Characterization of convex $\mu$-compact sets

TL;DR

This paper characterizes convex $$-compact sets in locally convex spaces by properties of functions on these sets, extending known results from compact sets to a broader class including many noncompact sets.

Contribution

It provides a new characterization of convex $$-compact sets based on the continuity of the convex closure operation with respect to certain function sequences.

Findings

Convex $$-compact sets include all compact sets and many noncompact sets used in applications.

The class can be characterized by the continuity of the double Fenchel transform operation.

The results generalize properties known for compact sets to a wider class of convex sets.

Abstract

The class of $\mu$-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which the particular results well known for compact sets can be generalized. This class contains all compact sets as well as many noncompact sets widely used in applications. In this paper we give a characterization of a convex $\mu$-compact set in terms of properties of functions defined on this set. Namely, we prove that the class of convex $\mu$-compact sets can be characterized by continuity of the operation of convex closure of a function (= the double Fenchel transform) with respect to monotonic pointwise converging sequences of continuous bounded and of lower semicontinuous lower bounded functions.