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.
yanyq@pub.sz.jsinfo.net
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 ifCN 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 ballBX ={x∈X :kxk ≤1}or its unit sphereSX ={x∈X :kxk= 1}.A Banach spaceX is called uniformly non-square if there exists aδ ∈ (0,1)such that for anyx, y ∈SX eitherkx+yk/2≤1−δorkx−yk/2≤1−δ.The constant
J(X) = sup{min(kx+yk,kx−yk) :x, y ∈SX}
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
CN J(X) = sup
kx+yk2+kx−yk2
2(kxk2+kyk2) :x, y ∈X, not both zero
.
Clearly, 1 ≤ CN J(X) ≤ 2. and X is a Hilbert space if and only if CN J(X) = 1. Kato, Maligranda and Takahashi [4] proved that for any non-trivial Banach spaceX(dimX ≥2),
(1.1) 1
2J(X)2 ≤CN 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)≥
CN 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.
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, . . . , xn ∈ C, norm-one functionals x∗1, . . . , x∗n ∈X∗and nonnegative numberλ1, . . . , λnsuch thatPn
i=1λi = 1, x∗i(xi) = kxik=r(C)
fori= 1, . . . , nandPn
i=1λix∗i(x) = 0wheneverλx∈Cfor someλ >0.
Without loss of generality, we assumer(C) = 1thereforeC ⊂BX.
Theorem 2.2. LetXbe a non-trivial Banach space with the Neumann-Jordan constantCN J(X).
Then
(2.1) N(X)≥ 2
p8CN 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, . . . , xn ∈C,norm-one functionals x∗1, . . . , x∗n ∈ X∗ and nonnegative numbersλ1, . . . , λn such thatPn
i=1λi = 1, x∗i(xi) = 1and Pn
j=1λjx∗j(xi) = 0fori= 1, . . . , n.
Define
(2.2) xi,j = 1
d(xi−xj), yi,j =xi
i, j = 1, . . . , n.Clearlyxi,j, yi,j ∈ BX andxi,j +yi,j = (1 + 1/d)xi −(1/d)xj, xi,j −yi,j = (−1 + 1/d)xi−(1/d)xj.It follows that
n
X
i,j=1
λiλj
kxi,j +yi,jk2+kxi,j−yi,jk2
≥
n
X
j=1
λj
n
X
i=1
λi[x∗i(xi,j +yi,j)]2+
n
X
i=1
λi
n
X
j=1
λj
x∗j(xi,j−yi,j)2
=
n
X
j=1
λj
n
X
i=1
λi
1 + 1 d − 1
dx∗i(xj) 2
+
n
X
i=1
λi
n
X
j=1
λj 1
d +
1− 1 d
x∗j(xi)
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
kxi,j+yi,jk2+kxi,j −yi,jk2 ≥
1 + 1 d
2
+ 1 d2. From the definition of Neumann-Jordan constant we see that
(2.3) CN J(X)≥ kxi,j +yi,jk2+kxi,j−yi,jk2
4 ≥ 1
4
"
1 + 1
d 2
+ 1 d2
# .
This inequality is equivalent to the following one
(2.4) d≥ 2
p8CN 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
p8CN J(X)−1−1
when1 < CN 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
2CN J(X) + 1− (p
2CN J(X) + 1)2−412
< 2
p8CN J(X)−1−1 when1< CN J(X)<5/4.SinceJ(X)≤p
2CN 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 constantCN J(X) and normal structure coefficientN(X). Then
(2.6) CN J(X)≥
qN2(X)
4 +N21(X) +N(X)− N(X)1 2
+N21(X)
2
1 +qN2(X)
4 +N21(X) +N(X)− N(X2 )2. Moreover, ifCN 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) xi,j = 1
d(xi−xj), yi,j =txi witht >0.Thenkxi,jk ≤1,kyi,jk=t.Similar to (2.3), we obtain
(2.8) CN 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) CN J(X)≥
q
d2
4 +d12 +d− 1d 2
+ d12
2
"
1 + q
d2
4 + d12 +d− 2d 2#.
Since the function
c=g(d) :=
q
d2
4 + d12 +d− 1d 2
+ d12
2
"
1 + q
d2
4 + d12 +d− 2d 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(CN J(X)).
It follows by take the infimum ofd thatN(X) ≥ g−1(CN J(X)).Equivalently, we have (2.6).
From the above statements of monotony property, we deduce thatN(X) = 1is corresponding toCN J(X) = (3 +√
5)/4.Therefore, ifCN J(X) < (3 +√
5)/4,then N(X) > 1.Since the non-square constantJ(X)≤√
2CN X,we have in other word that ifJ(X)<(1 +√
5)/2,then
N(X)>1.
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