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

TL;DR

The paper proves that strongly convex functions can be approximated arbitrarily closely by smooth, strongly convex functions in both uniform norm and measure, extending the classical Lusin approximation.

Contribution

It introduces a new approximation result for strongly convex functions by smooth strongly convex functions, with precise control over approximation error and measure.

Findings

Any strongly convex function can be approximated uniformly by a smooth strongly convex function.

The approximation can be made to differ from the original on an arbitrarily small measure.

The result extends classical Lusin approximation to the class of strongly convex functions.

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 $\Vert u-v\Vert_\infty<\varepsilon$.