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function) , and let be a positive real number. The open (metric) ball of radius centered at a point in , usually denoted by or , is defined the same way as a Euclidean ball, as the set of points in of distance less than away from ,

The closed (metric) ball, sometimes denoted or , is likewise defined as the set of points of distance less than or equal to away from ,

In particular, a ball (open or closed) always includes itself, since the definition requires . A unit ball (open

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