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On Geometric Constants for Discrete Morrey Spaces

Adam Adam, Hendra Gunawan∗
Analysis and Geometry Group, Faculty of Mathematics and Natural Sciences,

Bandung Institute of Technology, Bandung 40132, Indonesia
adam_adam@students.itb.ac.id, hgunawan@math.itb.ac.id

∗Correspondence: hgunawan@math.itb.ac.id

Abstract. In this paper we prove that the n-th Von Neumann-Jordan constant and the n-th Jamesconstant for discrete Morrey spaces `pq where 1 ≤ p < q < ∞ are both equal to n. This resulttells us that the discrete Morrey spaces are not uniformly non-`1, and hence they are not uniformly
n-convex.

1. Introduction
Let n ≥ 2 be a non-negative integer and (X,‖·‖) be a Banach space. The n-th Von Neumann-

Jordan constant for X [6] is defined by
C
(n)
NJ
(X) := sup

{∑
±‖u1 ±u2 ±···±un‖

2
X

2n−1
∑n
i=1‖ui‖X

: ui 6=0, i =1,2, . . . ,n
}

and the n-th James constant for X [7] is defined by
C
(n)
J
(X) := sup{min‖u1 ±u2 ±···±un‖ : ui ∈ SX, i =1,2, . . . ,n}.

Note that in the definition of C(n)
NJ
(X), the sum ∑± is taken over all possible combinations of ±signs. Similarly, in the definition of C(n)
J
(X), the minimum is taken over all possible combinationsof ± signs, while the supremum is taken over all ui’s in the unit sphere SX := {u ∈ X : ‖u‖=1}.These constants measure some sort of convexity of a Banach space.We say that X is uniformly n-convex [2] if for every ε ∈ (0,n] there exists a δ ∈ (0,1) such thatfor every u1,u2, . . . ,un ∈ SX with ‖u1 ±u2 ±···±un‖≥ ε for all combinations of ± signs exceptfor ‖u1 +u2 + · · ·+un‖, we have

‖u1 +u2 + · · ·+un‖≤ n(1−δ).

Received: 31 Aug 2021.
Key words and phrases. n-th Von Neumann-Jordan constant; n-th James constant; discrete Morrey spaces; uniformlynon-`1 spaces; uniformly n-convex spaces. 1

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https://doi.org/10.28924/ada/ma.2.2
https://orcid.org/0000-0001-7879-8321


Eur. J. Math. Anal. 10.28924/ada/ma.2.2 2
Meanwhile, we say that X is uniformly non-`1n [1, 5, 8] if there exists a δ ∈ (0,1) such that for every
u1,u2, . . . ,un ∈ SX we have

min‖u1 ±u2 ±···±un‖≤ n(1−δ).

Note that for n =2, uniformly non-`1n spaces are known as uniformly nonsquare spaces, while for
n = 3 they are known as uniformly non-octahedral spaces. One may verify that if X is uniformly
n-convex, then X is uniformly non-`1n [2].Now a few remarks about the two constants, and their associations with the uniformly non-`1nand uniformly n-convex properties.

• 1≤ C(n)
NJ
(X)≤ n and C(n)

NJ
(X)=1 if and only if X is a Hilbert space [6].

• 1 ≤ C(n)
J
(X) ≤ n. If dim(X) = ∞, then √n ≤ C(n)

J
(X) ≤ n. Moreover, if X is a Hilbertspace, then C(n)

J
(X)=

√
n [7].

• X is uniformly non-`1n if and only if C(n)NJ(X) < n [6].
• X is uniformly non-`1n if and only if C(n)J (X) < n [7].

The last two statements tell us that if C(n)
NJ
(X)= n or C(n)

J
(X)= n, then X is not uniformly non-`1nand hence not uniformly n-convex.In this paper, we shall compute the value of the two constants for discrete Morrey spaces. Let

ω :=N∪{0} and m =(m1,m2, . . . ,md)∈Zd . Define
Sm,N := {k ∈Zd : ‖k −m‖∞ ≤ N}

where N ∈ ω and ‖m‖∞ = max{|mi| : 1 ≤ i ≤ d}. Denote by |Sm,N| the cardinality of Sm,N for
m ∈Zd and N ∈ ω. Then we have |Sm,N|=(2N +1)d .Now let 1 ≤ p ≤ q < ∞. Define `pq = `pq(Zd) to be the discrete Morrey space as introducedin [3], which consists of all sequences x :Zd →R with

‖x‖`pq := sup
m∈Zd,N∈ω

|Sm,N|
1
q
−1
p

( ∑
k∈Sm,N

|xk|p
)1
p

< ∞,

where x := (xk) with k ∈ Zd . One may observe that these discrete Morrey spaces are Banachspaces [3]. Note, in particular, that for p = q, we have `pq = `q.From [4] we already know that CNJ(`pq) = CJ(`pq) = 2 for 1 ≤ p < q < ∞, which impliesthat `pq are not uniformly nonsquares for those p’s and q’s. In this paper, we shall show that
C
(n)
NJ
(`
p
q) = C

(n)
J
(`
p
q) = n for 1 ≤ p < q < ∞, which leads us to the conclusion that `pq arenot uniformly non-`1n for those p’s and q’s, which is sharper than the existing result. (If X is notuniformly non-`1n, then X is not uniformly non-`1n−1, provided that n ≥ 3.)

https://doi.org/10.28924/ada/ma.2.2


Eur. J. Math. Anal. 10.28924/ada/ma.2.2 3
2. Main Results

The value of the n-th Von Neumann-Jordan constant and the n-th James constant for discreteMorrey spaces are stated in the following theorems. To understand the idea of the proof, we firstpresent the result for n =3.
Theorem 2.1. For 1≤ p < q < ∞, we have C(3)

NJ
(`
p
q(Zd))= C

(3)
J
(`
p
q(Zd))=3.

Proof. To prove the theorem, it suffices for us to find x(1),x(2),x(3) ∈ `pq such that∑
±‖x

(1) ±x(2) ±x(3)‖2
`
p
q

22
∑3
i=1‖x(i)‖`pq

=3

for the Von Neumann-Jordan constant, and
min‖x(1) ±x(2) ±x(3)‖`pq =3

for the James constant.
Case 1: d =1. Let j ∈Z be a nonnegative, even integer such that j > 4 qq−p −1, or equivalently

(j +1)
1
q
−1
p < 4

−1
p .

Construct x(1),x(2),x(3) ∈ `pq(Z) as follows:
• x(1) =(x(1)

k
)k∈Z is defined by

x
(1)
k
=

1, k =0, j,2j,3j,
0, otherwise;

• x(2) =(x(2)
k
)k∈Z is defined by

x
(2)
k
=


1, k =0, j,

−1, k =2j,3j,

0, otherwise;
• x(3) =(x(3)

k
)k∈Z is defined by

x
(3)
k
=


1, k =0,2j,

−1, k = j,3j,

0, otherwise.

https://doi.org/10.28924/ada/ma.2.2


Eur. J. Math. Anal. 10.28924/ada/ma.2.2 4
The three sequences are in the unit sphere of `pq(Z). Indeed, for the first sequence, we have

‖x(1)‖`pq = sup
m∈Z,N∈ω

|Sm,N|
1
q
−1
p

( ∑
k∈Sm,N

|x(1)
k
|p
)1
p

= sup
m∈Z∩[0,3j],N∈Z∩[0,3j/2]

|Sm,N|
1
q
−1
p

( ∑
k∈Sm,N

|x(1)
k
|p
)1
p

=max{1,(j +1)
1
q
−1
p2

1
p ,(2j +1)

1
q
−1
p3

1
p ,(3j +1)

1
q
−1
p4

1
p}.

Since (3j +1)1q−1p < (2j +1)1q−1p < (j +1)1q−1p < 4−1p , we get ‖x(1)‖`pq = 1. Similarly, one mayobserve that ‖x(2)‖`pq = ‖x(3)‖`pq =1.Next, we observe that
x
(1)
k
+x

(2)
k
+x

(3)
k
=



3, k =0,

1, k = j,2j,

−1, k =3j,

0, otherwise;

x
(1)
k
+x

(2)
k
−x(3)
k
=



3, k = j,

1, k =0,3j,

−1, k =2j,

0, otherwise;

x
(1)
k
−x(2)
k
+x

(3)
k
=



3, k =2j,

1, k =0,3j,

−1, k = j,

0, otherwise;

x
(1)
k
−x(2)
k
−x(3)
k
=



3, k =3j,

1, k = j,2j,

−1, k =0,

0, otherwise.We first compute that
‖x(1)+x(2)+x(3)‖`pq =max{3,(j+1)

1
q
−1
p(3p+1)

1
p ,(2j+1)

1
q
−1
p(3p+2)

1
p ,(3j+1)

1
q
−1
p(3p+3)

1
p}.

Notice that
• (j +1)

1
q
−1
p(3p +1)

1
p <

(
3p+1p

4

)1
p
< (3p)

1
p =3.

• (2j +1)
1
q
−1
p(3p +2)

1
p < (j +1)

1
q
−1
p(3p +2)

1
p <

(
3p+2
4

)1
p
< 3.

https://doi.org/10.28924/ada/ma.2.2


Eur. J. Math. Anal. 10.28924/ada/ma.2.2 5
• (3j +1)

1
q
−1
p(3p +3)

1
p < (j +1)

1
q
−1
p(3p +3)

1
p <

(
3p+3
4

)1
p
< 3.

Hence, we obtain ‖x(1) +x(2) +x(3)‖`pq =3.Similarly, we have
‖x(1) ±x(2) ±x(3)‖`pq = sup

m∈Z∩[0,3j],N∈Z∩[0,3j/2]
|Sm,N|

1
q
−1
p

( ∑
k∈Sm,N

|x(1)
k
±x(2)
k
±x(3)
k
|p
)1
p

=3

for every combination of ± signs.
Consequently, ∑±‖x(1)±x(2)±x(3)‖2`pq

22
∑3
i=1‖x(i)‖`pq

= 3 and min‖x(1) ± x(2) ± x(3)‖`pq = 3, so we come to theconclusion that
C
(3)
NJ
(`pq(Z))= C

(3)
J
(`pq(Z))=3.

Case 2: d > 1. Let j ∈ Z be a nonnegative, even integer such that j > 4 qd(q−p) −1, which isequivalent to
(j +1)

d(1
q
−1
p
)
< 4

−1
p .

We then construct x(1),x(2),x(3) ∈ `pq(Zd) as follows:
• x(1) =(x(1)

k
)k∈Zd is defined by

x
(1)
k
=

1, k =(0,0, . . . ,0),(j,0, . . . ,0),(2j,0, . . . ,0),(3j,0, . . . ,0),
0, otherwise;

• x(2) =(x(2)
k
)k∈Zd is defined by

x
(2)
k
=


1, k =(0,0, . . . ,0),(j,0, . . . ,0),

−1, k =(2j,0, . . . ,0),(3j,0, . . . ,0),

0, otherwise;
• x(3) =(x(3)

k
)k∈Zd is defined by

x
(3)
k
=


1, k =(0,0, . . . ,0),(2j,0, . . . ,0),

−1, k =(j,0, . . . ,0),(3j,0, . . . ,0),

0, otherwise.
As in the case where d =1, one may observe that

‖x(1)‖`pq = sup
m∈Zd,N∈ω

|Sm,N|
1
q
−1
p

( ∑
k∈Sm,N

|x(1)
k
|p
)1
p

=max{1,(j +1)d(
1
q
−1
p
)
2
1
p ,(2j +1)

d(1
q
−1
p
)
3
1
p ,(3j +1)

d(1
q
−1
p
)
4
1
p}

=1.

https://doi.org/10.28924/ada/ma.2.2


Eur. J. Math. Anal. 10.28924/ada/ma.2.2 6
We also get ‖x(2)‖`pq = ‖x(3)‖`pq =1. Moreover, through similar observation as in the 1-dimensionalcase, we have

‖x(1) ±x(2) ±x(3)‖`pq =3for every possible combinations of ± signs. It thus follows that
C
(3)
J
(`pq(Z

d))= sup{min‖x1 ±x2 ±x3‖`pq : x1,x2,x3 ∈ S`pq}=3

and
C
(3)
NJ
(`pq(Z

d))= sup

{∑
±‖x1 ±x2 ±x3‖

2
`
p
q

22
∑3
i=1‖xi‖`pq

: xi 6=0, i =1,2,3
}
=3.

�

We now state the general result for n ≥ 3. (The proof is also valid for n =2, which amounts tothe work of [3].)
Theorem 2.2. For 1≤ p < q < ∞, we have C(n)

NJ
(`
p
q(Zd))= C

(n)
J
(`
p
q(Zd))= n.

Proof. As for n =3, we shall consider the case where d =1 first, and then the case where d > 1later.
Case 1: d =1. Let j ∈Z be a nonnegative, even integer such that j > 2(n−1)( qq−p)−1, which isequivalent to

(j +1)
1
q
−1
p < 2

−(n−1)
p .

We construct x(i) ∈ `pq ∈Z for i =1,2, . . . ,n as follows:
• x(1) =(x(1)

k
)k∈Z is defined by

x
(1)
k
=

1, k ∈ S
(1)
1 ,

0, otherwise,
where

S
(1)
1 = {0, j,2j,3j, . . . ,(2

n−1 −1)j};

• x(i) =(x(i)
k
)k∈Z for 2≤ i ≤ n is defined by

x
(i)
k
=


1, k ∈ S(i)1 ,

−1, k ∈ S(i)−1,

0, otherwise,
with the following rules: Write P = {0, j,2j, . . . ,(2n−1 −1)j} as

P = P
(i)
1 ∪P

(i)
2 ∪·· ·∪P

(i)

2i−1

https://doi.org/10.28924/ada/ma.2.2


Eur. J. Math. Anal. 10.28924/ada/ma.2.2 7
where P(i)1 consists of the first 2n−12i−1 terms of P , P(i)2 consists of the next 2n−12i−1 terms of P ,and so on. Then S(i)1 and S(i)−1 are given by

S
(i)
1 = P

(i)
1 ∪P

(i)
3 ∪·· ·∪P

(i)

2i−1−1,

S
(i)
−1 = P

(i)
2 ∪P

(i)
4 ∪·· ·∪P

(i)

2i−1
.

For example, for i =2, x(2) =(x(2)
k
)k∈Z is defined by

x
(2)
k
=


1, k ∈ S(2)1 ,

−1, k ∈ S(2)−1,

0, otherwise,
where

S
(2)
1 =

{
0, j,2j,3j, . . . ,

(2n−1
2
−1
)
j

}
S
(2)
−1 =

{(2n−1
2

)
j,
(2n−1
2
+1
)
j, . . . ,(2n−1 −1)j

}
;

Note that the largest absolute value of the terms of x(i) in the above construction will beequal to 1 for each i = 1, . . . ,n. Next, since the number of possible combinations of ± signs in
x(1) ± x(2) ± ···± x(n) is 2n−1, the above construction will give us 1+1+ · · ·+1 = n as thelargest absolute value of x(1)±x(2)±···±x(n) for every combination of ± signs. This means that,if x(1) ±x(2) ±···±x(n) =(xk)k∈Z, then max

k∈Z
|xk|= n.Let us now compute the norms. For x(1), we have

‖x(1)‖`pq = sup
m∈Z,N∈ω

|Sm,N|
1
q
−1
p

( ∑
k∈Sm,N

|x(1)
k
|p
)1
p

= sup
m∈Z∩[0,(2n−1−1)j],N∈Z∩[0,(2n−1−1)j/2]

|Sm,N|
1
q
−1
p

( ∑
k∈Sm,N

|x(1)
k
|p
)1
p

=max{1,(j +1)
1
q
−1
p2

1
p ,(2j +1)

1
q
−1
p3

1
p , . . . ,((2n−1 −1)j +1)

1
q
−1
p2

n−1
p }.

For each r =1,2, . . . ,2n−1 −1, we have (rj +1)1q−1p ≤ (j +1)1q−1p and (r +1)1p ≤ 2n−1p , so that
(rj +1)

1
q
−1
p(r +1)

1
p ≤ (j +1)

1
q
−1
p2

n−1
p < 2

−n−1
p 2

n−1
p =1.

Hence we obtain ‖x(1)‖`pq =1. Similarly, one may verify that
‖x(2)‖`pq = ‖x

(3)‖`pq = · · ·= ‖x
(n)‖`pq =1.

https://doi.org/10.28924/ada/ma.2.2


Eur. J. Math. Anal. 10.28924/ada/ma.2.2 8
Next, we shall compute the norms of x(1)±x(2)±···±x(n). Write x(1)+x(2)+· · ·+x(n) =(xk)k∈Zwhere

xk :=



a1, k =0,

a2, k = j,

a3, k =2j,...
a2n−1, k =(2

n−1 −1)j,

0, otherwise,
with a1 = n and |ai| < n for i =2,3, . . . ,(2n−1)j. Accordingly, we have
‖x(1) +x(2) + · · ·+x(n)‖`pq = sup

m∈Z,N∈ω
|Sm,N|

1
q
−1
p

( ∑
k∈Sm,N

|xk|p
)1
p

= sup
m∈Z∩[0,(2n−1−1)j],N∈Z∩[0,(2n−1−1)j/2]

|Sm,N|
1
q
−1
p

( ∑
k∈Sm,N

|xk|p
)1
p

=max
{
n,(j +1)

1
q
−1
p(np +a

p
2)
1
p ,(2j +1)

1
q
−1
p(np +a

p
2 +a

p
3)
1
p ,

. . . ,((2n−1 −1)j +1)
1
q
−1
p
(
np +

2n−1∑
i=2

a
p
i

)1
p

}
.

Since (rj +1)1q−1p ≤ (j +1)1q−1p for each r =1,2, . . . ,2n−1 −1, we obtain
(rj +1)

1
q
−1
p
(
np +

r+1∑
i=2

a
p
i

)1
p ≤ (j +1)

1
q
−1
p
(
np +

r+1∑
i=2

a
p
i

)1
p

< 2
−(n−1)

p
(
np +

r+1∑
i=2

a
p
i

)1
p

< 2
−(n−1)

p (np +np + · · ·+np︸ ︷︷ ︸
r +1 times

)
1
p

=2
−(n−1)

p (r +1)
1
p(np)

1
p

≤ 2−
(n−1)
p 2

(n−1)
p n

= n.

It thus follows that
‖x(1) +x(2) + · · ·+x(n)‖`pq = n.

As we have remarked earlier, the largest absolute value of x(1) ± x(2) ±···± x(n) is equal to
n for every combination of ± signs. Moreover, it is clear that for k /∈ {0,2j, . . . ,(2n−1 −1)j}, the

https://doi.org/10.28924/ada/ma.2.2


Eur. J. Math. Anal. 10.28924/ada/ma.2.2 9
k-th term of x(1) ±x(2) ±···±x(n) is equal to 0. Hence, we obtain
‖x(1) ±x(2) ±···±x(n)‖`pq = sup

m∈Z,N∈ω
|Sm,N|

1
q
−1
p

( ∑
k∈Sm,N

|x(1)
k
±x(2)
k
±···±x(n)

k
|p
)1
p

= sup
m∈Z∩[0,(2n−1−1)j],N∈Z∩[0,(2n−1−1)j/2]

|Sm,N|
1
q
−1
p

( ∑
k∈Sm,N

|x(1)
k
±x(2)
k
±···±x(n)

k
|p
)1
p

= n.

Consequently, we get ∑
±‖x

(1) ±x(2) ±···±x(n)‖2
`
p
q

2n−1
∑n
i=1‖xi‖`pq

=
2n−1n2

2n−1n
= n

and
min‖x(1) ±x(2) ±···±x(n)‖`pq = n,whence
C
(n)
NJ
(`pq(Z))= C

(n)
J
(`pq(Z))= n.

Case 2: d > 1. Here we choose j ∈ Z to be a nonnegative, even integer such that j >
2
(n−1
d
)(

q
q−p) −1 or, equivalently,

(j +1)
d(1
q
−1
p
)
< 2

−(n−1)
p .

Then, using the sequences
x(i) =(x

(i)
k1
)k1∈Z ∈ `

p
q(Z), i =1, . . . ,n,

in the case where d =1, we now define x(i) := (x(i)
k
)k∈Zd ∈ `

p
q(Zd) for i =1, . . . ,n, where

x
(i)
k
=

x
(i)
k1
, k =(k1,0,0, . . . ,0),

0, otherwise.
We shall then obtain

C
(n)
NJ
(`pq(Z

d))= C
(n)
J
(`pq(Z

d))= n,

as desired. �
Corollary 2.2.1. For 1≤ p < q < ∞, the space `pq is not uniformly non-`1n.

Corollary 2.2.2. For 1≤ p < q < ∞, the space `pq is not uniformly n-convex.

Acknowledgement. The work is part of the first author’s thesis. Both authors are supported byP2MI 2021 Program of Bandung Institute of Technology.

https://doi.org/10.28924/ada/ma.2.2


Eur. J. Math. Anal. 10.28924/ada/ma.2.2 10
References

[1] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, 2nd Ed., North Holland, Amsterdam- NewYork-Oxford, 1985. https://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=
PASCAL82X0319279.[2] H. Gunawan, D.I. Hakim, A.S. Putri, On geometric properties of Morrey spaces, Ufimsk. Mat. Zh. 13 (2021) 131–136.
https://doi.org/10.13108/2021-13-1-131.[3] H. Gunawan, E. Kikianty, C. Schwanke, Discrete Morrey spaces and their inclusion properties, Math. Nachr. 291(2018) 1283–1296. https://doi.org/10.1002/mana.201700054.[4] H. Gunawan, E. Kikianty, Y. Sawano, and C. Schwanke, Three geometric constants for Morrey spaces, Bull. Korean.Math. Soc. 56 (2019) 1569-1575. https://doi.org/10.4134/BKMS.b190010.[5] R.C. James, Uniformly non-square banach spaces, Ann. Math. 80 (1964) 542-550. https://doi.org/10.2307/
1970663.[6] M. Kato, Y. Takahashi, and K. Hashimoto, On n-th Von Neumann-Jordan constants for Banach spaces, Bull. KyushuInst. Tech. 45 (1998), 25-33. https://ci.nii.ac.jp/naid/110000079659.[7] L. Maligranda, L. Nikolova, L.-E. Persson, T. Zachariades, On n-th James and Khintchine constants of Banachspaces, Math. Inequal. Appl. 1 (2007) 1–22. https://doi.org/10.7153/mia-11-01.[8] W.A. Wojczynski, Geometry and martingales in Banach spaces, Part II, in: Probability in Banach Spaces IV, J.Kuelbs, ed., Marcel-Dekker, 1978, 267–517. https://doi.org/10.1201/9780429462153.

https://doi.org/10.28924/ada/ma.2.2
https://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=PASCAL82X0319279
https://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=PASCAL82X0319279
https://doi.org/10.13108/2021-13-1-131
https://doi.org/10.1002/mana.201700054
https://doi.org/10.4134/BKMS.b190010
https://doi.org/10.2307/1970663
https://doi.org/10.2307/1970663
https://ci.nii.ac.jp/naid/110000079659
https://doi.org/10.7153/mia-11-01
https://doi.org/10.1201/9780429462153

	1. Introduction
	2. Main Results
	References

