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True or false: The closed balls of a metric space are precisely those subsets such that every proper superset has strictly greater diameter.

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I think that any closed convex set will satisfy this. – Tomás Jan 19 '13 at 15:59
    
To prove one direction, I think it's easiest to do by contradiction. – Clayton Jan 19 '13 at 15:59
    
1- how about non-empty interior. 2- how about every proper subset having smaller diameter? – Maesumi Jan 19 '13 at 16:00
up vote 4 down vote accepted

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 $\mathrm{diam}\ X=\mathrm{diam}\ B=4$.

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