Analytic characterizations of Mazur's intersection property via convex functions

Abstract

In this paper, we present analytical characterizations of Mazur's intersection property (MIP), the CIP and the MIP* via a specific class of convex functions and their conjugates. More precisely, let X be a Banach space and X* be its dual. Then X has the MIP if and only if for every extended real-valued lower semi-continuous convex function f defined on X with bounded domain, f is the supremum of all functions g <= f of the form: g(x) = r(0) - root R-2 - parallel to x - x(0)parallel to(2), if parallel to x - x(0)parallel to <= R; = + infinity, otherwise for some x(0) is an element of X (X*) and r(0) is an element of R, R > 0. And X has the CIP if and only if for every extended real-valued lower semi-continuous convex function on X with relatively compact domain, f* is the infimum of all functions h >= f* which are of the form: h(x*) = R-0 root 1 + parallel to x*parallel to(2) + (x*, x(0)) + r(0), for all x* is an element of X*. (C) 2012 Elsevier Inc. All rights reserved.