Normal structure and invariance of Chebyshev center under isometries

Abstract

A normed space is said to have the weak fixed point property (WFPP) if every nonexpansive self map on a weakly compact convex set has a fixed point. Kirk proved that if a normed space has normal structure, then it has WFPP. It is known that if a normed space is uniformly convex in every direction (UCED), then it has normal structure. Also known is that every normed space that is uniformly convex in all but countably many directions has normal structure. We show that a normed space X has normal structure if the set of all directions in which it is not uniformly convex is contained in a countable union of n-dimensional subspaces of X for some positive integer n. We also show that in such a space, the Chebyshev center C(K) of a weakly compact convex set K is a common invariant set for the collection of all isometries that map K into K and also that there is a common fixed point in C(K) for this collection of maps. This was previously known to be true only in the case of a normed space that is UCED. Another observation made in this paper is that a Banach space X has normal structure if the set of all directions in which it is not uniformly convex is contained in a linear subspace with a countable Hamel basis.