Set-Valued Analysis

, Volume 2, Issue 1–2, pp 77–94 | Cite as

Wijsman convergence: A survey

  • Gerald Beer
Article

Abstract

A net 〈Aλ〉 of nonempty closed sets in a metric space 〈X, d〉 is declaredWijsman convergent to a nonempty closed setA provided for eachx εX, we haved(x, A)=limλd(x, A). Interest in this convergence notion originates from the seminal work of R. Wijsman, who showed in finite dimensions that the conjugate map for proper lower semicontinuous convex functions preserves convergence in this sense, where functions are identified with their epigraphs. In this paper, we review the attempts over the last 25 years to produce infinite-dimensional extensions of Wijsman's theorem, and we look closely at the topology of Wijsman convergence in an arbitrary metric space as well. Special emphasis is given to the developments of the past five years, and several new limiting counterexamples are presented.

Mathematics Subject Classifications (1991)

54B20 40A30 52A41 46N10 

Key words

Wijsman convergence hyperspace conjugate function convex function Mosco convergence Attouch-Wets convergence slice convergence 

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Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Gerald Beer
    • 1
  1. 1.Department of MathematicsCalifornia State University at Los AngelesLos AngelesUSA

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