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 non-empty finite subset $F$ of $X$ and closed convex subset $V$ of $X$, $ra{d}_V(F)=ra{d}_X(F)+d(V,cen{t}_X(F))$, where we denote $ra{d}_X(F)$, $ra{d}_V(F)$, $cen{t}_X(F)$ and $d(V,cen{t}_X(F))$ 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 $cen{t}_X(F)$ 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.