Approximation of Lipschitz functions by $\Delta$-convex functions in Banach spaces

Abstract

In this paper we give some results about the approximation of a Lipschitz function on a Banach space by means of $\Delta$-convex functions. In particular, we prove that the density of $\Delta$-convex functions in the set of Lipschitz functions for the topology of uniform convergence on bounded sets characterizes the superreflexivity of the Banach space. We also show that Lipschitz functions on superreflexive Banach spaces are uniform limits on the whole space of $\Delta$-convex functions.