Best approximation in metric spaces

Abstract:

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