Some geometrical characterizations of $L_1$-predual spaces

Abstract

. Let X be a real Banach space. For a non-empty finite subset F and closed convex subset V of X , we denote by rad X ( F ) , rad V ( F ) , cent X ( F ) and d ( V, cent X ( F )) 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 ( F ) respectively. We prove that X is an L 1 -predual space if and only if for each four-point subset F of X and non-empty closed convex subset V of X , rad V ( F ) = rad X ( F ) + d ( V, cent X ( F )) . Moreover, we explicitly describe the Chebyshev centers of a compact subset of an L 1 - predual space. Various new characterizations of ideals in an L 1 -predual space are also obtained. In particular, for a compact Hausdorff space S and a subspace A of C ( S ) which contains the constant function 1 and separates the points of S , we prove that the state space of A is a Choquet simplex if and only if d ( A , cent C ( S ) ( F )) = 0 for every four-point subset F of A . We also derive characterizations for a compact convex subset of a locally convex topological vector space to be a Choquet simplex.