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

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