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

We prove that, if $W\subset {\mathbb{R}}^n$ is a locally strongly convex body (not necessarily compact), then for any open set $V\supset \mathrm{\partial }W$ and $\epsilon >0$, and $V\supset \mathrm{\partial }W$ is open, then there exists a $C^2$ locally strongly convex body $W_{\epsilon ,V}$ such that ${\mathcal{H}}^{n-1}(\mathrm{\partial }{W}_{\epsilon ,V}\mathrm{△}\mathrm{\partial }W)<\epsilon$ and $\mathrm{\partial }{W}_{\epsilon ,V}\subset V$. Moreover, if $W$ is strongly convex, then $W_{\epsilon ,V}$ is strongly convex as well.