Uniformly normal structure and related coefficients

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. 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 } n ι ' 1 ), i.e. the distance from the nth element of the sequence to the convex hull of the preceding elements, approaches the diameter of the sequence as n approaches infinity.(For normal type structures and their applications to fixed point theory, we refer to the exhaustive survey of Kirk [9].)The first constant we consider, N( X), is the already known constant of uniformity of normal structure.It has a clear geometrical meaning, for it is the supremum, taken with respect to the convex bounded subsets C of X, of the ratio between r(C,C) and the diameter of C. Hence N(X) < 1 characterizes uniformly normal structure.The second constant, D( X), is a sequence coefficient which controls the behavior, as n approaches infinity, of d(x n ,co{XjY λ ~x)\ more precisely, D(X) describes how closely this distance can approach the diameter of the sequence.For the two constants we give some evaluations and estimates.We prove they both must be one in nonreflexive spaces, thus answering in the affirmative the following question raised in [9].Does uniformly normal structure imply reflexivity?This question follows naturally from the fact that in [4] it was proved, without requiring any hypothesis of reflexivity, that a space X with N{ X) < 1 has the fixed point property for nonexpansive mappings.We prove also that, in infinite-dimensional spaces, N(X) >: 2~1 /2 , thus showing that the best value of N(X) is achieved by the space / 2 , and