CONVERGENCE OF LIPSCHITZ REGULARIZATIONS OF CONVEX-FUNCTIONS

For a sequence or net of convex functions on a Banach space, we study pointwise convergence of their Lipschitz regularizations and convergen ce of their epigraphs. The Lipschitz regularizations we will consider are the infimal convolutions of the functions with appropriate multipl es of the norm. For a sequence of convex functions on a separable Bana ch space we show that both pointwise convergence of their Lipschitz re gularizations and Wijsman convergence of their epigraphs are equivalen t to variants of two conditions used by Attouch and Beer to characteri ze 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 no rm topologies agree on tile dual sphere. This extends some known resul ts and answers an open question. (C) 1995 Academic Press, Inc.