Inner and outer smooth approximation of convex hypersurfaces. When is it possible?

TL;DR

This paper characterizes when convex hypersurfaces can be approximated by real-analytic convex hypersurfaces, establishing conditions related to the absence of lines or rays, and extends results to convex functions with smooth approximations.

Contribution

It provides necessary and sufficient conditions for approximating convex hypersurfaces and functions by smooth convex ones, including real-analytic and strongly convex cases.

Findings

Convex hypersurfaces without lines can be approximated by real-analytic convex hypersurfaces.

Convex hypersurfaces without rays can be approximated from outside by real-analytic convex hypersurfaces.

Convex functions can be approximated in the $C^0$-fine topology by smooth convex functions from above or below.

Abstract

Let $S$ be a convex hypersurface (the boundary of a closed convex set $V$ with nonempty interior) in $\mathbb{R}^n$. We prove that $S$ contains no lines if and only if for every open set $U\supset S$ there exists a real-analytic convex hypersurface $S_{U} \subset U\cap \textrm{int}(V) $. We also show that $S$ contains no rays if and only if for every open set $U\supset S$ there exists a real-analytic convex hypersurface $S_{U}\subset U\setminus V$. Moreover, in both cases, $S_U$ can be taken strongly convex. We also establish similar results for convex functions defined on open convex subsets of $\mathbb{R}^n$, completely characterizing the class of convex functions that can be approximated in the $C^0$-fine topology by smooth convex functions from above or from below. We also provide similar results for $C^1$-fine approximations