Self-dual Smooth Approximations of Convex Functions via the Proximal Average

TL;DR

This paper introduces a new self-dual smoothing operator for convex functions using the proximal average, simplifying proofs and providing practical smoothing techniques for norms.

Contribution

It expresses Goebel's self-dual smoothing operator via the proximal average and introduces a novel self-dual smoothing operator.

Findings

Simplified proof of self-duality for smoothing operators

New self-dual smoothing operator derived from proximal average

Illustrations include smoothing of the norm

Abstract

The proximal average of two convex functions has proven to be a useful tool in convex analysis. In this note, we express Goebel's self-dual smoothing operator in terms of the proximal average, which allows us to give a simple proof of self duality. We also provide a novel self-dual smoothing operator. Both operators are illustrated by smoothing the norm.