Normal structure coefficients for Banach spaces

This paper introduces several geometric concepts in Banach spaces related to normal structure and uses these ideas to generalize a recent theorem of Baillon for fixed points of nonexpansive mappings.1* Definitions.The concepts introduced in this paper are phrased in terms of reflexive Banach spaces.This is not a serious restriction, but rather one of technical convenience.All of the concepts discussed here deal with closed bounded convex subsets of a reflexive Banach space, so one could replace "closed bounded and convex" with "weakly compact and convex", if desired.For the following definitions, X will denote a reflexive Banach space and C will denote a closed bounded convex subset of X.For each x in C, let r(x, C) = sup{||ίc -y\\: y in C) and let R(C) denote the Cebysev radius of C [11, p. 178]: R(C) = min {r(x, C): x in C} .This minimum is achieved because of the weak compactness of C.