Analytic characterizations of Mazur’s intersection property via convex functions

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 semicontinuous convex function f defined on X with bounded domain, f is the supremum of all functions g f of the form: g(x) = r0 − R2 − x −x0 2, if x −x0 R; =+∞, otherwise, for some x0 ∈ X(X∗) and r0 ∈ 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∗ = R0 1+ x∗ 2 + x∗, x0 + r0, for all x∗ ∈ X∗. © 2012 Elsevier Inc. All rights reserved.