C²-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 $V \supset \partial W$ and $\varepsilon>0$, there exists a $C^2$ locally strongly convex body $W_{\varepsilon, V}$ such that $\mathcal{H}^{n-1}(\partial W_{\varepsilon, V}\triangle\,\partial W)<\varepsilon$ and $\partial W_{\varepsilon, V}\subset V$. Moreover, if $W$ is strongly convex, then $W_{\varepsilon, V}$ is strongly convex as well.