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volume 5, issue 1, article 10, 2004.

Received 22 August, 2003;

accepted 09 January, 2004.

Communicated by:C.P. Niculescu

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Journal of Inequalities in Pure and Applied Mathematics

ON ESTIMATES OF NORMAL STRUCTURE COEFFICIENTS OF BANACH SPACES

Y. Q. YAN

Department of Mathematics Suzhou University

Suzhou, Jiangsu, P.R. China, 215006.

EMail:yanyq@pub.sz.jsinfo.net

c

2000Victoria University ISSN (electronic): 1443-5756 114-03

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On Estimates of Normal Structure Coefficients of

Banach Spaces Y. Q. Yan

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J. Ineq. Pure and Appl. Math. 5(1) Art. 10, 2004

http://jipam.vu.edu.au

Abstract

We obtained the estimates of Normal structure coefficientN(X)by Neumann- Jordan constantCNJ(X)of a Banach spaceXand found thatXhas uniform normal structure ifC_NJ(X)<(3 +√

5)/4.These results improved both Prus’

[6] and Kato, Maligranda and Takahashi’s [4] work.

2000 Mathematics Subject Classification:46B20, 46E30.

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

1. Introduction

LetX = (X,k · k)be a real Banach space. Geometrical properties of a Banach spaceX are determined by its unit ballBX = {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 ofX in the sense of James. It is well-known that√

2≤J(X)≤2ifdimX ≥2.The Neumann-Jordan constantCN J(X)of a Banach spaceX 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.andX is a Hilbert space if and only ifC_{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.

A Banach spaceX is said to have normal structure ifr(K)< diam(K)for every non-singleton closed bounded convex subsetK ofX,where diam(K) = sup{kx−yk : x, y ∈ K} is the diameter of K and r(K) = inf{sup{kx−

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yk : y ∈ K} : x ∈ K}is the Chebyshev radius of K. The normal structure coefficient ofXis 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 spaceX is reflexive, then the infimum in the definition of N(X)can be taken over all convex hulls of finite subsets ofX. The spaceXis said to have uniform normal structure if N(X) > 1.IfX has uniform normal structure, thenX 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 spaceX,

(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 ifCN 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.

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2. Main Results

Our proofs are based on the idea due to Prus [6], who estimated N(X) by modulus of convexity of X. Let C be a convex hull of a finite subset of a Banach spaceX.SinceC is compact, there exists an elementy ∈ C such that sup{kx−yk:x∈C}=r(C).Translating the setCwe 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 thatsup{kxk : x ∈ C} = r(C).Then there exist points x₁, . . . , x_n ∈ C,norm-one functionalsx^∗₁, . . . , x^∗_n ∈ X^∗ and nonnegative numberλ1, . . . , λn

such thatPn

i=1λ_i = 1,

x^∗_i(x_i) =kx_ik=r(C) fori= 1, . . . , nandPn

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

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

Theorem 2.2. LetX be a non-trivial Banach space with the Neumann-Jordan constantCN 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 elements x₁, . . . , x_n ∈ C,norm-one functionalsx^∗₁, . . . , x^∗_n ∈ X^∗ and nonnegative num- bersλ₁, . . . , λ_nsuch thatPn

i=1λ_i = 1, x^∗_i(x_i) = 1andPn

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

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Define

(2.2) xi,j = 1

d(xi−xj), yi,j =xi

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

λ_i

n

X

j=1

λ_jx^∗_j(x_i) +

1− 1 d

2 n

X

i=1

λ_i

n

X

j=1

λ_j[x^∗_j(x_i)]²

≥

1 + 1 d

2

+ 1 d².

Therefore there existi, j such that

kx_i,j +y_i,jk² +kx_i,j −y_i,jk² ≥

1 + 1 d

2

+ 1 d².

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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

CN J(X)− ¹₄

< 2

p8C_{N J}(X)−1−1

when 1 < CN J(X) < 5/4.Therefore, the estimate of the above theorem im- proves (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

p8C_{N J}(X)−1−1 when 1 < CN J(X) < 5/4. Since J(X) ≤ p

2CN J(X), and the function x+ 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).

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Prus [6] gave the result that ifJ(X)<4/3,thenN(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 if C_{N J}(X) < 5/4, which implies J(X) < √

10/2, then N(X) > 1. The following theorem will give a wider interval ofC_{N J}(X)forXto have uniform normal structure.

Theorem 2.3. LetX be a non-trivial Banach space with the Neumann-Jordan constantCN 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(X)² ². Moreover, ifC_{N J}(X)< (3 +√

5)/4orJ(X) <(1 +√

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

Proof. We modify the first step in the proof of Theorem2.2. In (2.2), let

(2.7) xi,j = 1

d(xi−xj), yi,j =txi

witht >0.Thenkx_i,jk ≤1,ky_i,jk=t.Similar to (2.3), we obtain

(2.8) CN J(X)≥ t+¹_d2

+_d¹2

2(1 +t²)

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for anyt >0.The function

f(t) = t+¹_d2

+_d¹2

2(1 +t²)

reach the maximum at the point t₀ =

rd² 4 + 1

d² +d− 2 d.

It is decreasing ont > t₀ and 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 in- verse function d = g⁻¹(c) exists and must also be decreasing. Thus, we have from (2.9) that d ≥ g⁻¹(CN J(X)). It follows by take the infimum of d that

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N(X) ≥ g⁻¹(C_{N J}(X)). Equivalently, we have (2.6). From the above state- ments of monotony property, we deduce that N(X) = 1 is corresponding to C_{N J}(X) = (3 +√

5)/4.Therefore, ifC_{N J}(X)<(3 +√

5)/4,thenN(X)>1.

Since the non-square constant J(X)≤ √

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

5)/2,thenN(X)>1.

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References

[1] W.L. BYNUM, About some parameters of normed linear spaces, Pacific J.

Math., 86 (1980), 427–436.

[2] T. DOMINGUEZ BENAVIDES, Normal structued coefficients of L^p(Ω), Proc. Roy. Soc. Sect., A117 (1991), 299–303.

[3] J. GAO AND K.S. LAU, On two classes of Banach spaces with uniform normal structure, Studia Math., 99 (1991), 41–56.

[4] M. KATO, L. MALIGRANDA AND Y. TAKAHASHI, On James and Jordan-von Neumann constants and the normal structure coefficient of Ba- nach spaces, Studia Math., 144 (2001), 275–295.

[5] E. MALUTA, Uniformly normal structure and telated coefficients, Pacific J. Math., 111(1984), 357–369.

[6] S. PRUS, Some estimates for the normal structure coefficient in Banach spaces, Rend. Circ. Mat. Palermo, 40 (1991), 128–135.

[7] M.M. RAOANDZ.D. REN, Applications of Orlicz spaces, Marcel Dekker, New York, (2002), 49–53.