Best approximation in metric spaces

Abstract

A metric space ( X , d ) \left ( {X,d} \right ) is called an M M -space if for every x x and y y in X X and for every r ∈ [ 0 , λ ] r \in \left [ {0,\lambda } \right ] we have B [ x , r ] ∩ B [ y , λ − r ] = { z } B\left [ {x,r} \right ] \cap B\left [ {y,\lambda - r} \right ] = \left \{ z \right \} for some z ∈ X z \in X , where λ = d ( x , y ) \lambda = d\left ( {x,y} \right ) . It is the object of this paper to study M M -spaces in terms of proximinality properties of certain sets.