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In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n -ball is a ball in an n -dimensional Euclidean space. The volume of a n -ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n -ball of radius R is R n V n , {\displaystyle R^{n}V_{n},} where V n {\displaystyle V_{n}} is the volume of the unit n -ball, the n -ball of radius 1 .

The real number V n {\displaystyle V_{n}} can be expressed by a expression involving the gamma function. It can be computed with several recurrence relations. It can be expressed in terms of A n , {\displaystyle A_{n},} the area of the unit n -sphere.

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