On $C^{0}$-fine approximation of convex functions by real analytic convex functions

TL;DR

This paper investigates the conditions under which convex functions can be approximated by real analytic convex functions in the $C^0$ topology, revealing a dimension-dependent phenomenon and characterizing the approximable functions based on their global geometry.

Contribution

It provides a complete characterization of convex functions on $ ^d$ that can be approximated by real analytic convex functions in the $C^0$-fine topology, highlighting the role of global geometry.

Findings

Approximation by real analytic convex functions is possible in dimension 1.

In higher dimensions, only certain convex functions can be approximated, depending on their global shape.

The approximation capability is governed by global geometric properties, not local convexity or smoothness.

Abstract

We show that $C^0$-fine approximation of convex functions by smooth (or real analytic) convex functions on $\R^d$ 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 $\R^d$ 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 behavior. We give some applications concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.