Normal structure coefficient

http://jipam.vu.edu.au/

Volume 7, Issue 1, Article 18, 2006

ON ESTIMATES OF THE GENERALIZED JORDAN-VON NEUMANN CONSTANT OF BANACH SPACES

CHANGSEN YANG AND FENGHUI WANG DEPARTMENT OFMATHEMATICS

HENANNORMALUNIVERSITY

XINXIANG453007, CHINA

yangchangsen0991@sina.com DEPARTMENT OFMATHEMATICS

LUOYANGNORMALUNIVERSITY

LUOYANG471022, CHINA. wfenghui@163.com

Received 27 June, 2005; accepted 17 January, 2006 Communicated by S.S. Dragomir

ABSTRACT. In this paper, we study the generalized Jordan-von Neumann constant and obtain its estimates for the normal structure coefficientN(X),improving the known results of S. Dhom- pongsa.

Key words and phrases: Generalized Jordan-von Neumann constant; Normal structure coefficient.

2000 Mathematics Subject Classification. 46B20.

1. INTRODUCTION

It is well known that normal structure and uniform normal structure play an important role in fixed point theory. So it is worthwhile studying the relationship between uniform normal structure and other geometrical constants of Banach spaces. Recently J. Gao [5] proved that δ(1 +) > /2implies that a Banach space X has uniform normal structure. Kato et al. [6]

obtained

(1.1) N(X)≥

C_NJ(X)− 1 4

−¹₂

,

which implies that X has uniform normal structure if C_NJ(X) < 5/4.S. Dhompongsa et al.

[3, 4] proved thatC_NJ(X)<(3 +√

5)/4orC_NJ(a, X)<(1 +a)²/(1 +a²)for somea∈[0,1]

implies thatX has uniform normal structure. HoweverCNJ(a, X) <(1 +a)²/(1 +a²)is not

ISSN (electronic): 1443-5756 c

2006 Victoria University. All rights reserved.

Supported by Natural Science Fund of Henan Province (No.2003110006).

The authors would like to express their sincere thanks to the referee for his valuable suggestions.

194-05

a sharp condition forX to have uniform normal structure especially whenais close to 0. Our aim is to improve the result of S. Dhompongsa.

We shall assume throughout this paper that X is a Banach space and X^∗ its dual space.

We will use S_X to denote the unit sphere of X. A Banach space X is called non-trivial if dimX ≥2.A Banach spaceXis called uniformly nonsquare if for anyx, y ∈ S_X there exists δ > 0, such that either kx−yk/2 ≤ 1−δ, or kx+yk/2 ≤ 1−δ. Uniformly nonsquare spaces are superreflexive. Let C be a nonempty bounded convex subset of X. The number diamC = sup{kx − yk : x, y ∈ C} is called the diameter of C. The number r(C) = inf{sup{kx−yk : x ∈ C} : y ∈ C}is called the Chebyshev radius of C. By Z(C)we will denote the set of allx∈ C at which this infimum is attained. It is called the Chebyshev center ofC. Bynum [2] introduced the following normal structure coefficient

(1.2) N(X) = inf{diamC},

where the infimum is taken over all closed convex subsetsC ofX with r(C) = 1.Obviously 1≤N(X)≤2andXis said to have uniform normal structure providedN(X)>1.Moreover ifX is reflexive, then the infimum in the definition ofN(X)may as well be taken over all con- vex hulls of finite subsets ofX[1]. In connection with a famous work of Jordan-von Neumann concerning inner products, the Jordan-von Neumann constantC_NJ(X)ofXwas introduced by Clarkson as the smallest constantCfor which

1

C ≤ kx+yk²+kx−yk² 2(kxk²+kyk²) ≤C

holds for allx, ywith(x, y)6= (0,0).IfCis the best possible in the right hand side of the above inequality then so is1/C on the left. Hence

(1.3) C_NJ(X) = sup

kx+yk²+kx−yk²

2(kxk²+kyk²) :x, y ∈X not both zero

.

The statements without explicit reference have been taken from Kato et al. [6]. In [3] S.

Dhompongsa generalized this definition in the following sense.

(1.4) C_NJ(a, X)

= sup

kx+yk²+kx−zk²

2kxk²+kyk²+kzk² :x, y, z ∈Xnot all zero andky−zk ≤akxk

whereais a nonnegative parameter. Obviously,CNJ(X) = CNJ(0, X).

2. MAINRESULTS

Our proofs are based on an idea due to S. Prus [7]. LetCbe a convex hull of a finite subset ofX.SinceCis compact, there exists an elementy∈Csuch that

(2.1) sup

x∈C

kx−yk=r(C).

Translating the setC we can assume thaty= 0.The following result is [7, Theorem 2.1].

Proposition 2.1. Let C be a nonempty compact convex subset of a finite dimensional Banach space X and x0 ∈ C. If x0 ∈ Z(C), then there exist elements x1. . . , xn ∈ C, functionals x^∗₁, . . . , x^∗_n ∈SX^∗,and nonnegative scalarsλ1, . . . , λnsuch thatPn

i=1λi = 1, x^∗_i(x₀−x_i) = kx₀−x_ik=r(C)

fori= 1, . . . , nand

n

X

i=1

λ_ix^∗_i(x−x₀)≥0 for everyx∈C.

Theorem 2.2. LetX be a non-trivial Banach space with the normal structure constantN(X).

Then for eacha, 0≤a≤1,

(2.2) N(X)≥

s

maxr∈[a,1]f(r) C_NJ(a, X) , where

f(r) = (1 +r)²+ (1 +a)²

2(1 +r²) , r ∈[a,1].

Proof.

Case 1: IfC_NJ(a, X) = 2,it suffices to note that

a≤r≤1max f(r) = max

a≤r≤1

(1 +r)²+ (1 +a)²

2(1 +r²) ≤ max

a≤r≤1

(1 +r)²+ (1 +r)² 2(1 +r²) ≤2.

In this case our estimate is a trivial one.

Case 2: If C_NJ(a, X) < 2, thenX is uniformly nonsquare and therefore reflexive [3]. Now let C be a convex hull of a finite subset of X such that r(C) = 1 and diamC = d. We can assume that sup{kxk : x ∈ C} = 1 and by Proposition 2.1 we get elementsx₁. . . , x_n, norm-one functionalsx^∗₁, . . . , x^∗_n and nonnegative numbersλ1, . . . , λn such thatPn

i=1λi = 1, x^∗_i(−xi) = kxik = 1 for i = 1, . . . , n and Pn

i=1λix^∗_i(xj) ≥ 0 for j = 1, . . . , n. For any r∈[a,1], let us set

x_i,j = x_i−x_j

d , y_i,j = r

dx_i, z_i,j = (r−a)x_i+ax_j

d for i, j = 1, . . . , n.

Obviouslykx_i,jk ≤1, ky_i,jk ≤r, kz_i,jk ≤r, andky_i,j−z_i,jk=akx_i,jk.It follows that

n

X

i,j=1

λ_iλ_j

kx_i,j+y_i,jk²+kx_i,j−z_i,jk²

≥

n

X

j=1

λ_j

n

X

i=1

λ_i[x^∗_i(x_i,j +y_i,j)]²+

n

X

i=1

λ_i

n

X

j=1

λ_j[x^∗_j(x_i,j −z_i,j)]²

=

n

X

j=1

λ_j

n

X

i=1

λ_i

1 +r d +1

dx^∗_i(x_j) 2

+

n

X

j=1

λ_j

n

X

i=1

λ_i

1 +a

d + 1 +a−r d x^∗_j(x_i)

2

= (1 +r)²

d² + 2(1 +r) d²

n

X

j=1

λj n

X

i=1

λix^∗_i(xj) + (1/d²)

n

X

j=1

λj n

X

i=1

λi[x^∗_i(xj)]²

+ (1 +a)²

d² +2(1 +a)(1 +a−r) d²

n

X

i=1

λi n

X

j=1

λjx^∗_j(xi)

+ (1 +a−r)² d²

n

X

i=1

λi n

X

j=1

λj[x^∗_j(xi)]²

≥ (1 +r)²

d² +(1 +a)²

d² for anyr∈[a,1].

Therefore there existi, jsuch that

kx_i,j+y_i,jk²+kx_i,j −z_i,jk² ≥ (1 +r)²

d² + (1 +a)² d² .

From the definition of the generalized Jordan-von Neumann constant we obtain that C_NJ(a, X)≥ kx_i,j +y_i,jk²+kx_i,j−z_i,jk²

2kxi,jk²+kyi,jk²+kzi,jk² ≥ (1 +r)²+ (1 +a)² 2(1 +r²)d² , which implies

d≥ s

maxr∈[a,1]f(r) C_NJ(a, X) .

SinceC is arbitrary, we obtain the desired estimate (2.2).

Lemma 2.3. Let 0 ≤ a ≤ 1 and r₀ =

p4 + (1 +a)⁴−(1 +a)².

2. Then a ≤ r₀ if a∈

0,√ 2−1

anda≥r₀ ifa∈√

2−1,1 . Proof. Ifa∈[0,√

2−1]then

4 + (1 +a)⁴−[(1 +a)²+ 2a]² = 4(1−a−3a²−a³)

=−4(a+ 1)

a+ 1 +√

2 a+ 1−√ 2

≥0, which impliesp

4 + (1 +a)⁴ ≥(1 +a)²+ 2a.Therefore r₀−a=

p4 + (1 +a)⁴−(1 +a)²

2 −a≥0.

Thus we obtain thatr₀ ≥aifa ∈[0,√

2−1].Similarly we getr₀ ≤aifa∈[√

2−1,1].

Theorem 2.4. LetXbe a non-trivial Banach space with the generalized Jordan-von Neumann constantCNJ(a, X). If

(2.3) C_NJ(a, X)< 2 + (1 +a)²+p

4 + (1 +a)⁴

4 for somea ∈h

0,√ 2−1i

, or

(2.4) C_NJ(a, X)< (1 +a)²

1 +a² for somea∈h√

2−1,1i , thenX has uniform normal structure.

Proof. Let

f(r) := (1 +r)² + (1 +a)²

2(1 +r²) , r₀ =

p4 + (1 +a)⁴−(1 +a)²

2 .

First we note thatf(r)is increasing on[0, r₀],and decreasing on[r₀,1].

Case 1: Ifa∈ 0,√

2−1

,thenr0 ∈[a,1]by Lemma 2.3, which implies

r∈[a,1]maxf(r) = f(r₀) = 2 + (1 +a)²+p

4 + (1 +a)⁴

4 .

By (2.2) and (2.3) we obtain that

N(X)≥ s

max_r∈[a,1]f(r) CNJ(a, X) >1

and henceXhas uniform normal structure.

Case 2: Ifa ∈ √

2−1,1

,thenr₀ ≤ aby Lemma 2.3 and thus f(r)is decreasing on[a,1], which implies

r∈[a,1]max f(r) = f(a) = (1 +a)² 1 +a² . By (2.2) and (2.4) we obtain that

N(X)≥ s

maxr∈[a,1]f(r) C_NJ(a, X) >1

and henceXhas uniform normal structure.

Note that

2 + (1 +a)²+p

4 + (1 +a)⁴

4 > (1 +a)²

1 +a² for alla∈h 0,√

2−1 . Thus this gives a strong improvement of [3, Theorem 3.6] and [4, Corollary 3.8].

Corollary 2.5 ([3, Theorem 3.6]). X has uniform normal structure ifC_NJ(X)<(3 +√ 5)/4.

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