On selections of the metric projection and best proximity pairs in hyperconvex spaces

Abstract

In this work we present new results on nonexpansive retractions and best proximity pairs in hyperconvex metric spaces. We sharpen the main results of R. Esp´ınola et al. in [3] (Nonexpansive retracts in hyperconvex spaces, J. Math. Anal. Appl. 251 (2000), 557–570) on existence of nonexpansive selections of the metric projection. More precisely we characterize those subsets of a hyperconvex metric space with the property that the metric projection onto them admits a nonexpansive selection as a subclass of sets introduced in [3]. This is a rather exceptional property with a lot of applications in approximation theory, in particular we apply it to answer in the positive the main question posed by Kirk et al. in [5] (Proximinal retracts and best proximity pair theorems, Num. Funct. Anal. Opt. 24 (2003), 851–862).