On fine approximation of convex functions

Details On fine approximation of convex functions On fine approximation of convex functions

2012-01-23

arxiv.org/abs/1201.4760v3

Mathematics

Differential Geometry

14 pages. I added a proof of Theorem 1 (an easy consequence of Theorem 3, but not so immediate as we claimed in the previous v

Scientific paper

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 rather by its global geometrical behavior. We also show that every $C^{1}$ convex and proper function on an open convex subset $U$ of $\R^d$ can be approximated by $C^{\infty}$ convex functions, in the $C^{1}$-fine topology, and we give 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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