Global and fine approximation of convex functions

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.

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