Set-valued evenly convex functions: characterizations and c-conjugacy

TL;DR

This paper characterizes set-valued evenly convex functions in locally convex spaces, develops a conjugation framework, and extends Fenchel-Moreau duality to these functions, advancing the mathematical theory of set-valued analysis.

Contribution

It introduces a characterization of evenly convex set-valued functions via set-valued e-affine minorants and develops a conjugation pattern with a Fenchel-Moreau type theorem.

Findings

Characterization of evenly convex set-valued functions as supremum of e-affine minorants

Development of a conjugation framework for set-valued functions

Extension of Fenchel-Moreau duality to set-valued functions

Abstract

In this work we deal with set-valued functions with values in the power set of a separated locally convex space where a nontrivial pointed convex cone induces a partial order relation. A set-valued function is evenly convex if its epigraph is an evenly convex set, i.e., it is the intersection of an arbitrary family of open half-spaces. In this paper we characterize evenly convex set-valued functions as the pointwise supremum of its set-valued e-affine minorants. Moreover, a suitable conjugation pattern will be developed for these functions, as well as the counterpart of the biconjugation Fenchel-Moreau theorem.