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

TL;DR

This paper demonstrates that strongly convex bodies can be approximated by smooth, strongly convex bodies with boundaries arbitrarily close in measure, preserving strong convexity if present initially.

Contribution

It establishes a $C^2$ approximation method for strongly convex bodies, ensuring boundary proximity and convexity preservation.

Findings

Existence of $C^2$ strongly convex approximations within any boundary neighborhood.

Approximate bodies can be made arbitrarily close in boundary measure.

Strong convexity is preserved in the approximation if the original body is strongly convex.

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 $\varepsilon>0$, 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.