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

Abstract

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

16 pages. Final version