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A closed n-ball of radius r is the set of all points of distance less than or equal to r away from x. In Euclidean n -space, every ball is bounded by a hypersphere . The ball is a bounded interval when n = 1 , is a disk bounded by a circle when n = 2 , and is bounded by a sphere when n = 3 .
Intersecting the n-ball with the (n − 2)-dimensional plane defined by fixing a radius and an azimuth gives an (n − 2)-ball of radius √ R 2 − r 2. The volume of the ball can therefore be written as an iterated integral of the volumes of the (n − 2)-balls over the possible radii and azimuths:
An unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center. A sphere or ball with unit radius and center at the origin of the space is called the unit sphere or the unit ball.
At the other end of the spectrum, in an injective metric space such as the Manhattan distance in the plane, = /: any two closed balls of radius / centered at points of have a non-empty intersection, therefore all such balls have a common intersection, and a radius / ball centered at a point of this intersection contains all of .
An (+) -ball is closed if it includes the -sphere, and it is open if it does not include the -sphere. Specifically: A 1 {\displaystyle 1} - ball , a line segment , is the interior of a 0-sphere.
For every natural number n, the n-sphere is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.
Malcolm X was assassinated in 1965 when gunmen opened fire while he gave a speech in New York. A new lawsuit accuses the government of conspiracy.
The n-sphere of unit radius centered at the origin is denoted S n and is often referred to as "the" n-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space. In topology, the n-sphere is an example of a compact topological manifold without boundary.