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| {{redirect|N-ball|the video game|N-ball (game)}}
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| [[File:Sphere wireframe.svg|thumb|A '''ball''' is the inside of a sphere]]
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| In [[mathematics]], a '''ball''' is the space inside a [[sphere]]. It may be a '''closed ball''' (including the [[boundary points]] of the sphere) or an '''open ball''' (excluding them).
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| These concepts are defined not only in three-dimensional [[Euclidean space]] but also for lower and higher dimensions, and for [[metric space]]s in general. A ''ball'' in ''n'' dimensions is called an [[n-ball|'''''n''-ball''']] and is bounded by an [[N-sphere|'''''(n-1)''-sphere''']]. Thus, for example, a ball in the [[Euclidean plane]] is the same thing as a [[disk (mathematics)|disk]], the area bounded by a [[circle]]. In [[Euclidean space | Euclidean 3-space]], a ball is taken to be the [[volume]] bounded by a [[2-sphere|2-dimensional spherical shell]] boundary.
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| In other contexts, such as in [[Euclidean geometry]] and informal use, ''sphere'' is sometimes used to mean ''ball''.
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| ==Balls in general metric spaces==
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| Let (''M'',''d'') be a [[metric space]], namely a set ''M'' with a [[metric (mathematics)|metric]] (distance function) ''d''. The '''open''' ('''metric''') '''ball of radius''' ''r'' > 0 '''centered at''' a point ''p'' in ''M'', usually denoted by ''B''<sub>''r''</sub>(''p'') or ''B''(''p''; ''r''), is defined by
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| :<math>B_r(p) \triangleq \{ x \in M \mid d(x,p) < r \}.</math>
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| The '''closed''' ('''metric''') '''ball''', which may be denoted by ''B''<sub>''r''</sub>[''p''] or ''B''[''p''; ''r''], is defined by
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| :<math>B_r[p] \triangleq \{ x \in M \mid d(x,p) \le r \}.</math>
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| Note in particular that a ball (open or closed) always includes <var>p</var> itself, since the definition requires <var>r</var> > 0.
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| The [[closure (mathematics)|closure]] of the open ball ''B''<sub>''r''</sub>(''p'') is usually denoted <math>\overline{ B_r(p) }</math>. While it is always the case that <math>B_r(p) \subseteq \overline{ B_r(p) }</math> and <math>\overline{ B_r(p) } \subseteq B_r[p]</math>, it is ''not'' always the case that <math>\overline{ B_r(p) } = B_r[p]</math>. For example, in a metric space <math>X</math> with the [[discrete metric]], one has <math>\overline{B_1(p)} = \{p\}</math> and <math>B_1[p] = X</math>, for any <math>p \in X</math>.
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| A (open or closed) '''[[unit ball]]''' is a ball of radius 1.
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| A subset of a metric space is [[bounded set|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.
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| The open balls of a [[metric space]] are a [[basis (topology)|basis]] for a [[topological space]], whose open sets are all possible [[union (set theory)|union]]s of open balls. This space is called the '''topology induced by''' the metric ''d''.
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| ==Balls in normed vector spaces==
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| 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''<sub>''r''</sub>(''p'') is a copy of the unit ball ''B''<sub>1</sub>(0), scaled by ''r'' and translated by ''p''.
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| ===Euclidean norm===
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| In particular, if ''V'' is ''n''-dimensional [[Euclidean space]] with the ordinary [[Euclidean distance|(Euclidean) metric]], every ball is the interior of an [[hypersphere]] (a '''hyperball'''). That is a bounded [[interval (mathematics)|interval]] when ''n'' = 1, the interior of a [[circle]] (a '''[[disk (mathematics)|disk]]''') when ''n'' = 2, and the interior of a [[sphere]] when ''n'' = 3.
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| ===P-norm===
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| In [[Cartesian space]] <math>\R^n</math> with the [[p-norm]] ''L''<sub>''p''</sub>, an open ball is the set
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| : <math>B(r) = \left\{ x \in \R^n \,:\, \sum_{i=1}^n \left|x_i\right|^p < r^p \right\}.</math>
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| For ''n''=2, in particular, the balls of ''L''<sub>1</sub> (often called the ''[[taxicab geometry|taxicab]]'' or ''Manhattan'' metric) are squares with the diagonals parallel to the coordinate axes;
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| those of ''L''<sub>∞</sub> (the [[Chebyshev distance|Chebyshev]] metric) are squares with the sides parallel to the coordinate axes. For other values of ''p'', the balls are the interiors of [[Lamé curve]]s (hypoellipses or hyperellipses).
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| For ''n'' = 3, the balls of ''L''<sub>1</sub> are octahedra with axis-aligned body diagonals, those of ''L''<sub>∞</sub> are cubes with axis-aligned edges, and those of ''L''<sub>''p''</sub> with ''p'' > 2 are [[superegg|superellipsoids]].
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| ===General convex norm===
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| More generally, given any [[central symmetry|centrally symmetric]], [[bounded set|bounded]], [[open set|open]], and [[convex set|convex]] subset ''X'' of '''R'''<sup>''n''</sup>, one can define a [[Norm (mathematics)|norm]] on '''R'''<sup>''n''</sup> 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 '''R'''<sup>''n''</sup>.
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| ==Topological balls==
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| One may talk about balls in any [[topological space]] ''X'', not necessarily induced by a metric. An (open or closed) ''n'''''-dimensional topological ball''' of ''X'' is any subset of ''X'' 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 complex]]es.
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| Any open topological ''n''-ball is homeomorphic to the Cartesian space '''R'''<sup>''n''</sup> and to the open [[hypercube|unit ''n''-cube]] <math>(0,1)^n \subseteq \R^n</math>. Any closed topological ''n''-ball is homeomorphic to the closed ''n''-cube [0, 1]<sup>''n''</sup>.
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| An ''n''-ball is homeomorphic to an ''m''-ball if and only if ''n'' = ''m''. The homeomorphisms between an open ''n''-ball ''B'' and '''R'''<sup>''n''</sup> can be classified in two classes, that can be identified with the two possible [[orientation (mathematics)|topological orientation]]s of ''B''.
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| A topological ''n''-ball need not be [[differentiable manifold|smooth]]; if it is smooth, it need not be [[diffeomorphic]] to a Euclidean ''n''-ball.
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| ==See also==
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| *[[Ball]] - ordinary meaning
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| *[[Disk (mathematics)]]
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| *[[Neighborhood (mathematics)]]
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| *[[3-sphere]]
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| *[[n-sphere|''n''-sphere]], or hypersphere
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| *[[Alexander horned sphere]]
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| *[[Manifold]]
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| *[[Volume of an n-ball|Volume of an ''n''-ball]]
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| ==References==
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| {{Reflist}}
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| *D. J. Smith and M. K. Vamanamurthy, "How small is a unit ball?", ''[[Mathematics Magazine]]'', 62 (1989) 101–107.
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| *"Robin conditions on the Euclidean ball", J. S. Dowker [http://www.citebase.org/fulltext?format=application/pdf&identifier=oai:arXiv.org:hep-th/9506042]
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| *"Isometries of the space of convex bodies contained in a Euclidean ball", Peter M. Gruber[http://www.springerlink.com/content/0v74h15104232532/]
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| {{Expand section|date=December 2009}}
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| {{DEFAULTSORT:Ball (Mathematics)}}
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| [[Category:Balls]]
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| [[Category:Metric geometry]]
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| [[Category:Spheres]]
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| [[Category:Topology]]
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