TITLE: Characterisation of closed balls in terms of diameters
QUESTION [5 upvotes]: True or false: The closed balls of a metric space are precisely those subsets such that every proper superset has strictly greater diameter.

REPLY [4 votes]: False: take $X=\{(-2,0),(0,0),(2,0),(0,3)\}$ in $\mathbb R^2$ with the induced metric. Then the closed ball centered at $(0,0)$ of radius $2$ is the set $B=X\setminus\{(0,3)\}$, and we have $\mathrm{diam}\ X=\mathrm{diam}\ B=4$.