Convex regularization and subdifferential calculus

TL;DR

This paper explores conditions under which the regularization of sums of functions in locally convex spaces can be simplified using convex hulls and subdifferential calculus, extending classical convex analysis results.

Contribution

It provides new conditions and calculus rules for the sum of regularized functions and extends Rockafellar's convex integration theorem.

Findings

Conditions for sum of convex hulls in locally convex spaces

Epsilon-subdifferential calculus rules for sums

Extension of Rockafellar's convex integration theorem

Abstract

This paper deals with the regularization of the sum of functions defined on a locally convex spaces through their closed-convex hulls in the bidual space. Different conditions guaranteeing that the closed-convex hull of the sum is the sum of the corresponding closed-convex hulls are provided. These conditions are expressed in terms of some epsilon-subdifferential calculus rules for the sum. The case of convex functions is also studied, and exact calculus rules are given under additional continuity/qualifications conditions. As an illustration, a variant of the proof of the classical Rockafellar theorem on convex integration is proposed.