 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]
Wikimedia Foundation. 2010.
Look at other dictionaries:
Ball (disambiguation) — A ball is a spherical or ovoid object typically used in games. In games: * Bungee ball, a toy * No ball, in cricket, an illegal delivery of the ball to the batsman by the bowler * Ball (baseball), a pitch at which the batter does not swing and… … Wikipedia
Ball State University — Motto Education Redefined Established 1918 (details) Type Public coeducational Endowment … Wikipedia
Mathematics and science partnerships — (MSP) is education policy from Title 2, Part B, Sections 2201 2203 of the No Child Left Behind Act of 2001. The purpose of MSP is to increase student achievement in science and mathematics by partnering IHE science, math, and engineering… … Wikipedia
Mathematics (song) — Mathematics Single by Mos Def from the album Black on Both Sides Released … Wikipedia
Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity. Computer scientist Manindra Agrawal of the… … Universalium
mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… … Universalium
Mathematics and Science Academy UTB — The Mathematics and Science Academy (MSA), a high school located in Brownsville, Texas, was established by the 79th Texas Legislature in May 2005. It was designed as a commuter program at the University of Texas at Brownsville and Texas Southmost … Wikipedia
Disk (mathematics) — For other uses, see Disc (disambiguation). A disk is the region bounded by a circle. An open disk is the interior of the disk excluding the bounding circle, while a closed disk (see closed set) is the open disk together with the bounding circle.… … Wikipedia
List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… … Wikipedia
Nball (game) — This article is about the video game. For the mathematical concept of a ball in higher dimensional spaces, see ball (mathematics). N Ball Developer(s) Rag Doll Software Platform(s) Microsoft Windows, Apple Macintosh OSX … Wikipedia