Analytic characterizations of Mazurʼs intersection property via convex functions

In this paper, we present analytical characterizations of Mazurʼs 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⁎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⩽fg⩽f of the form:g(x)=r0−R2−‖x−x0‖2,if ‖x−x0‖⩽R;=+∞,otherwise, for some x0∈X(X⁎)x0∈X(X⁎) and r0∈Rr0∈R, R>0R>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⁎f⁎ is the infimum of all functions h⩾f⁎h⩾f⁎ which are of the form:h(x⁎)=R01+‖x⁎‖2+〈x⁎,x0〉+r0,for all x⁎∈X⁎.