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Characterization of metrics such that the volume of balls doesn't depend on their centers?

Given a finite metric space $(X,d)$, when does it hold that for all $y\in X$ and $r>0$, $\#B(y,r)$ does not depend on $y$? Here $ B(y,r):=\{x\in X: d(x,y)\le r\} $ denotes a ball of radius $r$ centere …