  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
$\varepsilon>0$, and $V \supset \partial W$ is open, then 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.
