Wijsman convergence: A survey

Abstract

A net 〈Aλ〉 of nonempty closed sets in a metric space 〈X, d〉 is declared Wijsman convergent to a nonempty closed set A provided for each x εX, we have d(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. © 1994 Kluwer Academic Publishers.