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 }

No abstract

Abstract. In reflexive Banach spaces with some degree of uniform convexity, we obtain estimates for Kottman's separation constant in terms of the corresponding modulus.

We show that infinite dimensional geometric moduli introduced by Milman are strongly related to nearly uniform convexity and nearly uniform smoothness. An application of those moduli to fixed point theory is given.

A.M.S. subject classification: 46B20.We prove that property (β) of Rolewicz implies normal structure of the dual space and we characterize spaces which are duals of spaces with property (β).