 Ball (mathematics)

Nball redirects here. For the video game, see Nball (game).
In mathematics, a ball is the space inside a sphere. It may be a closed ball (including the boundary points) or an open ball (excluding them).
These concepts are defined not only in threedimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in the Euclidean plane, for example, is the same thing as a disk, the area bounded by a circle.
In mathematical contexts where ball is used, a sphere is usually assumed to be the boundary points only (namely, a spherical surface in threedimensional space). In other contexts, such as in Euclidean geometry and informal use, sphere sometimes means ball.
Contents
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 B_{r}(p) or B(p; r), is defined by
The closed (metric) ball, which may be denoted by B_{r}[p] or B[p; r], 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 B_{r}(p) is usually denoted . While it is always the case that and , it is not always the case that . For example, in a metric space X with the discrete metric, one has and B_{1}[p] = X, 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 V with norm · is also a metric space, with the metric d(x, y) = x − y. In such spaces, every ball B_{r}(p) is a copy of the unit ball B_{1}(0), scaled by r and translated by p.
Euclidean norm
In particular, if V is ndimensional 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.
Pnorm
In Cartesian space with the pnorm L_{p}, an open ball is the set
For n=2, in particular, the balls of L_{1} (often called the taxicab or Manhattan metric) are squares with the diagonals parallel to the coordinate axes; those of L_{∞} (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 L_{1} are octahedra with axisaligned body diagonals, those of L_{∞} are cubes with axisaligned edges, and those of L_{p} with p > 2 are superellipsoids.
General convex norm
More generally, given any centrally symmetric, bounded, open, and convex subset X of Rn, one can define a norm on Rn where the balls are all translated and uniformly scaled copies of X. 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 Rn.
Topological balls
One may talk about balls in any topological space X, not necessarily induced by a metric. An (open or closed) ndimensional topological ball of X is any subset of X which is homeomorphic to an (open or closed) Euclidean nball. Topological nballs are important in combinatorial topology, as the building blocks of cell complexes.
Any open topological nball is homeomorphic to the Cartesian space R^{n} and to the open unit ncube . Any closed topological nball is homeomorphic to the closed ncube [0, 1]^{n}.
An nball is homeomorphic to an mball if and only if n = m. The homeomorphisms between an open nball B and R^{n} can be classified in two classes, that can be identified with the two possible topological orientations of B.
A topological nball need not be smooth; if it is smooth, it need not be diffeomorphic to a Euclidean nball.
See also
 Ball  ordinary meaning
 Disk (mathematics)
 Neighborhood (mathematics)
 3sphere
 nsphere, or hypersphere
 Alexander horned sphere
 Manifold
References
 D. J. Smith and M. K. Vamanamurthy, "How small is a unit ball?", Mathematics Magazine, 62 (1989) 101–107.
 "Robin conditions on the Euclidean ball", J. S. Dowker [1]
 "Isometries of the space of convex bodies contained in a Euclidean ball", Peter M. Gruber[2]
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