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Ball (mathematics)
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In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general.
closed (metric) ball, which may be denoted by or , is defined by
Note in particular that a ball (open or closed) always includes p itself, since the definition requires r > 0.
The closure of the open ball is usually denoted .
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In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general.
Balls in general metric spaces
Let (M,d) be a metric space, namely a set M with a metric (distance function) d. The open (metric) ball of radius r > 0 centered at a point p in M, usually denoted by or , is defined by
The closed (metric) ball, which may be denoted by or , is defined by
Note in particular that a ball (open or closed) always includes p itself, since the definition requires r > 0.
The closure of the open ball is usually denoted . While it is always the case that and , it is not always the case that . For example, in a metric space with the discrete metric, one has and , for any .
An (open or closed) unit ball is a ball of radius 1.
A subset of a metric space is bounded if it is contained in some ball. A set is totally bounded if, given any positive radius, it is covered by finitely many balls of that radius.
The open balls of a metric space are a basis for a topological space, whose open sets are all possible unions of open balls. This space is called the topology induced by the metric d.
Balls in normed vector spaces
Any normed vector space with norm is also a metric space, with the metric . In such spaces, every ball is a copy of the unit ball , scaled by and translated by .
Euclidean norm
In particular, if is n-dimensional Euclidean space with the ordinary (Euclidean) metric, every ball is the interior of an hypersphere (a hyperball). That is a bounded interval when n=1, the interior of a circle (a disk) when n=2, and the interior of a sphere when n=3.
P-norm
In Cartesian space with the p-norm Lp, an open ball is the set
-
For n=2, in particular, the balls of L1 (often called the taxicab or Manhattan metric) are squares with the diagonals parallel to the coordinate axes;
those of L8 (the Chebyshev metric) are squares with the sides parallel to the coordinate axes. For other values of p, the balls are the interiors of Lamé curves (hypoellipses or hyperellipses).
For n=3, the balls of L1 are octahedra with axis-aligned body diagonals, those of L8 are cubes with axis-aligned edges, and those of Lp with p> 2 are superellipsoids.
General convex norm
More generally, given any centrally symmetric, bounded, open, and convex subset of , one can define a norm on where the balls are all translated and uniformly scaled copies of . Note this theorem does not hold if "open" subset is replaced by "closed" subset, because the origin point qualifies but does not define a norm on .
Topological balls
One may talk about balls in any topological space , not necessarily induced by a metric. An (open or closed) n-dimensional topological ball of is any subset of which is homeomorphic to an (open or closed) Euclidean n-ball. Topological n-balls are important in combinatorial topology, as the building blocks of cell complexes.
Any open topological n-ball is homeomorphic to the Cartesian space and to the open unit n-cube . Any closed topological n-ball is homeomorphic to the closed n-cube .
An n-ball is homeomorphic to an m-ball if and only if n=m. The homeomorphisms between an open n-ball and can be classified in two classes, that can be identified with the two possible topological orientations of .
A topological n-ball need not be smooth; if it is smooth, it need not be diffeomorphic to an Euclidean n-ball.
See also
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