.	PUNCT
ejpam-2541	95	1	the	the	DET
ejpam-2541	95	2	proof	proof	NOUN
ejpam-2541	95	3	is	be	AUX
ejpam-2541	95	4	direct	direct	ADJ
ejpam-2541	95	5	in	in	ADP
ejpam-2541	95	6	the	the	DET
ejpam-2541	95	7	light	light	NOUN
ejpam-2541	95	8	of	of	ADP
ejpam-2541	95	9	theorem	theorem	ADJ
ejpam-2541	95	10	1	1	NUM
ejpam-2541	95	11	.	.	PUNCT
ejpam-2541	95	12	notice	notice	VERB
ejpam-2541	95	13	that	that	SCONJ
ejpam-2541	95	14	if	if	SCONJ
ejpam-2541	95	15	ai	ai	VERB
ejpam-2541	95	16	⊂	⊂	PROPN
ejpam-2541	95	17	wi	wi	PROPN
ejpam-2541	95	18	,	,	PUNCT
ejpam-2541	95	19	i	i	PRON
ejpam-2541	95	20	=	=	NOUN
ejpam-2541	96	1	1,2	1,2	NUM
ejpam-2541	96	2	is	be	AUX
ejpam-2541	96	3	an	an	DET
ejpam-2541	96	4	open	open	ADJ
ejpam-2541	96	5	subset	subset	NOUN
ejpam-2541	96	6	such	such	ADJ
ejpam-2541	96	7	that	that	DET
ejpam-2541	96	8	a1	a1	NOUN
ejpam-2541	96	9	×	×	PROPN
ejpam-2541	96	10	a2	a2	PROPN
ejpam-2541	96	11	is	be	AUX
ejpam-2541	96	12	convex	convex	PROPN
ejpam-2541	96	13	,	,	PUNCT
ejpam-2541	96	14	then	then	ADV
ejpam-2541	96	15	ai	ai	VERB
ejpam-2541	96	16	,	,	PUNCT
ejpam-2541	96	17	i	i	PRON
ejpam-2541	96	18	=	=	NOUN
ejpam-2541	96	19	1,2	1,2	NUM
ejpam-2541	96	20	is	be	AUX
ejpam-2541	96	21	not	not	PART
ejpam-2541	96	22	necessarily	necessarily	ADV
ejpam-2541	96	23	convex	convex	ADJ
ejpam-2541	96	24	.	.	PUNCT
ejpam-2541	97	1	the	the	DET
ejpam-2541	97	2	following	following	ADJ
ejpam-2541	97	3	example	example	NOUN
ejpam-2541	97	4	indicates	indicate	VERB
ejpam-2541	97	5	this	this	DET
ejpam-2541	97	6	claim	claim	NOUN
ejpam-2541	97	7	.	.	PUNCT
ejpam-2541	98	1	example	example	NOUN
ejpam-2541	99	1	1	1	NUM
ejpam-2541	99	2	.	.	PUNCT
ejpam-2541	99	3	let	let	VERB
ejpam-2541	99	4	a1	a1	NOUN
ejpam-2541	99	5	=	=	SYM
ejpam-2541	99	6	s1	s1	NOUN
ejpam-2541	99	7	=	=	SYM
ejpam-2541	99	8	{	{	PUNCT
ejpam-2541	99	9	(	(	PUNCT
ejpam-2541	99	10	x	x	INTJ
ejpam-2541	99	11	,	,	PUNCT
ejpam-2541	99	12	y	y	PROPN
ejpam-2541	99	13	)	)	PUNCT
ejpam-2541	99	14	:	:	PUNCT
ejpam-2541	100	1	x2	x2	X
ejpam-2541	100	2	+	+	CCONJ
ejpam-2541	100	3	y2	y2	ADJ
ejpam-2541	100	4	≤	≤	NOUN
ejpam-2541	100	5	1}\{(0,0	1}\{(0,0	NOUN
ejpam-2541	100	6	)	)	PUNCT
ejpam-2541	100	7	}	}	PUNCT
ejpam-2541	100	8	and	and	CCONJ
ejpam-2541	100	9	a2	a2	PROPN
ejpam-2541	100	10	=	=	PUNCT
ejpam-2541	101	1	[	[	X
ejpam-2541	101	2	0,1]\{12	0,1]\{12	NUM
ejpam-2541	101	3	}	}	PUNCT
ejpam-2541	101	4	.	.	PUNCT
ejpam-2541	102	1	clearly	clearly	ADV
ejpam-2541	102	2	a1	a1	VERB
ejpam-2541	102	3	×	×	PROPN
ejpam-2541	102	4	a2	a2	PROPN
ejpam-2541	102	5	is	be	AUX
ejpam-2541	102	6	a	a	DET
ejpam-2541	102	7	convex	convex	NOUN
ejpam-2541	102	8	subset	subset	NOUN
ejpam-2541	102	9	of	of	ADP
ejpam-2541	102	10	r3	r3	PROPN
ejpam-2541	102	11	while	while	SCONJ
ejpam-2541	102	12	a1	a1	NOUN
ejpam-2541	102	13	and	and	CCONJ
ejpam-2541	102	14	a2	a2	PROPN
ejpam-2541	102	15	are	be	AUX
ejpam-2541	102	16	non	non	ADJ
ejpam-2541	102	17	-	-	ADJ
ejpam-2541	102	18	convex	convex	ADJ
ejpam-2541	102	19	.	.	PUNCT
ejpam-2541	103	1	the	the	DET
ejpam-2541	103	2	relationship	relationship	NOUN
ejpam-2541	103	3	between	between	ADP
ejpam-2541	103	4	global	global	ADJ
ejpam-2541	103	5	supporting	supporting	NOUN
ejpam-2541	103	6	and	and	CCONJ
ejpam-2541	103	7	convexity	convexity	NOUN
ejpam-2541	103	8	in	in	ADP
ejpam-2541	103	9	the	the	DET
ejpam-2541	103	10	cartesian	cartesian	ADJ
ejpam-2541	103	11	product	product	NOUN
ejpam-2541	103	12	of	of	ADP
ejpam-2541	103	13	riemannian	riemannian	ADJ
ejpam-2541	103	14	manifolds	manifold	NOUN
ejpam-2541	103	15	without	without	ADP
ejpam-2541	103	16	conjugate	conjugate	ADJ
ejpam-2541	103	17	points	point	NOUN
ejpam-2541	103	18	is	be	AUX
ejpam-2541	103	19	given	give	VERB
ejpam-2541	103	20	in	in	ADP
ejpam-2541	103	21	the	the	DET
ejpam-2541	103	22	following	following	NOUN
ejpam-2541	103	23	theorem	theorem	NOUN
ejpam-2541	103	24	:	:	PUNCT
ejpam-2541	103	25	theorem	theorem	NOUN
ejpam-2541	103	26	6	6	NUM
ejpam-2541	103	27	.	.	PUNCT
ejpam-2541	104	1	let	let	VERB
ejpam-2541	104	2	a1	a1	PROPN
ejpam-2541	104	3	⊂w1	⊂w1	PROPN
ejpam-2541	104	4	and	and	CCONJ
ejpam-2541	104	5	a2	a2	PROPN
ejpam-2541	104	6	⊂w2	⊂w2	NOUN
ejpam-2541	104	7	be	be	AUX
ejpam-2541	104	8	open	open	ADJ
ejpam-2541	104	9	subsets	subset	NOUN
ejpam-2541	104	10	whose	whose	DET
ejpam-2541	104	11	boundary	boundary	ADJ
ejpam-2541	104	12	∂	∂	NOUN
ejpam-2541	104	13	a1	a1	NOUN
ejpam-2541	104	14	and	and	CCONJ
ejpam-2541	104	15	∂	∂	NUM
ejpam-2541	104	16	a2	a2	NOUN
ejpam-2541	104	17	are	be	AUX
ejpam-2541	104	18	smooth	smooth	ADJ
ejpam-2541	104	19	hypersurface	hypersurface	NOUN
ejpam-2541	104	20	,	,	PUNCT
ejpam-2541	104	21	respectively	respectively	ADV
ejpam-2541	104	22	.	.	PUNCT
ejpam-2541	105	1	then	then	ADV
ejpam-2541	105	2	,	,	PUNCT
ejpam-2541	105	3	a	a	DET
ejpam-2541	105	4	=	=	NOUN
ejpam-2541	105	5	a1	a1	NOUN
ejpam-2541	105	6	×	×	PROPN
ejpam-2541	105	7	a2	a2	PROPN
ejpam-2541	105	8	⊂	⊂	PROPN
ejpam-2541	105	9	w1	w1	NOUN
ejpam-2541	105	10	×w2	×w2	PROPN
ejpam-2541	105	11	is	be	AUX
ejpam-2541	105	12	convex	convex	ADJ
ejpam-2541	105	13	if	if	SCONJ
ejpam-2541	105	14	and	and	CCONJ
ejpam-2541	105	15	only	only	ADV
ejpam-2541	105	16	if	if	SCONJ
ejpam-2541	105	17	a1	a1	NOUN
ejpam-2541	105	18	and	and	CCONJ
ejpam-2541	105	19	a2	a2	PROPN
ejpam-2541	105	20	are	be	AUX
ejpam-2541	105	21	globally	globally	ADV
ejpam-2541	105	22	supported	support	VERB
ejpam-2541	105	23	at	at	ADP
ejpam-2541	105	24	each	each	DET
ejpam-2541	105	25	boundary	boundary	ADJ
ejpam-2541	105	26	point	point	NOUN
ejpam-2541	105	27	.	.	PUNCT
ejpam-2541	106	1	proof	proof	NOUN
ejpam-2541	106	2	.	.	PUNCT
ejpam-2541	107	1	let	let	VERB
ejpam-2541	107	2	a1	a1	NOUN
ejpam-2541	107	3	and	and	CCONJ
ejpam-2541	107	4	a2	a2	PROPN
ejpam-2541	107	5	be	be	AUX
ejpam-2541	107	6	globally	globally	ADV
ejpam-2541	107	7	supported	support	VERB
ejpam-2541	107	8	at	at	ADP
ejpam-2541	107	9	each	each	DET
ejpam-2541	107	10	boundary	boundary	ADJ
ejpam-2541	107	11	point	point	NOUN
ejpam-2541	107	12	,	,	PUNCT
ejpam-2541	107	13	then	then	ADV
ejpam-2541	107	14	by	by	ADP
ejpam-2541	107	15	using	use	VERB
ejpam-2541	107	16	theorem	theorem	ADJ
ejpam-2541	107	17	2	2	NUM
ejpam-2541	107	18	we	we	PRON
ejpam-2541	107	19	have	have	AUX
ejpam-2541	107	20	that	that	DET
ejpam-2541	107	21	a1	a1	NOUN
ejpam-2541	107	22	and	and	CCONJ
ejpam-2541	107	23	a2	a2	PROPN
ejpam-2541	107	24	are	be	AUX
ejpam-2541	107	25	convex	convex	ADJ
ejpam-2541	107	26	which	which	PRON
ejpam-2541	107	27	implies	imply	VERB
ejpam-2541	107	28	that	that	SCONJ
ejpam-2541	107	29	a1×a2	a1×a2	PROPN
ejpam-2541	107	30	is	be	AUX
ejpam-2541	107	31	convex	convex	ADJ
ejpam-2541	107	32	.	.	PUNCT
ejpam-2541	108	1	now	now	ADV
ejpam-2541	108	2	,	,	PUNCT
ejpam-2541	108	3	let	let	VERB
ejpam-2541	108	4	a=	a=	ADV
ejpam-2541	108	5	a1×a2	a1×a2	VERB
ejpam-2541	108	6	be	be	AUX
ejpam-2541	108	7	a	a	DET
ejpam-2541	108	8	convex	convex	NOUN
ejpam-2541	108	9	,	,	PUNCT
ejpam-2541	108	10	then	then	ADV
ejpam-2541	108	11	a1	a1	NOUN
ejpam-2541	108	12	and	and	CCONJ
ejpam-2541	108	13	a2	a2	PROPN
ejpam-2541	108	14	are	be	AUX
ejpam-2541	108	15	convex	convex	ADJ
ejpam-2541	108	16	,	,	PUNCT
ejpam-2541	108	17	by	by	ADP
ejpam-2541	108	18	using	use	VERB
ejpam-2541	108	19	theorem	theorem	ADJ
ejpam-2541	108	20	2	2	NUM
ejpam-2541	108	21	,	,	PUNCT
ejpam-2541	108	22	a1	a1	NOUN
ejpam-2541	108	23	and	and	CCONJ
ejpam-2541	108	24	a2	a2	PROPN
ejpam-2541	108	25	are	be	AUX
ejpam-2541	108	26	globally	globally	ADV
ejpam-2541	108	27	supported	support	VERB
ejpam-2541	108	28	at	at	ADP
ejpam-2541	108	29	each	each	DET
ejpam-2541	108	30	boundary	boundary	ADJ
ejpam-2541	108	31	point	point	NOUN
ejpam-2541	108	32	.	.	PUNCT
ejpam-2541	109	1	corollary	corollary	ADJ
ejpam-2541	109	2	1	1	NUM
ejpam-2541	109	3	.	.	PUNCT
ejpam-2541	110	1	let	let	VERB
ejpam-2541	110	2	a1	a1	PROPN
ejpam-2541	110	3	⊂w1	⊂w1	PROPN
ejpam-2541	110	4	and	and	CCONJ
ejpam-2541	110	5	a2	a2	PROPN
ejpam-2541	110	6	⊂w2	⊂w2	NOUN
ejpam-2541	110	7	be	be	AUX
ejpam-2541	110	8	open	open	ADJ
ejpam-2541	110	9	subsets	subset	NOUN
ejpam-2541	110	10	whose	whose	DET
ejpam-2541	110	11	boundary	boundary	ADJ
ejpam-2541	110	12	∂	∂	NOUN
ejpam-2541	110	13	a1	a1	NOUN
ejpam-2541	110	14	and	and	CCONJ
ejpam-2541	110	15	∂	∂	NUM
ejpam-2541	110	16	a2	a2	NOUN
ejpam-2541	110	17	are	be	AUX
ejpam-2541	110	18	smooth	smooth	ADJ
ejpam-2541	110	19	hypersurface	hypersurface	NOUN
ejpam-2541	110	20	of	of	ADP
ejpam-2541	110	21	w1	w1	NOUN
ejpam-2541	110	22	and	and	CCONJ
ejpam-2541	110	23	w2	w2	NOUN
ejpam-2541	110	24	,	,	PUNCT
ejpam-2541	110	25	respectively	respectively	ADV
ejpam-2541	110	26	.	.	PUNCT
ejpam-2541	111	1	then	then	ADV
ejpam-2541	111	2	,	,	PUNCT
ejpam-2541	111	3	a=	a=	ADV
ejpam-2541	111	4	a1×a2	a1×a2	NOUN
ejpam-2541	111	5	is	be	AUX
ejpam-2541	111	6	convex	convex	ADJ
ejpam-2541	111	7	if	if	SCONJ
ejpam-2541	111	8	and	and	CCONJ
ejpam-2541	111	9	only	only	ADV
ejpam-2541	111	10	if	if	SCONJ
ejpam-2541	111	11	every	every	DET
ejpam-2541	111	12	maximal	maximal	ADJ
ejpam-2541	111	13	tangent	tangent	NOUN
ejpam-2541	111	14	geodesic	geodesic	NOUN
ejpam-2541	111	15	of	of	ADP
ejpam-2541	111	16	∂	∂	NUM
ejpam-2541	111	17	a1	a1	NOUN
ejpam-2541	111	18	and	and	CCONJ
ejpam-2541	111	19	∂	∂	NUM
ejpam-2541	111	20	a2	a2	PROPN
ejpam-2541	111	21	have	have	VERB
ejpam-2541	111	22	an	an	DET
ejpam-2541	111	23	empty	empty	ADJ
ejpam-2541	111	24	intersection	intersection	NOUN
ejpam-2541	111	25	with	with	ADP
ejpam-2541	111	26	a1	a1	NOUN
ejpam-2541	111	27	and	and	CCONJ
ejpam-2541	111	28	a2	a2	PROPN
ejpam-2541	111	29	.	.	PUNCT
ejpam-2541	112	1	4	4	X
ejpam-2541	112	2	.	.	X
ejpam-2541	112	3	starshapedness	starshapedness	NOUN
ejpam-2541	112	4	in	in	ADP
ejpam-2541	112	5	riemannian	riemannian	ADJ
ejpam-2541	112	6	manifolds	manifold	NOUN
ejpam-2541	112	7	product	product	NOUN
ejpam-2541	112	8	in	in	ADP
ejpam-2541	112	9	this	this	DET
ejpam-2541	112	10	section	section	NOUN
ejpam-2541	112	11	,	,	PUNCT
ejpam-2541	112	12	we	we	PRON
ejpam-2541	112	13	aim	aim	VERB
ejpam-2541	112	14	to	to	PART
ejpam-2541	112	15	establish	establish	VERB
ejpam-2541	112	16	some	some	DET
ejpam-2541	112	17	properties	property	NOUN
ejpam-2541	112	18	of	of	ADP
ejpam-2541	112	19	starhapedness	starhapedness	NOUN
ejpam-2541	112	20	in	in	ADP
ejpam-2541	112	21	riemannian	riemannian	ADJ
ejpam-2541	112	22	manifolds	manifold	NOUN
ejpam-2541	112	23	product	product	NOUN
ejpam-2541	112	24	.	.	PUNCT
ejpam-2541	113	1	theorem	theorem	ADJ
ejpam-2541	113	2	7	7	NUM
ejpam-2541	113	3	.	.	PUNCT
ejpam-2541	114	1	let	let	VERB
ejpam-2541	114	2	a	a	PRON
ejpam-2541	114	3	be	be	AUX
ejpam-2541	114	4	a	a	DET
ejpam-2541	114	5	non	non	ADJ
ejpam-2541	114	6	-	-	ADJ
ejpam-2541	114	7	empty	empty	ADJ
ejpam-2541	114	8	closed	closed	ADJ
ejpam-2541	114	9	subset	subset	NOUN
ejpam-2541	114	10	of	of	ADP
ejpam-2541	114	11	w.	w.	PROPN
ejpam-2541	114	12	if	if	SCONJ
ejpam-2541	114	13	∂	∂	NOUN
ejpam-2541	114	14	a	a	PRON
ejpam-2541	114	15	is	be	AUX
ejpam-2541	114	16	starshaped	starshape	VERB
ejpam-2541	114	17	,	,	PUNCT
ejpam-2541	114	18	then	then	ADV
ejpam-2541	114	19	ker(∂	ker(∂	PROPN
ejpam-2541	114	20	a	a	PRON
ejpam-2541	114	21	)	)	PUNCT
ejpam-2541	114	22	⊂	⊂	PROPN
ejpam-2541	114	23	kera	kera	PROPN
ejpam-2541	114	24	.	.	PUNCT
ejpam-2541	115	1	w.	w.	PROPN
ejpam-2541	115	2	saleh	saleh	PROPN
ejpam-2541	115	3	,	,	PUNCT
ejpam-2541	115	4	a.	a.	NOUN
ejpam-2541	115	5	kiliccman	kiliccman	PROPN
ejpam-2541	115	6	/	/	SYM
ejpam-2541	115	7	eur	eur	PROPN
ejpam-2541	115	8	.	.	PUNCT
ejpam-2541	116	1	j.	j.	PROPN
ejpam-2541	116	2	pure	pure	PROPN
ejpam-2541	116	3	appl	appl	PROPN
ejpam-2541	116	4	.	.	PROPN
ejpam-2541	116	5	math	math	PROPN
ejpam-2541	116	6	,	,	PUNCT
ejpam-2541	116	7	9	9	NUM
ejpam-2541	116	8	(	(	PUNCT
ejpam-2541	116	9	2016	2016	NUM
ejpam-2541	116	10	)	)	PUNCT
ejpam-2541	116	11	,	,	PUNCT
ejpam-2541	116	12	57	57	NUM
ejpam-2541	116	13	-	-	SYM
ejpam-2541	116	14	63	63	NUM
ejpam-2541	116	15	61	61	NUM
ejpam-2541	116	16	proof	proof	NOUN
ejpam-2541	116	17	.	.	PUNCT
ejpam-2541	117	1	let	let	VERB
ejpam-2541	117	2	∂	∂	NOUN
ejpam-2541	117	3	a	a	PRON
ejpam-2541	117	4	be	be	AUX
ejpam-2541	117	5	starshaped	starshape	VERB
ejpam-2541	117	6	with	with	ADP
ejpam-2541	117	7	respect	respect	NOUN
ejpam-2541	117	8	to	to	ADP
ejpam-2541	117	9	x	x	PRON
ejpam-2541	117	10	,	,	PUNCT
ejpam-2541	117	11	i.e.	i.e.	X
ejpam-2541	117	12	,	,	PUNCT
ejpam-2541	117	13	x	x	SYM
ejpam-2541	117	14	∈	∈	PROPN
ejpam-2541	117	15	ker(∂	ker(∂	VERB
ejpam-2541	117	16	a	a	PRON
ejpam-2541	117	17	)	)	PUNCT
ejpam-2541	117	18	.	.	PUNCT
ejpam-2541	118	1	suppose	suppose	VERB
ejpam-2541	118	2	that	that	SCONJ
ejpam-2541	118	3	x	x	PRON
ejpam-2541	118	4	is	be	AUX
ejpam-2541	118	5	not	not	PART
ejpam-2541	118	6	in	in	ADP
ejpam-2541	118	7	kera	kera	PROPN
ejpam-2541	118	8	,	,	PUNCT
ejpam-2541	118	9	i.e.	i.e.	X
ejpam-2541	118	10	,	,	PUNCT
ejpam-2541	118	11	there	there	PRON
ejpam-2541	118	12	is	be	VERB
ejpam-2541	118	13	a	a	DET
ejpam-2541	118	14	point	point	NOUN
ejpam-2541	118	15	y	y	PROPN
ejpam-2541	118	16	∈	∈	PROPN
ejpam-2541	118	17	a	a	DET
ejpam-2541	118	18	such	such	ADJ
ejpam-2541	118	19	that	that	SCONJ
ejpam-2541	118	20	γ[x	γ[x	PROPN
ejpam-2541	118	21	y	y	PROPN
ejpam-2541	118	22	]	]	PUNCT
ejpam-2541	118	23	is	be	AUX
ejpam-2541	118	24	not	not	PART
ejpam-2541	118	25	contained	contain	VERB
ejpam-2541	118	26	in	in	ADP
ejpam-2541	118	27	a.	a.	NOUN
ejpam-2541	118	28	since	since	SCONJ
ejpam-2541	118	29	a	a	PRON
ejpam-2541	118	30	is	be	AUX
ejpam-2541	118	31	closed	closed	ADJ
ejpam-2541	118	32	,	,	PUNCT
ejpam-2541	118	33	there	there	PRON
ejpam-2541	118	34	is	be	VERB
ejpam-2541	118	35	a	a	DET
ejpam-2541	118	36	point	point	NOUN
ejpam-2541	118	37	y1	y1	NOUN
ejpam-2541	118	38	∈	∈	PROPN
ejpam-2541	118	39	∂	∂	NOUN
ejpam-2541	118	40	a∩	a∩	PROPN
ejpam-2541	118	41	γ[x	γ[x	PROPN
ejpam-2541	118	42	y	y	PROPN
ejpam-2541	118	43	]	]	PUNCT
ejpam-2541	118	44	such	such	ADJ
ejpam-2541	118	45	that	that	SCONJ
ejpam-2541	118	46	γ(x	γ(x	PROPN
ejpam-2541	118	47	y1	y1	PROPN
ejpam-2541	118	48	)	)	PUNCT
ejpam-2541	118	49	∩a=	∩a=	PUNCT
ejpam-2541	118	50	φ	φ	PROPN
ejpam-2541	118	51	.	.	PUNCT
ejpam-2541	119	1	thus	thus	ADV
ejpam-2541	119	2	,	,	PUNCT
ejpam-2541	119	3	x	x	PRON
ejpam-2541	119	4	does	do	AUX
ejpam-2541	119	5	not	not	PART
ejpam-2541	119	6	see	see	VERB
ejpam-2541	119	7	y1	y1	NOUN
ejpam-2541	119	8	via	via	ADP
ejpam-2541	119	9	∂	∂	NUM
ejpam-2541	119	10	a.	a.	NOUN
ejpam-2541	119	11	this	this	PRON
ejpam-2541	119	12	contradicts	contradict	VERB
ejpam-2541	119	13	the	the	DET
ejpam-2541	119	14	fact	fact	NOUN
ejpam-2541	119	15	that	that	SCONJ
ejpam-2541	119	16	∂	∂	NOUN
ejpam-2541	119	17	a	a	PRON
ejpam-2541	119	18	is	be	AUX
ejpam-2541	119	19	starshaped	starshape	VERB
ejpam-2541	119	20	with	with	ADP
ejpam-2541	119	21	respect	respect	NOUN
ejpam-2541	119	22	to	to	ADP
ejpam-2541	119	23	x	x	X
ejpam-2541	119	24	.	.	PUNCT
ejpam-2541	120	1	therefore	therefore	ADV
ejpam-2541	120	2	,	,	PUNCT
ejpam-2541	120	3	we	we	PRON
ejpam-2541	120	4	can	can	AUX
ejpam-2541	120	5	state	state	VERB
ejpam-2541	120	6	the	the	DET
ejpam-2541	120	7	following	following	ADJ
ejpam-2541	120	8	result	result	NOUN
ejpam-2541	120	9	as	as	ADV
ejpam-2541	120	10	well	well	ADV
ejpam-2541	120	11	.	.	PUNCT
ejpam-2541	121	1	theorem	theorem	ADJ
ejpam-2541	121	2	8	8	NUM
ejpam-2541	121	3	.	.	PUNCT
ejpam-2541	122	1	let	let	VERB
ejpam-2541	122	2	a1	a1	NOUN
ejpam-2541	122	3	be	be	AUX
ejpam-2541	122	4	a	a	DET
ejpam-2541	122	5	non	non	ADJ
ejpam-2541	122	6	-	-	ADJ
ejpam-2541	122	7	empty	empty	ADJ
ejpam-2541	122	8	closed	closed	ADJ
ejpam-2541	122	9	subset	subset	NOUN
ejpam-2541	122	10	of	of	ADP
ejpam-2541	122	11	w1	w1	NOUN
ejpam-2541	122	12	,	,	PUNCT
ejpam-2541	122	13	and	and	CCONJ
ejpam-2541	122	14	a2	a2	PROPN
ejpam-2541	122	15	be	be	VERB
ejpam-2541	122	16	a	a	DET
ejpam-2541	122	17	non	non	ADJ
ejpam-2541	122	18	-	-	ADJ
ejpam-2541	122	19	empty	empty	ADJ
ejpam-2541	122	20	closed	closed	ADJ
ejpam-2541	122	21	subset	subset	NOUN
ejpam-2541	122	22	of	of	ADP
ejpam-2541	122	23	w2	w2	NOUN
ejpam-2541	122	24	.	.	PUNCT
ejpam-2541	123	1	if	if	SCONJ
ejpam-2541	123	2	∂	∂	NUM
ejpam-2541	123	3	a1	a1	NOUN
ejpam-2541	123	4	and	and	CCONJ
ejpam-2541	123	5	∂	∂	NUM
ejpam-2541	123	6	a2	a2	NOUN
ejpam-2541	123	7	are	be	AUX
ejpam-2541	123	8	starshaped	starshape	VERB
ejpam-2541	123	9	,	,	PUNCT
ejpam-2541	123	10	then	then	ADV
ejpam-2541	123	11	ker(∂	ker(∂	PROPN
ejpam-2541	123	12	a1	a1	PROPN
ejpam-2541	123	13	×	×	PROPN
ejpam-2541	123	14	∂	∂	NUM
ejpam-2541	123	15	a2	a2	PROPN
ejpam-2541	123	16	)	)	PUNCT
ejpam-2541	124	1	⊂	⊂	PROPN
ejpam-2541	125	1	ker(a1	ker(a1	PROPN
ejpam-2541	125	2	×	×	PROPN
ejpam-2541	125	3	a2	a2	PROPN
ejpam-2541	125	4	)	)	PUNCT
ejpam-2541	125	5	.	.	PUNCT
ejpam-2541	126	1	proof	proof	NOUN
ejpam-2541	126	2	.	.	PUNCT
ejpam-2541	127	1	since	since	SCONJ
ejpam-2541	127	2	ker(∂	ker(∂	PROPN
ejpam-2541	127	3	a1	a1	PROPN
ejpam-2541	127	4	×	×	PROPN
ejpam-2541	127	5	∂	∂	NUM
ejpam-2541	127	6	a2	a2	PROPN
ejpam-2541	127	7	)	)	PUNCT
ejpam-2541	128	1	=	=	NOUN
ejpam-2541	128	2	ker(∂	ker(∂	PROPN
ejpam-2541	128	3	a1)×	a1)×	PROPN
ejpam-2541	128	4	ker(∂	ker(∂	PROPN
ejpam-2541	128	5	a2	a2	PROPN
ejpam-2541	128	6	)	)	PUNCT
ejpam-2541	128	7	⊂	⊂	PROPN
ejpam-2541	129	1	kera1	kera1	NOUN
ejpam-2541	129	2	×	×	NOUN
ejpam-2541	129	3	kera2	kera2	NOUN
ejpam-2541	130	1	=	=	SYM
ejpam-2541	130	2	ker(a1	ker(a1	PROPN
ejpam-2541	130	3	×	×	PROPN
ejpam-2541	130	4	a2	a2	PROPN
ejpam-2541	130	5	)	)	PUNCT
ejpam-2541	130	6	.	.	PUNCT
ejpam-2541	131	1	then	then	ADV
ejpam-2541	131	2	,	,	PUNCT
ejpam-2541	131	3	ker(∂	ker(∂	PROPN
ejpam-2541	131	4	a1	a1	VERB
ejpam-2541	131	5	×	×	PROPN
ejpam-2541	131	6	∂	∂	NUM
ejpam-2541	131	7	a2	a2	PROPN
ejpam-2541	131	8	)	)	PUNCT
ejpam-2541	131	9	⊂	⊂	PROPN
ejpam-2541	132	1	ker(a1	ker(a1	PROPN
ejpam-2541	132	2	×	×	PROPN
ejpam-2541	132	3	a2	a2	PROPN
ejpam-2541	132	4	)	)	PUNCT
ejpam-2541	132	5	.	.	PUNCT
ejpam-2541	133	1	theorem	theorem	NOUN
ejpam-2541	133	2	9	9	NUM
ejpam-2541	133	3	.	.	PUNCT
ejpam-2541	134	1	let	let	VERB
ejpam-2541	134	2	s	s	NOUN
ejpam-2541	134	3	=	=	VERB
ejpam-2541	134	4	s1	s1	PROPN
ejpam-2541	134	5	×	×	PROPN
ejpam-2541	134	6	s2	s2	NOUN
ejpam-2541	134	7	be	be	VERB
ejpam-2541	134	8	an	an	DET
ejpam-2541	134	9	open	open	ADJ
ejpam-2541	134	10	starshaped	starshape	VERB
ejpam-2541	134	11	subset	subset	VERB
ejpam-2541	134	12	with	with	ADP
ejpam-2541	134	13	respect	respect	NOUN
ejpam-2541	134	14	to	to	ADP
ejpam-2541	134	15	some	some	DET
ejpam-2541	134	16	point	point	NOUN
ejpam-2541	134	17	p	p	X
ejpam-2541	134	18	=	=	SYM
ejpam-2541	134	19	(	(	PUNCT
ejpam-2541	134	20	p1	p1	PROPN
ejpam-2541	134	21	,	,	PUNCT
ejpam-2541	134	22	p2	p2	X
ejpam-2541	134	23	)	)	PUNCT
ejpam-2541	134	24	∈	∈	PROPN
ejpam-2541	134	25	s	s	NOUN
ejpam-2541	134	26	,	,	PUNCT
ejpam-2541	134	27	then	then	ADV
ejpam-2541	134	28	the	the	DET
ejpam-2541	134	29	closure	closure	NOUN
ejpam-2541	134	30	s̄	s̄	NOUN
ejpam-2541	134	31	=	=	SYM
ejpam-2541	134	32	s1	s1	NOUN
ejpam-2541	134	33	×	×	NOUN
ejpam-2541	134	34	s2	s2	NOUN
ejpam-2541	134	35	is	be	AUX
ejpam-2541	134	36	also	also	ADV
ejpam-2541	134	37	starshaped	starshape	VERB
ejpam-2541	134	38	with	with	ADP
ejpam-2541	134	39	respect	respect	NOUN
ejpam-2541	134	40	to	to	ADP
ejpam-2541	134	41	the	the	DET
ejpam-2541	134	42	same	same	ADJ
ejpam-2541	134	43	point	point	NOUN
ejpam-2541	134	44	p	p	X
ejpam-2541	134	45	=	=	SYM
ejpam-2541	134	46	(	(	PUNCT
ejpam-2541	134	47	p1	p1	PROPN
ejpam-2541	134	48	,	,	PUNCT
ejpam-2541	134	49	p2	p2	PROPN
ejpam-2541	134	50	)	)	PUNCT
ejpam-2541	134	51	.	.	PUNCT
ejpam-2541	135	1	proof	proof	NOUN
ejpam-2541	135	2	.	.	PUNCT
ejpam-2541	136	1	suppose	suppose	VERB
ejpam-2541	136	2	that	that	SCONJ
ejpam-2541	136	3	s	s	VERB
ejpam-2541	136	4	=	=	PUNCT
ejpam-2541	136	5	s1	s1	PROPN
ejpam-2541	136	6	×	×	PROPN
ejpam-2541	136	7	s2	s2	NOUN
ejpam-2541	136	8	is	be	AUX
ejpam-2541	136	9	starshaped	starshape	VERB
ejpam-2541	136	10	with	with	ADP
ejpam-2541	136	11	respect	respect	NOUN
ejpam-2541	136	12	to	to	ADP
ejpam-2541	136	13	the	the	DET
ejpam-2541	136	14	point	point	NOUN
ejpam-2541	136	15	p	p	X
ejpam-2541	136	16	=	=	X
ejpam-2541	136	17	(	(	PUNCT
ejpam-2541	136	18	p1	p1	PROPN
ejpam-2541	136	19	,	,	PUNCT
ejpam-2541	136	20	p2	p2	PROPN
ejpam-2541	136	21	)	)	PUNCT
ejpam-2541	136	22	,	,	PUNCT
ejpam-2541	136	23	then	then	ADV
ejpam-2541	136	24	s1	s1	PROPN
ejpam-2541	136	25	and	and	CCONJ
ejpam-2541	136	26	s2	s2	PROPN
ejpam-2541	136	27	are	be	AUX
ejpam-2541	136	28	starshaped	starshape	VERB
ejpam-2541	136	29	with	with	ADP
ejpam-2541	136	30	respect	respect	NOUN
ejpam-2541	136	31	to	to	ADP
ejpam-2541	136	32	the	the	DET
ejpam-2541	136	33	points	point	NOUN
ejpam-2541	136	34	p1	p1	NOUN
ejpam-2541	136	35	,	,	PUNCT
ejpam-2541	136	36	and	and	CCONJ
ejpam-2541	136	37	p2	p2	PROPN
ejpam-2541	136	38	,	,	PUNCT
ejpam-2541	136	39	respectively	respectively	ADV
ejpam-2541	136	40	.	.	PUNCT
ejpam-2541	137	1	this	this	PRON
ejpam-2541	137	2	implies	imply	VERB
ejpam-2541	137	3	,	,	PUNCT
ejpam-2541	137	4	by	by	ADP
ejpam-2541	137	5	using	use	VERB
ejpam-2541	137	6	theorem	theorem	ADJ
ejpam-2541	137	7	3	3	NUM
ejpam-2541	137	8	,	,	PUNCT
ejpam-2541	137	9	to	to	ADP
ejpam-2541	137	10	s̄1	s̄1	NOUN
ejpam-2541	137	11	and	and	CCONJ
ejpam-2541	137	12	s̄1	s̄1	NOUN
ejpam-2541	137	13	are	be	AUX
ejpam-2541	137	14	starshaped	starshape	VERB
ejpam-2541	137	15	with	with	ADP
ejpam-2541	137	16	respect	respect	NOUN
ejpam-2541	137	17	to	to	ADP
ejpam-2541	137	18	the	the	DET
ejpam-2541	137	19	points	point	NOUN
ejpam-2541	137	20	p1	p1	NOUN
ejpam-2541	137	21	and	and	CCONJ
ejpam-2541	137	22	p2	p2	NOUN
ejpam-2541	137	23	,	,	PUNCT
ejpam-2541	137	24	respectively	respectively	ADV
ejpam-2541	137	25	.	.	PUNCT
ejpam-2541	138	1	then	then	ADV
ejpam-2541	138	2	,	,	PUNCT
ejpam-2541	138	3	s̄1	s̄1	NOUN
ejpam-2541	138	4	×	×	PROPN
ejpam-2541	138	5	s̄2	s̄2	NOUN
ejpam-2541	138	6	=	=	SYM
ejpam-2541	139	1	s1	s1	PROPN
ejpam-2541	139	2	×	×	PROPN
ejpam-2541	139	3	s2	s2	NOUN
ejpam-2541	139	4	is	be	AUX
ejpam-2541	139	5	starshaped	starshape	VERB
ejpam-2541	139	6	with	with	ADP
ejpam-2541	139	7	respect	respect	NOUN
ejpam-2541	139	8	to	to	ADP
ejpam-2541	139	9	the	the	DET
ejpam-2541	139	10	point	point	NOUN
ejpam-2541	139	11	p	p	X
ejpam-2541	139	12	=	=	X
ejpam-2541	139	13	(	(	PUNCT
ejpam-2541	139	14	p1	p1	PROPN
ejpam-2541	139	15	,	,	PUNCT
ejpam-2541	139	16	p2	p2	PROPN
ejpam-2541	139	17	)	)	PUNCT
ejpam-2541	139	18	.	.	PUNCT
ejpam-2541	140	1	corollary	corollary	ADJ
ejpam-2541	140	2	2	2	NUM
ejpam-2541	140	3	.	.	PUNCT
ejpam-2541	141	1	let	let	VERB
ejpam-2541	141	2	s	s	NOUN
ejpam-2541	141	3	=	=	VERB
ejpam-2541	141	4	s1	s1	PROPN
ejpam-2541	141	5	×	×	PROPN
ejpam-2541	141	6	s2	s2	PROPN
ejpam-2541	141	7	⊂w1	⊂w1	NOUN
ejpam-2541	141	8	×w2	×w2	PRON
ejpam-2541	141	9	be	be	VERB
ejpam-2541	141	10	an	an	DET
ejpam-2541	141	11	open	open	ADJ
ejpam-2541	141	12	starshaped	starshape	VERB
ejpam-2541	141	13	subset	subset	NOUN
ejpam-2541	141	14	.	.	PUNCT
ejpam-2541	142	1	then	then	ADV
ejpam-2541	142	2	,	,	PUNCT
ejpam-2541	142	3	the	the	DET
ejpam-2541	142	4	kernel	kernel	NOUN
ejpam-2541	142	5	of	of	ADP
ejpam-2541	142	6	s	s	PROPN
ejpam-2541	142	7	is	be	AUX
ejpam-2541	142	8	contained	contain	VERB
ejpam-2541	142	9	the	the	DET
ejpam-2541	142	10	kernel	kernel	NOUN
ejpam-2541	142	11	of	of	ADP
ejpam-2541	142	12	s̄(ker(s1	s̄(ker(s1	NUM
ejpam-2541	142	13	×	×	PROPN
ejpam-2541	142	14	s2	s2	PROPN
ejpam-2541	142	15	)	)	PUNCT
ejpam-2541	142	16	⊂	⊂	PROPN
ejpam-2541	142	17	ker(s1	ker(s1	PROPN
ejpam-2541	142	18	×	×	PROPN
ejpam-2541	142	19	s2	s2	PROPN
ejpam-2541	142	20	)	)	PUNCT
ejpam-2541	142	21	)	)	PUNCT
ejpam-2541	142	22	.	.	PUNCT
ejpam-2541	143	1	proof	proof	NOUN
ejpam-2541	143	2	.	.	PUNCT
ejpam-2541	144	1	let	let	VERB
ejpam-2541	144	2	p	p	NOUN
ejpam-2541	144	3	=	=	X
ejpam-2541	144	4	(	(	PUNCT
ejpam-2541	144	5	p1	p1	PROPN
ejpam-2541	144	6	,	,	PUNCT
ejpam-2541	144	7	p2	p2	NOUN
ejpam-2541	144	8	)	)	PUNCT
ejpam-2541	144	9	∈	∈	NOUN
ejpam-2541	144	10	kers	ker	NOUN
ejpam-2541	144	11	,	,	PUNCT
ejpam-2541	144	12	i.e.	i.e.	X
ejpam-2541	144	13	,	,	PUNCT
ejpam-2541	144	14	p	p	PROPN
ejpam-2541	144	15	∈	∈	PROPN
ejpam-2541	144	16	ker(s1	ker(s1	PROPN
ejpam-2541	144	17	×	×	PROPN
ejpam-2541	144	18	s2	s2	PROPN
ejpam-2541	144	19	)	)	PUNCT
ejpam-2541	144	20	,	,	PUNCT
ejpam-2541	144	21	then	then	ADV
ejpam-2541	144	22	s	s	VERB
ejpam-2541	144	23	is	be	AUX
ejpam-2541	144	24	starshaped	starshape	VERB
ejpam-2541	144	25	with	with	ADP
ejpam-2541	144	26	respect	respect	NOUN
ejpam-2541	144	27	to	to	ADP
ejpam-2541	144	28	p	p	NOUN
ejpam-2541	144	29	=	=	SYM
ejpam-2541	144	30	(	(	PUNCT
ejpam-2541	144	31	p1	p1	PROPN
ejpam-2541	144	32	,	,	PUNCT
ejpam-2541	144	33	p2	p2	PROPN
ejpam-2541	144	34	)	)	PUNCT
ejpam-2541	144	35	.	.	PUNCT
ejpam-2541	145	1	by	by	ADP
ejpam-2541	145	2	theorem	theorem	NOUN
ejpam-2541	145	3	9	9	NUM
ejpam-2541	145	4	,	,	PUNCT
ejpam-2541	145	5	s̄	s̄	NOUN
ejpam-2541	145	6	=	=	SYM
ejpam-2541	145	7	s1	s1	NOUN
ejpam-2541	145	8	×	×	NOUN
ejpam-2541	145	9	s2	s2	NOUN
ejpam-2541	145	10	is	be	AUX
ejpam-2541	145	11	also	also	ADV
ejpam-2541	145	12	starshaped	starshape	VERB
ejpam-2541	145	13	with	with	ADP
ejpam-2541	145	14	respect	respect	NOUN
ejpam-2541	145	15	to	to	ADP
ejpam-2541	145	16	p	p	NOUN
ejpam-2541	145	17	=	=	SYM
ejpam-2541	145	18	(	(	PUNCT
ejpam-2541	145	19	p1	p1	PROPN
ejpam-2541	145	20	,	,	PUNCT
ejpam-2541	145	21	p2	p2	PROPN
ejpam-2541	145	22	)	)	PUNCT
ejpam-2541	145	23	,	,	PUNCT
ejpam-2541	145	24	which	which	PRON
ejpam-2541	145	25	implies	imply	VERB
ejpam-2541	145	26	that	that	SCONJ
ejpam-2541	145	27	p	p	PROPN
ejpam-2541	145	28	∈	∈	NOUN
ejpam-2541	145	29	kers̄.	kers̄.	PROPN
ejpam-2541	145	30	hence	hence	ADV
ejpam-2541	145	31	,	,	PUNCT
ejpam-2541	145	32	ker(s1	ker(s1	PROPN
ejpam-2541	145	33	×	×	PROPN
ejpam-2541	145	34	s2	s2	PROPN
ejpam-2541	145	35	)	)	PUNCT
ejpam-2541	146	1	⊂	⊂	PROPN
ejpam-2541	147	1	ker(s1	ker(s1	PROPN
ejpam-2541	147	2	×	×	PROPN
ejpam-2541	147	3	s2	s2	PROPN
ejpam-2541	147	4	)	)	PUNCT
ejpam-2541	147	5	.	.	PUNCT
ejpam-2541	148	1	the	the	DET
ejpam-2541	148	2	relation	relation	NOUN
ejpam-2541	148	3	between	between	ADP
ejpam-2541	148	4	ker(s1	ker(s1	PROPN
ejpam-2541	148	5	×	×	PROPN
ejpam-2541	148	6	s2	s2	PROPN
ejpam-2541	148	7	)	)	PUNCT
ejpam-2541	148	8	and	and	CCONJ
ejpam-2541	148	9	ker(s1	ker(s1	PROPN
ejpam-2541	148	10	×	×	PROPN
ejpam-2541	148	11	s2	s2	PROPN
ejpam-2541	148	12	)	)	PUNCT
ejpam-2541	148	13	for	for	ADP
ejpam-2541	148	14	any	any	PRON
ejpam-2541	148	15	arbitrary	arbitrary	ADJ
ejpam-2541	148	16	open	open	NOUN
ejpam-2541	148	17	starshaped	starshape	VERB
ejpam-2541	148	18	subset	subset	NOUN
ejpam-2541	148	19	s	s	PART
ejpam-2541	148	20	=	=	SYM
ejpam-2541	148	21	s1	s1	PROPN
ejpam-2541	148	22	×	×	PROPN
ejpam-2541	148	23	s2	s2	NOUN
ejpam-2541	148	24	⊂w1	⊂w1	NOUN
ejpam-2541	148	25	×w2	×w2	PROPN
ejpam-2541	148	26	is	be	AUX
ejpam-2541	148	27	given	give	VERB
ejpam-2541	148	28	in	in	ADP
ejpam-2541	148	29	the	the	DET
ejpam-2541	148	30	following	following	NOUN
ejpam-2541	148	31	theorem	theorem	NOUN
ejpam-2541	148	32	:	:	PUNCT
ejpam-2541	148	33	theorem	theorem	NOUN
ejpam-2541	148	34	10	10	NUM
ejpam-2541	148	35	.	.	PUNCT
ejpam-2541	149	1	let	let	VERB
ejpam-2541	149	2	s	s	PRON
ejpam-2541	149	3	=	=	PUNCT
ejpam-2541	150	1	s1×s2	s1×s2	ADJ
ejpam-2541	150	2	⊂w1×w2	⊂w1×w2	NOUN
ejpam-2541	150	3	be	be	AUX
ejpam-2541	150	4	an	an	DET
ejpam-2541	150	5	open	open	ADJ
ejpam-2541	150	6	starshaped	starshape	VERB
ejpam-2541	150	7	subset	subset	VERB
ejpam-2541	150	8	such	such	ADJ
ejpam-2541	150	9	that	that	DET
ejpam-2541	150	10	∂	∂	NOUN
ejpam-2541	150	11	s	s	NOUN
ejpam-2541	150	12	is	be	AUX
ejpam-2541	150	13	a	a	DET
ejpam-2541	150	14	smooth	smooth	ADJ
ejpam-2541	150	15	hypersurface	hypersurface	NOUN
ejpam-2541	150	16	.	.	PUNCT
ejpam-2541	151	1	then	then	ADV
ejpam-2541	151	2	,	,	PUNCT
ejpam-2541	151	3	ker(s1	ker(s1	PROPN
ejpam-2541	151	4	×	×	PROPN
ejpam-2541	151	5	s2	s2	PROPN
ejpam-2541	151	6	)	)	PUNCT
ejpam-2541	151	7	=	=	PUNCT
ejpam-2541	152	1	ker(s1	ker(s1	PROPN
ejpam-2541	152	2	×	×	PROPN
ejpam-2541	152	3	s2	s2	PROPN
ejpam-2541	152	4	)	)	PUNCT
ejpam-2541	152	5	.	.	PUNCT
ejpam-2541	153	1	proof	proof	NOUN
ejpam-2541	153	2	.	.	PUNCT
ejpam-2541	154	1	since	since	SCONJ
ejpam-2541	154	2	ker(s1	ker(s1	PROPN
ejpam-2541	154	3	×	×	PROPN
ejpam-2541	154	4	s2	s2	PROPN
ejpam-2541	154	5	)	)	PUNCT
ejpam-2541	154	6	=	=	NUM
ejpam-2541	154	7	ker(s̄1	ker(s̄1	PROPN
ejpam-2541	154	8	×	×	PROPN
ejpam-2541	154	9	s̄2	s̄2	NOUN
ejpam-2541	154	10	)	)	PUNCT
ejpam-2541	155	1	=	=	NOUN
ejpam-2541	155	2	ker(s̄1)×	ker(s̄1)×	NOUN
ejpam-2541	155	3	ker(s̄2	ker(s̄2	PROPN
ejpam-2541	155	4	)	)	PUNCT
ejpam-2541	156	1	=	=	PROPN
ejpam-2541	156	2	ker(s1)×	ker(s1)×	PROPN
ejpam-2541	156	3	ker(s2	ker(s2	NOUN
ejpam-2541	156	4	)	)	PUNCT
ejpam-2541	156	5	=	=	NOUN
ejpam-2541	156	6	kers1	kers1	X
ejpam-2541	156	7	×	×	NOUN
ejpam-2541	156	8	kers2	kers2	NOUN
ejpam-2541	156	9	references	reference	VERB
ejpam-2541	156	10	62	62	NUM
ejpam-2541	156	11	=	=	NOUN
ejpam-2541	156	12	ker(s1	ker(s1	PROPN
ejpam-2541	156	13	×	×	PROPN
ejpam-2541	156	14	s2	s2	PROPN
ejpam-2541	156	15	)	)	PUNCT
ejpam-2541	156	16	.	.	PUNCT
ejpam-2541	157	1	then	then	ADV
ejpam-2541	157	2	,	,	PUNCT
ejpam-2541	157	3	ker(s1	ker(s1	PROPN
ejpam-2541	157	4	×	×	PROPN
ejpam-2541	157	5	s2	s2	PROPN
ejpam-2541	157	6	)	)	PUNCT
ejpam-2541	157	7	=	=	PUNCT
ejpam-2541	158	1	ker(s1	ker(s1	PROPN
ejpam-2541	158	2	×	×	PROPN
ejpam-2541	158	3	s2	s2	PROPN
ejpam-2541	158	4	)	)	PUNCT
ejpam-2541	158	5	.	.	PUNCT
ejpam-2541	159	1	corollary	corollary	ADJ
ejpam-2541	159	2	3	3	X
ejpam-2541	159	3	.	.	PUNCT
ejpam-2541	160	1	let	let	VERB
ejpam-2541	160	2	s	s	PRON
ejpam-2541	160	3	=	=	PUNCT
ejpam-2541	161	1	s1×s2	s1×s2	ADJ
ejpam-2541	161	2	⊂w1×w2	⊂w1×w2	NOUN
ejpam-2541	161	3	be	be	AUX
ejpam-2541	161	4	a	a	DET
ejpam-2541	161	5	closed	close	VERB
ejpam-2541	161	6	starshaped	starshape	VERB
ejpam-2541	161	7	such	such	ADJ
ejpam-2541	161	8	that	that	SCONJ
ejpam-2541	161	9	s	s	PART
ejpam-2541	161	10	is	be	AUX
ejpam-2541	161	11	smooth	smooth	ADJ
ejpam-2541	161	12	hypersurface	hypersurface	NOUN
ejpam-2541	161	13	.	.	PUNCT
ejpam-2541	162	1	then	then	ADV
ejpam-2541	162	2	,	,	PUNCT
ejpam-2541	162	3	kers	ker	NOUN
ejpam-2541	162	4	is	be	AUX
ejpam-2541	162	5	a	a	DET
ejpam-2541	162	6	closed	closed	ADJ
ejpam-2541	162	7	subset	subset	NOUN
ejpam-2541	162	8	.	.	PUNCT
ejpam-2541	163	1	proof	proof	NOUN
ejpam-2541	163	2	.	.	PUNCT
ejpam-2541	164	1	the	the	DET
ejpam-2541	164	2	proof	proof	NOUN
ejpam-2541	164	3	is	be	AUX
ejpam-2541	164	4	direct	direct	ADJ
ejpam-2541	164	5	from	from	ADP
ejpam-2541	164	6	theorem	theorem	NOUN
ejpam-2541	164	7	10	10	NUM
ejpam-2541	164	8	since	since	SCONJ
ejpam-2541	164	9	s	s	PART
ejpam-2541	164	10	=	=	SYM
ejpam-2541	164	11	s1	s1	PROPN
ejpam-2541	164	12	×	×	PROPN
ejpam-2541	164	13	s2	s2	NOUN
ejpam-2541	164	14	is	be	AUX
ejpam-2541	164	15	closed	close	VERB
ejpam-2541	164	16	if	if	SCONJ
ejpam-2541	164	17	and	and	CCONJ
ejpam-2541	164	18	only	only	ADV
ejpam-2541	164	19	if	if	SCONJ
ejpam-2541	164	20	s	s	NOUN
ejpam-2541	164	21	=	=	NOUN
ejpam-2541	164	22	s̄	s̄	NOUN
ejpam-2541	164	23	which	which	PRON
ejpam-2541	164	24	implies	imply	VERB
ejpam-2541	164	25	that	that	SCONJ
ejpam-2541	164	26	kers	ker	NOUN
ejpam-2541	164	27	=	=	SYM
ejpam-2541	164	28	kers	ker	NOUN
ejpam-2541	164	29	.	.	PUNCT
ejpam-2541	165	1	5	5	X
ejpam-2541	165	2	.	.	X
ejpam-2541	165	3	concluding	conclude	VERB
ejpam-2541	165	4	remarks	remark	NOUN
ejpam-2541	165	5	(	(	PUNCT
ejpam-2541	165	6	i	i	NOUN
ejpam-2541	165	7	)	)	PUNCT
ejpam-2541	165	8	all	all	DET
ejpam-2541	165	9	results	result	NOUN
ejpam-2541	165	10	in	in	ADP
ejpam-2541	165	11	this	this	DET
ejpam-2541	165	12	paper	paper	NOUN
ejpam-2541	165	13	are	be	AUX
ejpam-2541	165	14	valid	valid	ADJ
ejpam-2541	165	15	in	in	ADP
ejpam-2541	165	16	the	the	DET
ejpam-2541	165	17	cartesian	cartesian	ADJ
ejpam-2541	165	18	product	product	NOUN
ejpam-2541	165	19	of	of	ADP
ejpam-2541	165	20	euclidean	euclidean	NOUN
ejpam-2541	165	21	as	as	ADV
ejpam-2541	165	22	well	well	ADV
ejpam-2541	165	23	as	as	ADP
ejpam-2541	165	24	hyperbolic	hyperbolic	ADJ
ejpam-2541	165	25	spaces	space	NOUN
ejpam-2541	165	26	as	as	ADP
ejpam-2541	165	27	examples	example	NOUN
ejpam-2541	165	28	of	of	ADP
ejpam-2541	165	29	manifold	manifold	ADJ
ejpam-2541	165	30	without	without	ADP
ejpam-2541	165	31	conjugate	conjugate	ADJ
ejpam-2541	165	32	points	point	NOUN
ejpam-2541	165	33	.	.	PUNCT
ejpam-2541	166	1	moreover	moreover	ADV
ejpam-2541	166	2	,	,	PUNCT
ejpam-2541	166	3	these	these	DET
ejpam-2541	166	4	results	result	NOUN
ejpam-2541	166	5	are	be	AUX
ejpam-2541	166	6	valid	valid	ADJ
ejpam-2541	166	7	in	in	ADP
ejpam-2541	166	8	the	the	DET
ejpam-2541	166	9	case	case	NOUN
ejpam-2541	166	10	of	of	ADP
ejpam-2541	166	11	cartesian	cartesian	ADJ
ejpam-2541	166	12	product	product	NOUN
ejpam-2541	166	13	of	of	ADP
ejpam-2541	166	14	manifolds	manifold	NOUN
ejpam-2541	166	15	without	without	ADP
ejpam-2541	166	16	focal	focal	ADJ
ejpam-2541	166	17	points	point	NOUN
ejpam-2541	166	18	as	as	ADP
ejpam-2541	166	19	every	every	DET
ejpam-2541	166	20	manifold	manifold	NOUN
ejpam-2541	166	21	without	without	ADP
ejpam-2541	166	22	focal	focal	ADJ
ejpam-2541	166	23	points	point	NOUN
ejpam-2541	166	24	has	have	VERB
ejpam-2541	166	25	no	no	DET
ejpam-2541	166	26	conjugate	conjugate	ADJ
ejpam-2541	166	27	points	point	NOUN
ejpam-2541	166	28	.	.	PUNCT
ejpam-2541	167	1	(	(	PUNCT
ejpam-2541	167	2	ii	ii	X
ejpam-2541	167	3	)	)	PUNCT
ejpam-2541	167	4	the	the	DET
ejpam-2541	167	5	results	result	NOUN
ejpam-2541	167	6	will	will	AUX
ejpam-2541	167	7	be	be	AUX
ejpam-2541	167	8	more	more	ADV
ejpam-2541	167	9	interesting	interesting	ADJ
ejpam-2541	167	10	in	in	ADP
ejpam-2541	167	11	the	the	DET
ejpam-2541	167	12	cartesian	cartesian	ADJ
ejpam-2541	167	13	product	product	NOUN
ejpam-2541	167	14	of	of	ADP
ejpam-2541	167	15	general	general	ADJ
ejpam-2541	167	16	riemannian	riemannian	ADJ
ejpam-2541	167	17	manifolds	manifold	NOUN
ejpam-2541	167	18	.	.	PUNCT
ejpam-2541	168	1	(	(	PUNCT
ejpam-2541	168	2	iii	iii	X
ejpam-2541	168	3	)	)	PUNCT
ejpam-2541	168	4	the	the	DET
ejpam-2541	168	5	study	study	NOUN
ejpam-2541	168	6	has	have	AUX
ejpam-2541	168	7	been	be	AUX
ejpam-2541	168	8	established	establish	VERB
ejpam-2541	168	9	in	in	ADP
ejpam-2541	168	10	this	this	DET
ejpam-2541	168	11	paper	paper	NOUN
ejpam-2541	168	12	could	could	AUX
ejpam-2541	168	13	be	be	AUX
ejpam-2541	168	14	considered	consider	VERB
ejpam-2541	168	15	as	as	ADP
ejpam-2541	168	16	a	a	DET
ejpam-2541	168	17	base	base	NOUN
ejpam-2541	168	18	of	of	ADP
ejpam-2541	168	19	a	a	DET
ejpam-2541	168	20	study	study	NOUN
ejpam-2541	168	21	of	of	ADP
ejpam-2541	168	22	other	other	ADJ
ejpam-2541	168	23	concepts	concept	NOUN
ejpam-2541	168	24	such	such	ADJ
ejpam-2541	168	25	as	as	ADP
ejpam-2541	168	26	local	local	ADJ
ejpam-2541	168	27	convexity	convexity	NOUN
ejpam-2541	168	28	and	and	CCONJ
ejpam-2541	168	29	so	so	ADV
ejpam-2541	168	30	on	on	ADV
ejpam-2541	168	31	.	.	PUNCT
ejpam-2541	169	1	acknowledgements	acknowledgement	NOUN
ejpam-2541	169	2	the	the	DET
ejpam-2541	169	3	authors	author	NOUN
ejpam-2541	169	4	are	be	AUX
ejpam-2541	169	5	exceptionally	exceptionally	ADV
ejpam-2541	169	6	grateful	grateful	ADJ
ejpam-2541	169	7	to	to	ADP
ejpam-2541	169	8	the	the	DET
ejpam-2541	169	9	anonymous	anonymous	ADJ
ejpam-2541	169	10	referees	referee	NOUN
ejpam-2541	169	11	for	for	ADP
ejpam-2541	169	12	their	their	PRON
ejpam-2541	169	13	valuable	valuable	ADJ
ejpam-2541	169	14	suggestions	suggestion	NOUN
ejpam-2541	169	15	and	and	CCONJ
ejpam-2541	169	16	comments	comment	NOUN
ejpam-2541	169	17	,	,	PUNCT
ejpam-2541	169	18	which	which	PRON
ejpam-2541	169	19	helped	help	VERB
ejpam-2541	169	20	the	the	DET
ejpam-2541	169	21	authors	author	NOUN
ejpam-2541	169	22	to	to	PART
ejpam-2541	169	23	improve	improve	VERB
ejpam-2541	169	24	the	the	DET
ejpam-2541	169	25	work	work	NOUN
ejpam-2541	169	26	.	.	PUNCT
ejpam-2541	170	1	references	reference	NOUN
ejpam-2541	170	2	[	[	X
ejpam-2541	170	3	1	1	NUM
ejpam-2541	170	4	]	]	PUNCT
ejpam-2541	170	5	m.	m.	NOUN
ejpam-2541	170	6	beltagy	beltagy	PROPN
ejpam-2541	170	7	.	.	PUNCT
ejpam-2541	171	1	immersions	immersion	NOUN
ejpam-2541	171	2	into	into	ADP
ejpam-2541	171	3	manifolds	manifold	NOUN
ejpam-2541	171	4	without	without	ADP
ejpam-2541	171	5	conjugate	conjugate	ADJ
ejpam-2541	171	6	points	point	NOUN
ejpam-2541	171	7	.	.	PUNCT
ejpam-2541	172	1	phd	phd	NOUN
ejpam-2541	172	2	thesis	thesis	PROPN
ejpam-2541	172	3	,	,	PUNCT
ejpam-2541	172	4	university	university	NOUN
ejpam-2541	172	5	of	of	ADP
ejpam-2541	172	6	durham	durham	PROPN
ejpam-2541	172	7	,	,	PUNCT
ejpam-2541	172	8	england	england	PROPN
ejpam-2541	172	9	,	,	PUNCT
ejpam-2541	172	10	1982	1982	NUM
ejpam-2541	172	11	.	.	PUNCT
ejpam-2541	173	1	[	[	X
ejpam-2541	173	2	2	2	NUM
ejpam-2541	173	3	]	]	PUNCT
ejpam-2541	173	4	m.	m.	NOUN
ejpam-2541	173	5	beltagy	beltagy	NOUN
ejpam-2541	173	6	.	.	PUNCT
ejpam-2541	174	1	on	on	ADP
ejpam-2541	174	2	the	the	DET
ejpam-2541	174	3	geometry	geometry	NOUN
ejpam-2541	174	4	of	of	ADP
ejpam-2541	174	5	the	the	DET
ejpam-2541	174	6	cartesian	cartesian	ADJ
ejpam-2541	174	7	product	product	NOUN
ejpam-2541	174	8	of	of	ADP
ejpam-2541	174	9	manifolds	manifold	NOUN
ejpam-2541	174	10	.	.	PUNCT
ejpam-2541	175	1	bulletin	bulletin	NOUN
ejpam-2541	175	2	of	of	ADP
ejpam-2541	175	3	the	the	DET
ejpam-2541	175	4	calcutta	calcutta	PROPN
ejpam-2541	175	5	mathematical	mathematical	ADJ
ejpam-2541	175	6	society	society	NOUN
ejpam-2541	175	7	,	,	PUNCT
ejpam-2541	175	8	81(4):315–320	81(4):315–320	NOUN
ejpam-2541	175	9	,	,	PUNCT
ejpam-2541	175	10	1989	1989	NUM
ejpam-2541	175	11	.	.	PUNCT
ejpam-2541	176	1	[	[	X
ejpam-2541	176	2	3	3	X
ejpam-2541	176	3	]	]	PUNCT
ejpam-2541	176	4	m.	m.	NOUN
ejpam-2541	176	5	beltagy	beltagy	NOUN
ejpam-2541	176	6	.	.	PUNCT
ejpam-2541	177	1	convex	convex	PROPN
ejpam-2541	177	2	and	and	CCONJ
ejpam-2541	177	3	starshaped	starshape	VERB
ejpam-2541	177	4	subset	subset	VERB
ejpam-2541	177	5	in	in	ADP
ejpam-2541	177	6	manifolds	manifold	NOUN
ejpam-2541	177	7	product	product	NOUN
ejpam-2541	177	8	.	.	PUNCT
ejpam-2541	178	1	communications	communication	NOUN
ejpam-2541	178	2	faculty	faculty	NOUN
ejpam-2541	178	3	of	of	ADP
ejpam-2541	178	4	sciences	sciences	PROPN
ejpam-2541	178	5	university	university	PROPN
ejpam-2541	178	6	of	of	ADP
ejpam-2541	178	7	ankara	ankara	PROPN
ejpam-2541	178	8	series	series	PROPN
ejpam-2541	178	9	a	a	PROPN
ejpam-2541	178	10	,	,	PUNCT
ejpam-2541	178	11	41(1	41(1	NOUN
ejpam-2541	178	12	-	-	PUNCT
ejpam-2541	178	13	2):35–44	2):35–44	NUM
ejpam-2541	178	14	,	,	PUNCT
ejpam-2541	178	15	1992	1992	NUM
ejpam-2541	178	16	.	.	PUNCT
ejpam-2541	179	1	[	[	X
ejpam-2541	179	2	4	4	X
ejpam-2541	179	3	]	]	PUNCT
ejpam-2541	179	4	m.	m.	NOUN
ejpam-2541	179	5	beltagy	beltagy	NOUN
ejpam-2541	179	6	.	.	PUNCT
ejpam-2541	180	1	sufficient	sufficient	ADJ
ejpam-2541	180	2	conditions	condition	NOUN
ejpam-2541	180	3	for	for	ADP
ejpam-2541	180	4	convexity	convexity	NOUN
ejpam-2541	180	5	in	in	ADP
ejpam-2541	180	6	manifolds	manifold	NOUN
ejpam-2541	180	7	without	without	ADP
ejpam-2541	180	8	focal	focal	ADJ
ejpam-2541	180	9	points	point	NOUN
ejpam-2541	180	10	.	.	PUNCT
ejpam-2541	181	1	commentationes	commentatione	NOUN
ejpam-2541	181	2	mathematicae	mathematicae	VERB
ejpam-2541	181	3	universitatis	universitatis	PROPN
ejpam-2541	181	4	carolinae	carolinae	PROPN
ejpam-2541	181	5	,	,	PUNCT
ejpam-2541	181	6	34(3):443–449	34(3):443–449	PROPN
ejpam-2541	181	7	,	,	PUNCT
ejpam-2541	181	8	1993	1993	NUM
ejpam-2541	181	9	.	.	PUNCT
ejpam-2541	182	1	[	[	X
ejpam-2541	182	2	5	5	X
ejpam-2541	182	3	]	]	PUNCT
ejpam-2541	182	4	m.	m.	NOUN
ejpam-2541	182	5	beltagy	beltagy	PROPN
ejpam-2541	182	6	and	and	CCONJ
ejpam-2541	182	7	a.	a.	PROPN
ejpam-2541	182	8	el	el	PROPN
ejpam-2541	182	9	-	-	PROPN
ejpam-2541	182	10	araby	araby	PROPN
ejpam-2541	182	11	.	.	PUNCT
ejpam-2541	183	1	convexity	convexity	NOUN
ejpam-2541	183	2	in	in	ADP
ejpam-2541	183	3	special	special	ADJ
ejpam-2541	183	4	types	type	NOUN
ejpam-2541	183	5	of	of	ADP
ejpam-2541	183	6	riemannian	riemannian	ADJ
ejpam-2541	183	7	manifolds	manifold	NOUN
ejpam-2541	183	8	.	.	PUNCT
ejpam-2541	184	1	bulletin	bulletin	NOUN
ejpam-2541	184	2	of	of	ADP
ejpam-2541	184	3	the	the	DET
ejpam-2541	184	4	calcutta	calcutta	PROPN
ejpam-2541	184	5	mathematical	mathematical	ADJ
ejpam-2541	184	6	society	society	NOUN
ejpam-2541	184	7	,	,	PUNCT
ejpam-2541	184	8	94(3):153–162	94(3):153–162	NUM
ejpam-2541	184	9	,	,	PUNCT
ejpam-2541	184	10	2002	2002	NUM
ejpam-2541	184	11	.	.	PUNCT
ejpam-2541	185	1	[	[	X
ejpam-2541	185	2	6	6	NUM
ejpam-2541	185	3	]	]	PUNCT
ejpam-2541	185	4	m.	m.	NOUN
ejpam-2541	185	5	beltagy	beltagy	NOUN
ejpam-2541	185	6	and	and	CCONJ
ejpam-2541	185	7	a.	a.	PROPN
ejpam-2541	185	8	el	el	PROPN
ejpam-2541	185	9	-	-	PROPN
ejpam-2541	185	10	araby	araby	PROPN
ejpam-2541	185	11	.	.	PUNCT
ejpam-2541	185	12	starshaped	starshape	VERB
ejpam-2541	185	13	sets	set	NOUN
ejpam-2541	185	14	in	in	ADP
ejpam-2541	185	15	riemannian	riemannian	ADJ
ejpam-2541	185	16	manifolds	manifold	NOUN
ejpam-2541	185	17	without	without	ADP
ejpam-2541	185	18	conjugate	conjugate	ADJ
ejpam-2541	185	19	points	point	NOUN
ejpam-2541	185	20	.	.	PUNCT
ejpam-2541	186	1	far	far	PROPN
ejpam-2541	186	2	east	east	PROPN
ejpam-2541	186	3	journal	journal	PROPN
ejpam-2541	186	4	of	of	ADP
ejpam-2541	186	5	mathematical	mathematical	ADJ
ejpam-2541	186	6	sciences	science	NOUN
ejpam-2541	186	7	,	,	PUNCT
ejpam-2541	186	8	6(2):187–196	6(2):187–196	NUM
ejpam-2541	186	9	,	,	PUNCT
ejpam-2541	186	10	2002	2002	NUM
ejpam-2541	186	11	.	.	PUNCT
ejpam-2541	187	1	references	reference	NOUN
ejpam-2541	187	2	63	63	NUM
ejpam-2541	188	1	[	[	X
ejpam-2541	188	2	7	7	NUM
ejpam-2541	188	3	]	]	X
ejpam-2541	188	4	v.	v.	ADP
ejpam-2541	188	5	boltyanski	boltyanski	NOUN
ejpam-2541	188	6	,	,	PUNCT
ejpam-2541	188	7	h.	h.	PROPN
ejpam-2541	188	8	martini	martini	PROPN
ejpam-2541	188	9	,	,	PUNCT
ejpam-2541	188	10	and	and	CCONJ
ejpam-2541	188	11	p.s	p.s	PROPN
ejpam-2541	188	12	.	.	PUNCT
ejpam-2541	188	13	soltan	soltan	PROPN
ejpam-2541	188	14	.	.	PUNCT
ejpam-2541	189	1	excursions	excursion	NOUN
ejpam-2541	189	2	into	into	ADP
ejpam-2541	189	3	combinatorial	combinatorial	ADJ
ejpam-2541	189	4	geometry	geometry	NOUN
ejpam-2541	189	5	.	.	PUNCT
ejpam-2541	190	1	springer	springer	NOUN
ejpam-2541	190	2	,	,	PUNCT
ejpam-2541	190	3	berlin	berlin	PROPN
ejpam-2541	190	4	,	,	PUNCT
ejpam-2541	190	5	germany	germany	PROPN
ejpam-2541	190	6	,	,	PUNCT
ejpam-2541	190	7	1997	1997	NUM
ejpam-2541	190	8	.	.	PUNCT
ejpam-2541	191	1	[	[	X
ejpam-2541	191	2	8	8	NUM
ejpam-2541	191	3	]	]	X
ejpam-2541	191	4	l.	l.	PROPN
ejpam-2541	191	5	danzer	danzer	PROPN
ejpam-2541	191	6	,	,	PUNCT
ejpam-2541	191	7	b.	b.	PROPN
ejpam-2541	191	8	grünbaum	grünbaum	PROPN
ejpam-2541	191	9	,	,	PUNCT
ejpam-2541	191	10	and	and	CCONJ
ejpam-2541	191	11	v.	v.	ADP
ejpam-2541	191	12	klee	klee	PROPN
ejpam-2541	191	13	.	.	PUNCT
ejpam-2541	192	1	helly	helly	AUX
ejpam-2541	192	2	’s	’s	PART
ejpam-2541	192	3	theorem	theorem	NOUN
ejpam-2541	192	4	and	and	CCONJ
ejpam-2541	192	5	its	its	PRON
ejpam-2541	192	6	relatives	relative	NOUN
ejpam-2541	192	7	.	.	PUNCT
ejpam-2541	193	1	american	american	PROPN
ejpam-2541	193	2	mathematical	mathematical	PROPN
ejpam-2541	193	3	society	society	NOUN
ejpam-2541	193	4	,	,	PUNCT
ejpam-2541	193	5	providence	providence	NOUN
ejpam-2541	193	6	,	,	PUNCT
ejpam-2541	193	7	ri	ri	PROPN
ejpam-2541	193	8	,	,	PUNCT
ejpam-2541	193	9	1963	1963	NUM
ejpam-2541	193	10	.	.	PUNCT
ejpam-2541	194	1	[	[	X
ejpam-2541	194	2	9	9	NUM
ejpam-2541	194	3	]	]	PUNCT
ejpam-2541	194	4	r.	r.	PROPN
ejpam-2541	194	5	j.	j.	PROPN
ejpam-2541	194	6	dwilewicz	dwilewicz	PROPN
ejpam-2541	194	7	.	.	PUNCT
ejpam-2541	195	1	a	a	DET
ejpam-2541	195	2	short	short	ADJ
ejpam-2541	195	3	history	history	NOUN
ejpam-2541	195	4	of	of	ADP
ejpam-2541	195	5	convexity	convexity	NOUN
ejpam-2541	195	6	.	.	PUNCT
ejpam-2541	196	1	differential	differential	ADJ
ejpam-2541	196	2	geometry	geometry	NOUN
ejpam-2541	196	3	dynamical	dynamical	ADJ
ejpam-2541	196	4	systems	system	NOUN
ejpam-2541	196	5	,	,	PUNCT
ejpam-2541	196	6	11:112–129	11:112–129	NUM
ejpam-2541	196	7	,	,	PUNCT
ejpam-2541	196	8	2009	2009	NUM
ejpam-2541	196	9	.	.	PUNCT
ejpam-2541	197	1	[	[	X
ejpam-2541	197	2	10	10	NUM
ejpam-2541	197	3	]	]	PUNCT
ejpam-2541	197	4	s.	s.	PROPN
ejpam-2541	197	5	hosseini	hosseini	PROPN
ejpam-2541	197	6	and	and	CCONJ
ejpam-2541	197	7	m.	m.	PROPN
ejpam-2541	197	8	pouryayevali	pouryayevali	PROPN
ejpam-2541	197	9	.	.	PUNCT
ejpam-2541	198	1	on	on	ADP
ejpam-2541	198	2	the	the	DET
ejpam-2541	198	3	metric	metric	ADJ
ejpam-2541	198	4	projection	projection	NOUN
ejpam-2541	198	5	onto	onto	ADP
ejpam-2541	198	6	prox	prox	NOUN
ejpam-2541	198	7	-	-	PUNCT
ejpam-2541	198	8	regular	regular	ADJ
ejpam-2541	198	9	subsets	subset	NOUN
ejpam-2541	198	10	of	of	ADP
ejpam-2541	198	11	riemannian	riemannian	ADJ
ejpam-2541	198	12	manifolds	manifold	NOUN
ejpam-2541	198	13	.	.	PUNCT
ejpam-2541	198	14	proceedings	proceeding	NOUN
ejpam-2541	198	15	of	of	ADP
ejpam-2541	198	16	the	the	DET
ejpam-2541	198	17	american	american	PROPN
ejpam-2541	198	18	mathematical	mathematical	PROPN
ejpam-2541	198	19	society	society	NOUN
ejpam-2541	198	20	,	,	PUNCT
ejpam-2541	198	21	141(1):233	141(1):233	NUM
ejpam-2541	198	22	–	–	PUNCT
ejpam-2541	198	23	244	244	NUM
ejpam-2541	198	24	,	,	PUNCT
ejpam-2541	198	25	2013	2013	NUM
ejpam-2541	198	26	.	.	PUNCT
ejpam-2541	199	1	[	[	X
ejpam-2541	199	2	11	11	NUM
ejpam-2541	199	3	]	]	PUNCT
ejpam-2541	199	4	m.	m.	NOUN
ejpam-2541	199	5	a.	a.	NOUN
ejpam-2541	199	6	jiménez	jiménez	PROPN
ejpam-2541	199	7	,	,	PUNCT
ejpam-2541	199	8	g.	g.	PROPN
ejpam-2541	199	9	r.	r.	PROPN
ejpam-2541	199	10	garzón	garzón	PROPN
ejpam-2541	199	11	,	,	PUNCT
ejpam-2541	199	12	and	and	CCONJ
ejpam-2541	199	13	a.	a.	PROPN
ejpam-2541	199	14	r.	r.	PROPN
ejpam-2541	199	15	lizana	lizana	PROPN
ejpam-2541	199	16	.	.	PUNCT
ejpam-2541	200	1	optimality	optimality	NOUN
ejpam-2541	200	2	conditions	condition	NOUN
ejpam-2541	200	3	in	in	ADP
ejpam-2541	200	4	vector	vector	NOUN
ejpam-2541	200	5	optimization	optimization	NOUN
ejpam-2541	200	6	.	.	PUNCT
ejpam-2541	201	1	bentham	bentham	PROPN
ejpam-2541	201	2	science	science	NOUN
ejpam-2541	201	3	publishers	publisher	NOUN
ejpam-2541	201	4	,	,	PUNCT
ejpam-2541	201	5	2010	2010	NUM
ejpam-2541	201	6	.	.	PUNCT
ejpam-2541	202	1	[	[	X
ejpam-2541	202	2	12	12	NUM
ejpam-2541	202	3	]	]	PUNCT
ejpam-2541	202	4	a.	a.	NOUN
ejpam-2541	202	5	kiliçman	kiliçman	NOUN
ejpam-2541	202	6	and	and	CCONJ
ejpam-2541	202	7	w	w	NOUN
ejpam-2541	202	8	saleh	saleh	NOUN
ejpam-2541	202	9	.	.	PUNCT
ejpam-2541	203	1	a	a	DET
ejpam-2541	203	2	note	note	NOUN
ejpam-2541	203	3	on	on	ADP
ejpam-2541	203	4	starshaped	starshape	VERB
ejpam-2541	203	5	sets	set	NOUN
ejpam-2541	203	6	in2	in2	ADJ
ejpam-2541	203	7	-	-	ADJ
ejpam-2541	203	8	dimensional	dimensional	ADJ
ejpam-2541	203	9	manifolds	manifold	NOUN
ejpam-2541	203	10	without	without	ADP
ejpam-2541	203	11	conjugate	conjugate	ADJ
ejpam-2541	203	12	points	point	NOUN
ejpam-2541	203	13	.	.	PUNCT
ejpam-2541	204	1	journal	journal	NOUN
ejpam-2541	204	2	of	of	ADP
ejpam-2541	204	3	function	function	NOUN
ejpam-2541	204	4	spaces	space	NOUN
ejpam-2541	204	5	,	,	PUNCT
ejpam-2541	204	6	2014	2014	NUM
ejpam-2541	204	7	:	:	PUNCT
ejpam-2541	204	8	article	article	NOUN
ejpam-2541	204	9	i	i	PROPN
ejpam-2541	204	10	d	d	PROPN
ejpam-2541	204	11	675735	675735	NUM
ejpam-2541	204	12	,	,	PUNCT
ejpam-2541	204	13	3	3	NUM
ejpam-2541	204	14	pages	page	NOUN
ejpam-2541	204	15	,	,	PUNCT
ejpam-2541	204	16	2014	2014	NUM
ejpam-2541	204	17	.	.	PUNCT
ejpam-2541	205	1	[	[	X
ejpam-2541	205	2	13	13	NUM
ejpam-2541	205	3	]	]	X
ejpam-2541	205	4	h.	h.	PROPN
ejpam-2541	205	5	martini	martini	PROPN
ejpam-2541	205	6	and	and	CCONJ
ejpam-2541	205	7	k.	k.	PROPN
ejpam-2541	205	8	swanepoel	swanepoel	PROPN
ejpam-2541	205	9	.	.	PUNCT
ejpam-2541	206	1	generalized	generalize	VERB
ejpam-2541	206	2	convexity	convexity	NOUN
ejpam-2541	206	3	notions	notion	NOUN
ejpam-2541	206	4	and	and	CCONJ
ejpam-2541	206	5	combinatorial	combinatorial	ADJ
ejpam-2541	206	6	geometry	geometry	NOUN
ejpam-2541	206	7	.	.	PUNCT
ejpam-2541	207	1	congressus	congressus	PROPN
ejpam-2541	207	2	numerantium	numerantium	PROPN
ejpam-2541	207	3	,	,	PUNCT
ejpam-2541	207	4	164:65–93	164:65–93	NUM
ejpam-2541	207	5	,	,	PUNCT
ejpam-2541	207	6	2003	2003	NUM
ejpam-2541	207	7	.	.	PUNCT
ejpam-2541	208	1	[	[	X
ejpam-2541	208	2	14	14	NUM
ejpam-2541	208	3	]	]	X
ejpam-2541	208	4	h.	h.	PROPN
ejpam-2541	208	5	martini	martini	PROPN
ejpam-2541	208	6	and	and	CCONJ
ejpam-2541	208	7	k.j	k.j	PROPN
ejpam-2541	208	8	.	.	PROPN
ejpam-2541	208	9	swanepoel	swanepoel	PROPN
ejpam-2541	208	10	.	.	PUNCT
ejpam-2541	209	1	the	the	DET
ejpam-2541	209	2	geometry	geometry	NOUN
ejpam-2541	209	3	of	of	ADP
ejpam-2541	209	4	minkowski	minkowski	ADJ
ejpam-2541	209	5	spaces	space	NOUN
ejpam-2541	209	6	-	-	PUNCT
ejpam-2541	209	7	a	a	DET
ejpam-2541	209	8	survey	survey	NOUN
ejpam-2541	209	9	.	.	PUNCT
ejpam-2541	210	1	part	part	PROPN
ejpam-2541	210	2	ii	ii	PROPN
ejpam-2541	210	3	.	.	PUNCT
ejpam-2541	211	1	expositiones	expositione	NOUN
ejpam-2541	211	2	mathematicae	mathematicae	VERB
ejpam-2541	211	3	,	,	PUNCT
ejpam-2541	211	4	22(2):93–144	22(2):93–144	NUM
ejpam-2541	211	5	,	,	PUNCT
ejpam-2541	211	6	2004	2004	NUM
ejpam-2541	211	7	.	.	PUNCT
ejpam-2541	212	1	[	[	X
ejpam-2541	212	2	15	15	NUM
ejpam-2541	212	3	]	]	X
ejpam-2541	212	4	h.b	h.b	PROPN
ejpam-2541	212	5	.	.	PROPN
ejpam-2541	212	6	pandey	pandey	PROPN
ejpam-2541	212	7	.	.	PUNCT
ejpam-2541	213	1	cartesian	cartesian	ADJ
ejpam-2541	213	2	product	product	NOUN
ejpam-2541	213	3	of	of	ADP
ejpam-2541	213	4	two	two	NUM
ejpam-2541	213	5	manifolds	manifold	NOUN
ejpam-2541	213	6	.	.	PUNCT
ejpam-2541	214	1	indian	indian	ADJ
ejpam-2541	214	2	journal	journal	PROPN
ejpam-2541	214	3	of	of	ADP
ejpam-2541	214	4	pure	pure	ADJ
ejpam-2541	214	5	and	and	CCONJ
ejpam-2541	214	6	applied	applied	ADJ
ejpam-2541	214	7	mathematics	mathematic	NOUN
ejpam-2541	214	8	,	,	PUNCT
ejpam-2541	214	9	12(1):55–60	12(1):55–60	NUM
ejpam-2541	214	10	,	,	PUNCT
ejpam-2541	214	11	1981	1981	NUM
ejpam-2541	214	12	.	.	PUNCT
ejpam-2541	215	1	[	[	X
ejpam-2541	215	2	16	16	NUM
ejpam-2541	215	3	]	]	X
ejpam-2541	215	4	w.	w.	PROPN
ejpam-2541	215	5	saleh	saleh	PROPN
ejpam-2541	215	6	and	and	CCONJ
ejpam-2541	215	7	a.	a.	NOUN
ejpam-2541	215	8	kiliçman	kiliçman	NOUN
ejpam-2541	215	9	.	.	PUNCT
ejpam-2541	216	1	on	on	ADP
ejpam-2541	216	2	generalized	generalize	VERB
ejpam-2541	216	3	s	s	NOUN
ejpam-2541	216	4	-	-	ADJ
ejpam-2541	216	5	convex	convex	ADJ
ejpam-2541	216	6	functions	function	NOUN
ejpam-2541	216	7	on	on	ADP
ejpam-2541	216	8	fractal	fractal	ADJ
ejpam-2541	216	9	sets	set	NOUN
ejpam-2541	216	10	.	.	PUNCT
ejpam-2541	217	1	jp	jp	PROPN
ejpam-2541	217	2	journal	journal	PROPN
ejpam-2541	217	3	of	of	ADP
ejpam-2541	217	4	geometry	geometry	NOUN
ejpam-2541	217	5	and	and	CCONJ
ejpam-2541	217	6	topology	topology	NOUN
ejpam-2541	217	7	,	,	PUNCT
ejpam-2541	217	8	17(1):63–82	17(1):63–82	NUM
ejpam-2541	217	9	,	,	PUNCT
ejpam-2541	217	10	2015	2015	NUM
ejpam-2541	217	11	.	.	PUNCT
ejpam-2541	218	1	[	[	X
ejpam-2541	218	2	17	17	NUM
ejpam-2541	218	3	]	]	X
ejpam-2541	218	4	g.	g.	PROPN
ejpam-2541	218	5	santhanam	santhanam	PROPN
ejpam-2541	218	6	.	.	PUNCT
ejpam-2541	219	1	isoperimetric	isoperimetric	ADJ
ejpam-2541	219	2	upper	upper	ADJ
ejpam-2541	219	3	bounds	bound	NOUN
ejpam-2541	219	4	for	for	ADP
ejpam-2541	219	5	the	the	DET
ejpam-2541	219	6	first	first	ADJ
ejpam-2541	219	7	eigenvalue	eigenvalue	PROPN
ejpam-2541	219	8	.	.	PUNCT
ejpam-2541	220	1	proceedingsmathematical	proceedingsmathematical	ADJ
ejpam-2541	220	2	sciences	sciences	PROPN
ejpam-2541	220	3	,	,	PUNCT
ejpam-2541	220	4	122(3):375–384	122(3):375–384	NUM
ejpam-2541	220	5	,	,	PUNCT
ejpam-2541	220	6	2012	2012	NUM
ejpam-2541	220	7	.	.	PUNCT
ejpam-2541	221	1	[	[	X
ejpam-2541	221	2	18	18	NUM
ejpam-2541	221	3	]	]	PUNCT
ejpam-2541	221	4	c.	c.	NOUN
ejpam-2541	221	5	udriste	udriste	NOUN
ejpam-2541	221	6	.	.	PUNCT
ejpam-2541	222	1	convex	convex	NOUN
ejpam-2541	222	2	functions	function	NOUN
ejpam-2541	222	3	and	and	CCONJ
ejpam-2541	222	4	optimization	optimization	NOUN
ejpam-2541	222	5	methods	method	NOUN
ejpam-2541	222	6	on	on	ADP
ejpam-2541	222	7	riemannian	riemannian	ADJ
ejpam-2541	222	8	manifolds	manifold	NOUN
ejpam-2541	222	9	,	,	PUNCT
ejpam-2541	222	10	volume	volume	NOUN
ejpam-2541	222	11	297	297	NUM
ejpam-2541	222	12	.	.	PUNCT
ejpam-2541	223	1	springer	springer	NOUN
ejpam-2541	223	2	science	science	PROPN
ejpam-2541	223	3	&	&	CCONJ
ejpam-2541	223	4	business	business	NOUN
ejpam-2541	223	5	media	medium	NOUN
ejpam-2541	223	6	,	,	PUNCT
ejpam-2541	223	7	1994	1994	NUM
ejpam-2541	223	8	.	.	PUNCT
ejpam-2541	224	1	[	[	X
ejpam-2541	224	2	19	19	NUM
ejpam-2541	224	3	]	]	X
ejpam-2541	224	4	f.	f.	PROPN
ejpam-2541	224	5	m.	m.	PROPN
ejpam-2541	224	6	valentine	valentine	PROPN
ejpam-2541	224	7	.	.	PUNCT
ejpam-2541	225	1	convex	convex	PROPN
ejpam-2541	225	2	subsets	subset	NOUN
ejpam-2541	225	3	.	.	PUNCT
ejpam-2541	226	1	mcgraw	mcgraw	PROPN
ejpam-2541	226	2	-	-	PUNCT
ejpam-2541	226	3	hill	hill	PROPN
ejpam-2541	226	4	series	series	PROPN
ejpam-2541	226	5	higher	high	ADJ
ejpam-2541	226	6	math	math	NOUN
ejpam-2541	226	7	.	.	PUNCT
ejpam-2541	226	8	,	,	PUNCT
ejpam-2541	226	9	mcgraw	mcgraw	PROPN
ejpam-2541	226	10	-	-	PUNCT
ejpam-2541	226	11	hill	hill	PROPN
ejpam-2541	226	12	,	,	PUNCT
ejpam-2541	226	13	new	new	PROPN
ejpam-2541	226	14	york	york	PROPN
ejpam-2541	226	15	,	,	PUNCT
ejpam-2541	226	16	1964	1964	NUM
ejpam-2541	226	17	.	.	PUNCT