Global approximation of convex functions

TL;DR

This paper demonstrates that convex functions on open convex sets can be uniformly approximated by real analytic convex functions, preserving key properties, and introduces a transfer technique applicable to various geometric contexts.

Contribution

It introduces a novel transfer method for uniform approximation of convex functions from bounded to unbounded sets, preserving convexity, smoothness, and other properties, with applications to manifolds and Banach spaces.

Findings

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

The transfer method preserves convexity, smoothness, and local properties.

Counterexamples show limitations of smooth approximation in higher dimensions.

Abstract

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show 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 $U$. In doing so we provide a technique which transfers results on uniform approximation on bounded sets to results on uniform approximation on unbounded sets, in such a way that not only convexity and $C^k$ smoothness, but also local Lipschitz constants, minimizers, order, and strict or strong convexity, are preserved. This transfer method is quite general and it can also be used to obtain new results on approximation of convex functions defined on Riemannian manifolds or Banach spaces. We also provide a characterization of the class of convex functions which can be uniformly approximated on $\mathbb{R}^n$ by strongly convex functions. Finally, we give some counterexamples showing that $C^0$-fine approximation of convex functions by smooth convex functions is not possible on $\mathbb{R}^{n}$ whenever $n\geq 2$.