The Distance Between the Perturbation of a Convex Function and its $\Gamma$-regularization

TL;DR

This paper investigates the relationship between perturbations of convex functions and their $\Gamma$-regularizations, establishing the optimality of a known estimate in the context of convex analysis.

Contribution

It proves that the previously known $o(\epsilon)$ bound on the difference between perturbed functions and their $\Gamma$-regularizations is actually optimal.

Findings

The difference is at most $o(\epsilon)$ and this bound is sharp.

The result clarifies the limitations of perturbation estimates in convex analysis.

It confirms the expected strength of the $\Gamma$-regularization approximation.

Abstract

In the study of a non-convex minimization problem by Lachand-Robert and Peletier, they found that the difference between the compactly supported perturbation $u+\epsilon h$ of a strictly convex function $u$, and the $\Gamma$-regularization of $u+\epsilon h$, is at most $o(\epsilon)$. Here we find that this result is optimal, albeit they expected a much stronger estimate.