Smooth approximation of convex functions in Banach spaces

Abstract

This paper shows that every extended-real-valued lower semi-continuous proper (respectively Lipschitzian) convex function defined on an Asplund space can be represented as the point-wise limit (respectively uniform limit on every bounded set) of a sequence of Lipschitzian convex functions which are locally affine (hence, C ∞ ) at all points of a dense open subset; and shows an analogous for w ∗ -lower semi-continuous proper (respectively Lipschitzian) convex functions defined on dual spaces whose pre-duals have the Radon–Nikodym property.