I want to know if there is a proper term for the following kind of points, which I call it center(borrowed from graph theory).
Let $(X,d)$ be a metric space. $x\in X$ is a center of $A\subset X$ if $x\in A$ and $ \sup_{a\in A} d(x,a) = \inf_{b\in A} \sup_{a\in A} d(b,a) $.