Metrics that generate the same hyperspace convergence
Let \(c(X)\) be the family of all closed subsets of a metric space \((X,d)\). A net \((A_ j)\) in \(c(X)\) is Wijsman convergent (\(w_ d\)-convergent) to \(A\in c(X)\) if for any \(x\in X\) we have \(d(x,A)= \lim_ j d(x,A_ j)\). The convergence is topological and the authors give necessary and sufficient conditions for two equivalent metrics to generate the same Wijsman convergence. Some applications to normed linear spaces and to various classes of metrizable spaces are also included. One of typical results: Let \(\|\cdot\|_ 1\), \(\|\cdot\|_ 2\) be two equivalent non homothetic norms on a real linear space \(X\) and \(d_ 1\), \(d_ 2\) the metrics they respectively generate. Then \(w_{d_ 1}= w_{d_ 2}\) if and only if \(X\) is finite-dimensional.
scientific article; zbMATH DE number 465282
scientific article