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

Zichang Liu

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 $u+εh$ of a strictly convex function $u$, and the $Γ$-regularization of $u+εh$, is at most $o(ε)$. Here we find that this result is optimal, albeit they expected a much stronger estimate.

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

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 of a strictly convex function , and the -regularization of , is at most . Here we find that this result is optimal, albeit they expected a much stronger estimate.

Paper Structure

This paper contains 2 sections, 2 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Main Result

Key Result

Theorem 1.1

Let $u$ be a minimizer of 1 in $\mathcal{C}$, and $\Omega_1$ be an open convex subset of $\Omega$. If $f$ is nowhere convex on the convex hull of $\nabla u(\Omega_1)$, then $u$ is not strictly convex on $\Omega_1$.

Theorems & Definitions (3)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 2.1