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$\mathbf{C^2}$-Lusin approximation of strongly convex functions

Daniel Azagra, Marjorie Drake, Piotr Hajłasz

Abstract

We prove that if $u:\mathbb{R}^n\to\mathbb{R}$ is strongly convex, then for every $\varepsilon>0$ there is a strongly convex function $v\in C^2(\mathbb{R}^n)$ such that $|\{u\neq v\}|<\varepsilon$ and $\Vert u-v\Vert_\infty<\varepsilon$.

$\mathbf{C^2}$-Lusin approximation of strongly convex functions

Abstract

We prove that if is strongly convex, then for every there is a strongly convex function such that and .
Paper Structure (14 sections, 23 theorems, 131 equations)

This paper contains 14 sections, 23 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Outline of the proof of Theorem \ref{['T22']}
  4. Preliminaries for the proof of Theorem \ref{['T22']}
  5. Smooth and Sobolev functions
  6. Mollifiers
  7. Convex functions
  8. Strongly convex functions
  9. Smooth strongly convex functions
  10. $C^{1,1}$ approximation of strongly convex functions
  11. Smooth maximum
  12. Smooth approximation of strongly convex functions
  13. Pointwise inequality
  14. Proof of Theorem \ref{['T22']}

Key Result

Theorem 1.1

Let $U\subseteq\mathbb R^n$ be open and convex, and $u:U\to\mathbb R$ be locally strongly convex. Then for every $\varepsilon_o>0$ and for every continuous function $\varepsilon:U\to (0, 1]$ there is a locally strongly convex function $v\in C^2(U)$, such that Also, if $u$ is $\eta$-strongly convex on $U$, then for every $\widetilde{\eta}\in (0, \eta)$ there exists such a function $v$ which is $\w

Theorems & Definitions (42)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 32 more