Topologies related to the prox map and the restricted center map. (English) Zbl 0849.41010
Summary: In this paper we review some of the recently introduced hypertopologies on collections of nonempty closed subsets of a metrizable space in the context of continuity properties of the prox map and the restricted center map. We identify here suitable families \({\mathcal A}, {\mathcal B}\) of nonempty closed sets equipped with natural topologies, as coordinate spaces for the gap functional and the restricted radius functional, with a view to explore bivariate continuity of these functionals. This leads us to sharpenings of many known results as well as to some new results for upper semicontinuity of the prox map and the restricted center map. This, in turn, also leads us to improvements of certain best approximation results for convex-valued multifunctions and fixed point theorems for such multifunctions.
MSC:
| 41A28 | Simultaneous approximation |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
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