Analytic characterizations of Mazur's intersection property

Abstract

In this paper, we present analytical characterizations of Mazur's intersection property (MIP), the CIP and the MIPTeX Embedding failed! via a specific class of convex functions and their conjugates. Precisely, let TeX Embedding failed! be a Banach space and TeX Embedding failed! be its dual, and let TeX Embedding failed! be the indicator of the set TeX Embedding failed!. Then a sufficient and necessary condition for TeX Embedding failed! admitting the MIP (MIPTeX Embedding failed!) is that for every extended-real-valued (TeX Embedding failed!) lower semicontinuous convex function TeX Embedding failed! defined on TeX Embedding failed! and for every (TeX Embedding failed!) closed bounded convex set TeX Embedding failed! there exists a family TeX Embedding failed! of convex functions of the form:

for some TeX Embedding failed! and TeX Embedding failed! such that

and TeX Embedding failed! admits the CIP if and only if for every such lower semicontinuous proper convex function on TeX Embedding failed! with relatively compact domTeX Embedding failed!, there exists a family TeX Embedding failed! of Lipschitz convex functions of the form

where TeX Embedding failed! and TeX Embedding failed!, such that

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