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The Slice Topology for Convex Functions

  • Gerald Beer
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 268)

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).

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Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  • Gerald Beer
    • 1
  1. 1.Department of Mathematics and Computer ScienceCalifornia State UniversityLos AngelesUSA

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