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

Abstract

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