Global and fine approximation of convex functions

TL;DR

This paper demonstrates that convex functions on open convex sets can be globally approximated by real analytic convex functions, and characterizes when such approximation is possible in the fine topology, highlighting the role of global geometry.

Contribution

It proves the universal approximation of convex functions by real analytic convex functions on convex domains and characterizes approximation possibilities in higher dimensions.

Findings

Convex functions on convex sets can be uniformly approximated by real analytic convex functions.

Approximation in the $C^0$-fine topology is only generally possible in one dimension.

The global geometric behavior of functions determines approximation feasibility in higher dimensions.

Abstract

Let $U\subseteq\mathbb{R}^d$ be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. We also show that $C^0$-fine approximation of convex functions by smooth (or real analytic) convex functions on $\mathbb{R}^d$ is possible in general if and only if $d=1$. Nevertheless, for $d\geq 2$ we give a characterization of the class of convex functions on $\mathbb{R}^d$ which can be approximated by real analytic (or just smoother) convex functions in the $C^0$-fine topology. It turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function, but by its global geometrical behaviour. We also show that every $C^{1}$ convex and proper function on $U$ can be approximated by $C^{\infty}$ convex functions in the $C^{1}$-fine topology, and we provide some applications of these results, concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.