We obtained the estimates of Normal structure coefficient N(X) by Neumann- Jordan constantCN J(X)of a Banach spaceXand found thatXhas uniform normal structure ifCN J(X) &lt

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

Volume 5, Issue 1, Article 10, 2004

ON ESTIMATES OF NORMAL STRUCTURE COEFFICIENTS OF BANACH SPACES

Y. Q. YAN

DEPARTMENT OFMATHEMATICS, SUZHOUUNIVERSITY

SUZHOU, JIANGSU, P.R. CHINA, 215006.

[email protected]

Received 22 August, 2003; accepted 09 January, 2004 Communicated by C.P. Niculescu

ABSTRACT. We obtained the estimates of Normal structure coefficient N(X) by Neumann- Jordan constantCN J(X)of a Banach spaceXand found thatXhas uniform normal structure ifC_{N J}(X) < (3 +√

5)/4.These results improved both Prus’ [6] and Kato, Maligranda and Takahashi’s [4] work.

Key words and phrases: Normal structure coefficient, Neumann-Jordan constant, Non-square constants, Banach space.

2000 Mathematics Subject Classification. 46B20, 46E30.

1. INTRODUCTION

LetX = (X,k · k)be a real Banach space. Geometrical properties of a Banach spaceXare determined by its unit ballB_X ={x∈X :kxk ≤1}or its unit sphereS_X ={x∈X :kxk= 1}.A Banach spaceX is called uniformly non-square if there exists aδ ∈ (0,1)such that for anyx, y ∈S_X eitherkx+yk/2≤1−δorkx−yk/2≤1−δ.The constant

J(X) = sup{min(kx+yk,kx−yk) :x, y ∈S_X}

is called the non-square constant of X in the sense of James. It is well-known that √ 2 ≤ J(X) ≤ 2 if dimX ≥ 2. The Neumann-Jordan constant CN J(X) of a Banach space X is defined by

C_{N J}(X) = sup

kx+yk²+kx−yk²

2(kxk²+kyk²) :x, y ∈X, not both zero

.

Clearly, 1 ≤ C_{N J}(X) ≤ 2. and X is a Hilbert space if and only if C_{N J}(X) = 1. Kato, Maligranda and Takahashi [4] proved that for any non-trivial Banach spaceX(dimX ≥2),

(1.1) 1

2J(X)² ≤C_{N J}(X)≤ J(X)² (J(X)−1)²+ 1.

ISSN (electronic): 1443-5756

c 2004 Victoria University. All rights reserved.

114-03

A Banach space X is said to have normal structure if r(K) < diam(K) for every non- singleton closed bounded convex subsetK ofX,where diam(K) = sup{kx−yk: x, y ∈K}

is the diameter ofK andr(K) = inf{sup{kx−yk:y∈K}:x∈K}is the Chebyshev radius ofK.The normal structure coefficient ofX is the number

N(X) = inf{diam(K)/r(K) :K ⊂Xbounded and convex,diam(K)>0}.

Obviously,1≤N(X)≤2.It is known [5], [2] that if the spaceXis reflexive, then the infimum in the definition ofN(X)can be taken over all convex hulls of finite subsets of X. The space X is said to have uniform normal structure ifN(X) > 1.IfX has uniform normal structure, then X is reflexive and hence X has fixed point property. Gao and Lau [3] showed that if J(X) < 3/2, then X has uniform normal structure. Prus [6] gave an estimate of N(X) by J(X)which contains Gao-Lau’s [3] and Bynum’s [1] results: For any non-trivial Banach space X,

(1.2) N(X)≥J(X) + 1− {(J(X) + 1)²−4}¹². Kato, Maligranda and Takahashi [4] proved

(1.3) N(X)≥

C_{N J}(X)− 1 4

⁻¹₂ ,

which implies that ifC_{N J}(X)<5/4thenXhas uniform normal structure. This result is a little finer than Gao-Lau’s condition byJ(X). This paper is devoted to improving the above results.

2. MAINRESULTS

Our proofs are based on the idea due to Prus [6], who estimatedN(X)by modulus of con- vexity ofX. LetCbe a convex hull of a finite subset of a Banach spaceX.SinceCis compact, there exists an elementy ∈C such thatsup{kx−yk: x ∈C} =r(C).Translating the setC we can assume thaty = 0.Prus [6] gave the following

Proposition 2.1. Let C be a convex hull of a finite subset of a Banach space X such that sup{kxk : x ∈ C} = r(C). Then there exist points x₁, . . . , x_n ∈ C, norm-one functionals x^∗₁, . . . , x^∗_n ∈X^∗and nonnegative numberλ₁, . . . , λ_nsuch thatPn

i=1λ_i = 1, x^∗_i(x_i) = kx_ik=r(C)

fori= 1, . . . , nandPn

i=1λix^∗_i(x) = 0wheneverλx∈Cfor someλ >0.

Without loss of generality, we assumer(C) = 1thereforeC ⊂B_X.

Theorem 2.2. LetXbe a non-trivial Banach space with the Neumann-Jordan constantC_{N J}(X).

Then

(2.1) N(X)≥ 2

p8C_{N J}(X)−1−1.

Proof. LetC be a convex hull of a finite subset ofX such thatsup{kxk :x∈C}=r(C) = 1 and diamC =d.By Proposition 2.1 we obtain elementsx₁, . . . , x_n ∈C,norm-one functionals x^∗₁, . . . , x^∗_n ∈ X^∗ and nonnegative numbersλ₁, . . . , λ_n such thatPn

i=1λ_i = 1, x^∗_i(x_i) = 1and Pn

j=1λ_jx^∗_j(x_i) = 0fori= 1, . . . , n.

Define

(2.2) x_i,j = 1

d(x_i−x_j), y_i,j =x_i

i, j = 1, . . . , n.Clearlyx_i,j, y_i,j ∈ B_X andx_i,j +y_i,j = (1 + 1/d)x_i −(1/d)x_j, x_i,j −y_i,j = (−1 + 1/d)x_i−(1/d)x_j.It follows that

n

X

i,j=1

λ_iλ_j

kx_i,j +y_i,jk²+kx_i,j−y_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−y_i,j)2

=

n

X

j=1

λ_j

n

X

i=1

λ_i

1 + 1 d − 1

dx^∗_i(x_j) 2

+

n

X

i=1

λ_i

n

X

j=1

λ_j 1

d +

1− 1 d

x^∗_j(x_i)

2

=

1− 1 d

2

−2

1− 1 d

1 d

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 d² + 2

1− 1

d 1

d

n

X

i=1

λi n

X

j=1

λjx^∗_j(xi) +

1− 1 d

2 n

X

i=1

λi n

X

j=1

λj[x^∗_j(xi)]²

≥

1 + 1 d

2

+ 1 d². Therefore there existi, jsuch that

kx_i,j+y_i,jk²+kx_i,j −y_i,jk² ≥

1 + 1 d

2

+ 1 d². From the definition of Neumann-Jordan constant we see that

(2.3) C_{N J}(X)≥ kx_i,j +y_i,jk²+kx_i,j−y_i,jk²

4 ≥ 1

4

"

1 + 1

d 2

+ 1 d²

# .

This inequality is equivalent to the following one

(2.4) d≥ 2

p8C_{N J}(X)−1−1.

Therefore, we obtain the desired estimate (2.1) sinceC⊂X is arbitrary. The proof is finished.

It is easy to check that

1 q

C_{N J}(X)−¹₄

< 2

p8C_{N J}(X)−1−1

when1 < C_{N J}(X) < 5/4.Therefore, the estimate of the above theorem improves (1.3). It is also not difficult to check that

(2.5) p

2C_{N J}(X) + 1− (p

2C_{N J}(X) + 1)²−4¹₂

< 2

p8CN J(X)−1−1 when1< C_{N J}(X)<5/4.SinceJ(X)≤p

2C_{N J}(X),and the functionx+1−((x+1)²−4)^1/2 is decreasing, we have (1.2) from (2.1) and (2.5). So (1.2) becomes a corollary of (2.1).

Prus [6] gave the result that if J(X) < 4/3, then N(X) > 1. Gao and Lau [3] gave a condition that ifJ(X)< 3/2thenN(X) >1.Then they asked whether the estimateJ(X) <

3/2is sharp forXto have uniform normal structure. Kato, Maligranda and Takahashi [4] found

that ifCN J(X)<5/4,which impliesJ(X)<√

10/2,thenN(X)>1.The following theorem will give a wider interval ofCN J(X)forX to have uniform normal structure.

Theorem 2.3. LetXbe a non-trivial Banach space with the Neumann-Jordan constantC_{N J}(X) and normal structure coefficientN(X). Then

(2.6) C_{N J}(X)≥

qN²(X)

4 +_N2¹(X) +N(X)− _N(X)^{1 2}

+_N2¹(X)

2

1 +q_N2(X)

4 +_N2¹(X) +N(X)− _N(X² ₎². Moreover, ifC_{N J}(X)<(3 +√

5)/4orJ(X)<(1 +√

5)/2,thenN(X)>1and henceXhas uniform normal structure.

Proof. We modify the first step in the proof of Theorem 2.2. In (2.2), let

(2.7) x_i,j = 1

d(x_i−x_j), y_i,j =tx_i witht >0.Thenkx_i,jk ≤1,ky_i,jk=t.Similar to (2.3), we obtain

(2.8) C_{N J}(X)≥ t+_d¹2

+ _d¹2

2(1 +t²) for anyt >0.The function

f(t) = t+_d¹2

+_d¹2

2(1 +t²) reach the maximum at the point

t₀ = rd²

4 + 1

d² +d− 2 d.

It is decreasing ont > t₀and increasing on0< t < t₀.Therefore, we have

(2.9) C_{N J}(X)≥

q

d²

4 +_d¹2 +d− ¹_d 2

+ _d¹2

2

"

1 + q

d²

4 + _d¹2 +d− ²_d 2#.

Since the function

c=g(d) :=

q

d²

4 + _d¹2 +d− ¹_d 2

+ _d¹2

2

"

1 + q

d²

4 + _d¹2 +d− ²_d 2#

is strictly decreasing and continuous on 1 ≤ d ≤ 2, we know that the inverse function d = g⁻¹(c) exists and must also be decreasing. Thus, we have from (2.9) thatd ≥ g⁻¹(C_{N J}(X)).

It follows by take the infimum ofd thatN(X) ≥ g⁻¹(C_{N J}(X)).Equivalently, we have (2.6).

From the above statements of monotony property, we deduce thatN(X) = 1is corresponding toC_{N J}(X) = (3 +√

5)/4.Therefore, ifC_{N J}(X) < (3 +√

5)/4,then N(X) > 1.Since the non-square constantJ(X)≤√

2C_{N X},we have in other word that ifJ(X)<(1 +√

5)/2,then

N(X)>1.

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