# Modulus of Convexity, the CoeffcientR(1,X), and Normal Structure in Banach Spaces

Authors: Hongwei Jiao; Yunrui Guo; Fenghui Wang
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

LetδX(ϵ) and R(1,X) be the modulus of convexity and the Domínguez-Benavides coefficient, respectively. According to these two geometric parameters, we obtain a sufficient condition for normal structure, that is, a Banach space
X has normal structure if
2δX(1+ϵ)>max{(R(1,x)-1)ϵ,1-(1-ϵ/R(1,X)-1)} for some ϵ∈[0,1] which generalizes the known result by
Gao and Prus.


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## Body

## 1. Introduction

LetX be a Banach space. Throughout the paper, denote by SX,BX the unit sphere and unit ball of X, respectively. Recall
that a Banach space X is said to have normal structure (resp., weak,
normal, structure) if for every closed bounded (resp., weakly compact) convex
subset C of X with diam C>0, there exists x∈C such that sup{∥x−y∥:y∈C}<diamC, where diam C=sup{∥x−y∥:x,y∈C}. For a reflexive Banach space, the normal
structure and weak normal structure are the same.
Recently a good deal of investigations have focused on finding the sufficient
conditions with various geometrical constants for a Banach space to have normal
structure (see, e.g., [1–5]). The geometric condition sufficient for normal
structure in terms of the modulus of convexity is given by Goebel [6], who proved that X has normal structure provided that δX(1)>0. Here the function δX(ϵ):[0,2]→[0,1], defined by Clarkson [7] as(1.1)δX(ϵ)=inf{1−∥x+y∥2:x,y∈BX,∥x−y∥≥ϵ},is called
the modulus of
convexity of X. Later Gao and Prus generalized the above
results as the following (see [2, 8]).Theorem 1.1.
A Banach spaceX has normal structure provided that δX(1+ϵ)>ϵ/2 for some ϵ∈[0,1].In this paper, we obtain a class of Banach spaces with
normal structure, which involves the coefficientR(1,X). This coefficient is defined by
Domínguez Benavides [9] as(1.2)R(1,X)=sup{lim infn→∞∥x+xn∥},where the supremum is taken over
all x∈X with ∥x∥≤1 and all weakly null sequence (xn) in BX such that(1.3)D[(xn)]:=lim supn→∞(lim supm→∞∥xn−xm∥)≤1.Obviously, 1≤R(1,X)≤2.


## 2. Main Results

Let us begin this section with a sufficient condition
for a Banach spaceX having weak normal structure and the idea in
the following proof is due to [5, Lemma 5].Lemma 2.1.
LetX be a Banach space for which BX* is w*-sequentially compact. If X does not have weak normal structure, then for
any η>0,
there exist x1,x2∈SX and f1,f2∈SX*, such that
(1)
|∥x1−x2∥−1|<η;(2)
|fi(xj)|<η  for i≠j and fi(xi)=1,i,j=1,2;(3)
∥x1+x2∥≤R(1,X)(1+η).Proof.
Assume
thatX does not have weak normal structure. It is
well known that (see, e.g., [10]) there exists a sequence {xn} in X satisfying
(1)
xn is weakly convergent to 0;(2)
diam({xn}n=1∞)=1=limn∥xn−x∥ for all x∈clco{xn}n=1∞.
Since BX* is w*-sequentially compact, we can find {fn} in SX* satisfying
(3)
fn(xn)=∥xn∥ for all n;(4)
fn→w*f for some f∈BX*.
Letη∈(0,1) sufficiently small and ϵ=η/3. Then, by the properties of the sequence (xn),
we can choose n1∈ℕ such that(2.1)|f(xn1)|<ϵ2,1−ϵ≤∥xn1∥≤1.Note that the sequence {xn} is weakly null and verifies D[{xn}]=1. It follows from the definition of R(1,X) that(2.2)liminfn∥xn+xn1∥≤R(1,X).The rest of
the proof is similar to that of [5, Lemma 5].Theorem 2.2.
A Banach spaceX has normal structure provided that δX(1+ϵ)>f(ϵ) for some ϵ∈[0,1], where the function f(ϵ) is defined as(2.3)f(ϵ):={(R(1,X)−1)ϵ2,0≤ϵ≤1R(1,X),12(1−1−ϵR(1,X)−1),1R(1,X)<ϵ≤1.Proof.
Observe thatX is uniformly nonsquare [11] and then reflexive. Therefore
normal structure and weak normal structure coincide.
Assume first thatX fails to have weak normal structure. Fix η>0 sufficiently small and ϵ∈[0,1].
It follows that there exist x1,x2∈SX and f1,f2∈SX*, satisfying the condition in Lemma 2.1. Next,
denote by R:=R(1,X) and consider two cases for ϵ∈[0,1].
Case 1  (ϵ∈[0,1/R]).
Now let us put(2.4)x=x1−x21+η,y=(1−(R−1)ϵ)x1+ϵx21+η,and so x∈BX,(2.5)∥y∥=∥ϵ1+η(x1+x2)+1−Rϵ1+ηx1∥≤Rϵ+(1−Rϵ)=1,and also that(2.6)∥x−y∥=∥(R−1)ϵ1+ηx1−1+ϵ1+ηx2∥≥1+ϵ1+ηf2(x2)−(R−1)ϵ1+ηf2(x1)≥1+ϵ−η1+η,∥x+y∥=∥(2−(R−1))ϵ1+ηx1−1−ϵ1+ηx2∥≥(2−(R−1))ϵ1+ηf1(x1)−1−ϵ1+ηf1(x2)≥(1−2η1+η)(2−(R−1)ϵ).By the definition of modulus of
convexity,(2.7)(1−2η1+η)(2−(R−1)ϵ)≤∥x+y∥≤2(1−δX(∥x−y∥))≤2(1−δX(1+ϵ−η1+η)),or equivalently,(2.8)(2−(R−1)ϵ)≤2(1−δX(1+ϵ−η1+η))(1+2η1−η).Letting η→0, we have(2.9)2δX(1+ϵ)≤(R−1)ϵ,which contradicts our
hypothesis.
Case 2  (ϵ∈(1/R,1]).
In this
caseR>1, otherwise ϵ>1. Let(2.10)x′=x2−x11+η,y′=(1−(R−1)ϵ′)x1+ϵ′x21+η,where ϵ′=1−(R−1)ϵ∈[0,1/R).
It follows from Case 1 that x,y∈BX,(2.11)∥x−y∥≥(1−2η1+η)(2−(R−1)ϵ′),∥x+y∥≥1+ϵ′−η1+η.This implies
that(2.12)δX(2−(R−1)ϵ′)≤12(1−ϵ′),which is equivalent to(2.13)δX(1+ϵ)≤12(1−1−ϵR−1).This is a contradiction.Remark 2.3.
(1) It
is readily seen thatf(ϵ)≤ϵ/2 for any ϵ∈[0,1] and
Theorem 2.2 is therefore a generalization of
Theorem 1.1. Moreover this generalization is strict whenever X is the space with R(1,X)<2.
(2) Consider the spaceX=ℝ2 with the norm ∥(x,y)∥:=max(|x|,|y|,|x−y|). It is known that δX(ϵ)=max{0,(ϵ−1)/2} [8] and R(1,X)=1, then X has normal structure from Theorem 2.2, but lies
out of the scope of Theorem 1.1.Corollary 2.4.
LetX be a Banach space with R(1,X)=1 and δX(1+ϵ)>0forsomeϵ∈[0,1], then X has normal structure.Corollary 2.5.
IfX is a Banach space with(2.14)δX(1+1R(1,X))>12(1−1R(1,X)),then X has normal structure.Remark 2.6.
Corollary2.5 is equivalent to
[4, Corollary 24].



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