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

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 $\mathrm{\mid }\{u\mathrm{\ne }v\}\mathrm{\mid }<\epsilon$ and $\mathrm{\parallel }u-v{\mathrm{\parallel }}_{\mathrm{\infty }}<\epsilon$.