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

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 $S_U\subset U\cap \text{int}(V)$. 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

22 pages