A Generalization of Ekeland’s e-Variational Principle and Its Borwein]Preiss Smooth Variant1

We give a generalization of Ekeland’s e-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 b-smooth ‘‘bump function,’’ then every continuous convex function on an open subset U of X is densely b-differentiable in U. This generalizes the Borwein]Preiss theorem on the differentiability of convex functions.