  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.
