# Analytic characterizations of Mazur's intersection property

Cheng, Lixin

### Abstract

In this paper, we present analytical characterizations of Mazur's intersection property (MIP), the CIP and the MIP*TeX 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 (MIP*TeX 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:

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

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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 dom*TeX Embedding failed!*, there exists a family *TeX Embedding failed!* of Lipschitz convex functions of the form

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

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