Convergence of Lipschitz Regularizations of Convex Functions

Abstract

For a sequence or net of convex functions on a Banach space, we study pointwise convergence of their Lipschitz regularizations and convergence of their epigraphs. The Lipschitz regularizations we will consider are the infimal convolutions of the functions with appropriate multiples of the norm. For a sequence of convex functions on a separable Banach space we show that both pointwise convergence of their Lipschitz regularizations and Wijsman convergence of their epigraphs are equivalent to variants of two conditions used by Attouch and Beer to characterize slice convergence. Results for nonseparable spaces are obtained by separable reduction arguments. As a by-product, slice convergence for an arbitrary net of convex functions can be deduced from the pointwise convergence of their regularizations precisely when the w* and the norm topologies agree on the dual sphere. This extends some known results and answers an open question.