The Attouch-Wets Topology for Convex Functions

Abstract

A proper lower semicontinuous convex function f on a normed linear space X, identified with its epigraph, is a nonempty closed convex subset of X × R containing no vertical lines that recedes in the vertical direction: whenever (x,α) ∈ epi f, and β > α, then (x,β) ∈ epi f. Hyperspace topologies defined on C(X × R) naturally induce topologies on Γ(X). In this chapter, we look in some depth at an important topology on Γ(X) that arises in this way: the Attouch-Wets topology. For one thing, the convergence of a sequence of closed convex sets corresponds to the convergence of the associated sequences of indicator functions, support functions, and distance functions. Furthermore, convergence of a sequence of proper lower semicontinuous convex functions in the Attouch-Wets topology implies and is implied by the Attouch-Wets convergence of functions dual to the originals in Γ*(X*). Continuity of polarity for convex sets easily falls out of this result for convex functions. Thus, the Attouch-Wets topology is stable with respect to duality. Moreover, the strength of Attouch-Wets topology, as well as its overall tractability in terms of estimation, make it a highly potent convergence concept applicable to convex optimization and approximation problems.