On restricted centers of sets

Abstract

Given a normed linear space X, a family V of nonempty closed subsets of X, and a family F of nonempty closed and bounded subsets of X, we identify three properties (R 1), (R 2), and (R 3) of the triplets ( X, V, F ) where V ∈ V , and two properties (R 4), (R̃ 4) of the triplets ( X, V,F ), with a view to studying existence of restricted centers and stability of the restricted center map. This leads to a sharpening of many known results as well as to some new results for existence of restricted centers, and it also enables us to obtain some new continuity results for restricted center maps.