Restricted Chebyshev centers in $L_1$-predual spaces

Abstract

In this paper, we provide a necessary and sufficient condition for the existence of a restricted Chebyshev center of a compact subset of an L 1 -predual space in a closed convex subset of the L 1 -predual space. We also provide a geometrical characterization of an L 1 -predual space in terms of the restricted Chebyshev radius in the following manner. A real Banach space X is an L 1 -predual space if and only if for each nonempty finite subset F of X and closed convex subset V of X, rad V pF q " rad X pF q `dpV, cent X pF qq, where we denote rad X pF q, rad V pF q, cent X pF q and dpV, cent X pF qq to be the Chebyshev radius of F in X, the restricted Chebyshev radius of F in V , the set of Chebyshev centers of F in X and the distance between the sets V and cent X pF q respectively. Furthermore, we explicitly describe the Chebyshev centers of closed bounded subsets of an M -summand in the space of real-valued continuous functions on a compact Hausdorff space.