A Generalization of Ekeland's ϵ-Variational Principle and Its Borwein–Preiss Smooth Variant

Abstract

We give a generalization of Ekeland's ϵ-Variational Principle and of its Borwein–Preiss smooth variant, replacing the distance and the norm by a “gauge-type” lower semi-continuous function. As an application of this generalization, we show that if on a Banach space X there exists a Lipschitz β-smooth “bump function,” then every continuous convex function on an open subset U of X is densely β-differentiable in U. This generalizes the Borwein–Preiss theorem on the differentiability of convex functions.