The Slice Topology for Convex Functions

Abstract

Consistent with the presentation of Chapter 7, slice convergence of a net of lower semicontinuous proper convex functions defined on a normed linear space X means the convergence of their epigraphs with respect to the slice topology on C(X × R) as determined by the box norm on X × R. Similarly, we may speak of dual slice convergence of functions in Γ*(X*). A major result of this chapter shows that slice convergence in Γ(X) implies and is implied by dual slice convergence in Γ*(X*). From this result, and our presentation of the slice and dual slice topologies as weak topologies in Chapter 2, Wijsman convergence of a net of closed convex sets with respect to all norms equivalent to a given initial norm implies Wijsman convergence of the polars with respect to the dual norms. But most importantly, slice convergence is intrinsic to convex duality, and we present a number of characterizations of the topology in terms of the lower semicontinuity of natural geometrically defined multifunctions on Γ(X) and on C(X).