Global approximation of convex functions by differentiable convex functions on Banach spaces

TL;DR

This paper proves that in certain Banach spaces, any continuous convex function can be uniformly approximated by differentiable convex functions, extending the approximation theory in infinite-dimensional spaces.

Contribution

It establishes conditions under which continuous convex functions on Banach spaces can be globally approximated by differentiable convex functions, especially when the dual space has an LUR norm.

Findings

Existence of $C^1$ convex approximations in Banach spaces with LUR duals.

Reduction of global approximation problems to Lipschitz convex functions.

Extension of approximation results to $C^k$ smooth convex functions.

Abstract

We show that if $X$ is a Banach space whose dual $X^{*}$ has an equivalent locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$, for every $\varepsilon >0$, and for every continuous and convex function $f:U \rightarrow \mathbb{R}$ (not necessarily bounded on bounded sets) there exists a convex function $g:X \rightarrow \mathbb{R}$ of class $C^1(U)$ such that $f-\varepsilon\leq g\leq f$ on $U.$ We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) and convex functions by $C^k$ smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by $C^k$ smooth convex functions.