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

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.