Lipschitz Regularization and the Convergence of Convex Functions

Abstract

Let be a normed linear space,and f be a proper lower semicontinuous convex function defined on X. The Lipschitz reg-ularization of f with parameter μ is the infimal convolution of f with , defined by . In this article we express, in an arbitrary normed space, the twomost important epigraphical convergence notions for convex functions in terms of the convergence of associated Lipschitz regularizations with respect to classical function theoretic modes of convergence. In the case of slice convergence of epigraphs, the associated mode of convergence for the regularizations is pointwise convergence, whereas for Attouch-Wets convergence of epigraphs, the associated mode of convergence for the regularizations is uniform convergence on bounded subsets of XL.