Chebyshev centers and fixed point theorems

Abstract

Brodskii and Milman proved that there is a point in C(K), the set of all Chebyshev centers of K, which is fixed by every surjective isometry from K into K whenever K is a nonempty weakly compact convex subset having normal structure in a Banach space. Motivated by this result, Lim et al. raised the following question namely “does there exist a point in C(K) which is fixed by every isometry from K into K?”. In fact, Lim et al. proved that “if K is a nonempty weakly compact convex subset of a uniformly convex Banach space, then the Chebyshev center of K is fixed by every isometry T from K into K”. In this paper, we prove that if K is a nonempty weakly compact convex set having normal structure in a strictly convex Banach space and F is a commuting family of isometry mappings on K then there exists a point in C(K) which is fixed by every mapping in F.