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One approach to see what is going on here is via the notion of an open ball under a given distance. Given a distance $d$, the open ball of radius $r$ around a point $p$ is given by $B_r(p):=\{q\mid d(p,q)<r\}$, i.e. the set of all points with distance less than $r$ to the center of the ball, $p$. In the plane and under the Euclidean distance, a ball is an ...

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