 
Summary: arXiv:1201.4760v3[math.DG]6Feb2012
ON FINE APPROXIMATION OF CONVEX FUNCTIONS
DANIEL AZAGRA
Abstract. We show that C0
fine approximation of convex functions by
smooth (or real analytic) convex functions on Rd
is possible in general if
and only if d = 1. Nevertheless, for d 2 we give a characterization of
the class of convex functions on Rd
which can be approximated by real
analytic (or just smoother) convex functions in the C0
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 C1
convex and proper function on an open convex
subset U of Rd
can be approximated by C
convex functions, in the C1
