Uniformly normal structure and related coefficients

Abstract

It is shown that uniformly normal structure implies reflexivity. In spaces with uniformly normal structure some estimates are given for the uniformity constant and for a related coefficient.1. Introduction. Our aim is to study two constants of a Banach space X connected with normal structure. We recall that a normed space (or a convex subset) X is said to have normal structure if for every convex bounded non-empty non-singleton subset C of X, the Chebyshev radius of C relative to C, r(C, C), is strictly smaller than the diameter of C. This concept was introduced by Brodskii and MiΓman (1948), who also gave the following characterization in terms of sequences. A space X has normal structure if and only if there exists in X no bounded non-constant sequence {x n } such that d(x n ,co{x J }

Property (β) implies normal structure of the dual space

“…If moreover r(A)/ diam(A) is bounded away from 1, X is said to have uniformly normal structure. In [14] it was proved that, for a Banach space X, uniformly normal structure implies (normal structure and) reflexivity, and it was introduced a coefficient D(X) of sequentially uniform normal structure defined by…”

Geometrical Background of Metric Fixed Point Theory

“…These formulae can be also obtained in a more direct way (see [99] and [79]). From Dvoretsky's theorem, Lemma 61, and Theorem 64 we see that N (X) ≤ N (l 2 ) = √ 2 for any infinite dimensional Banach space X.…”

“…Indeed, if a space X is not reflexive, then N (X) = 1 (see [79] or [6]) and for reflexive spaces we can modify the formula for N (X).…”

On fixed point theorems of nonexpansive mappings in product spaces

“…[15]), e.g., E is uniformly convex, or uniformly smooth, or /c-uniformly rotund [17] for an integer k>\.…”

“…(iii) The Maluta's constant D(E) of E (see [15]) is less than one, e.g., E is nearly uniformly convex [5].…”