Wijsman convergence of convex sets under renorming

Abstract

LET (X, p> BE A metric space. A natural way to define convergence for a sequence of nonempty closed subsets (A,) of X to a nonempty closed subset A is to insist that we have pointwise convergence of the associated sequence of distance functionals for the sets A, to the distance functional for the set A: for each x E X, c&(x, A) = lim d,(x, A,). Convergence in this sense is just convergence with respect to the weak topology on the nonempty closed subsets CL(X) of X determined by the family of distance functionals (d,(x, e): x E X), the so-called Wijsman topology z,(,) [l-8] determined by the metric p. Here, of course, each functional of the form d,(x, 0) is a function of a closed set variable. Unfortunately, different metrics may give rise to different Wijsman topologies, even if the metrics are uniformly equivalent (see, e.g. [4, Section 41). It turns out [6, theorem 3.11 that if we vary our metrics over the class C(X) of metrics that give rise to the topology of X as well as varying x over X, the resulting family