View source for Ball (mathematics)

{{more footnotes|date=March 2013}}
[[File:Blue ball.png|thumb|In [[Euclidean space]], a '''ball''' is the volume bounded by a sphere]]

In [[mathematics]], a '''ball''' is the space bounded by a [[sphere]]. It may be a '''closed ball''' (including the [[boundary points]] that constitute the sphere) or an '''open ball''' (excluding them).

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'' or '''hyperball''' in {{mvar|n}} dimensions is called an '''{{mvar|n}}-ball''' and is bounded by an [[N-sphere|'''({{math|''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 sphere]]. In a [[one-dimensional space]], a ball is a [[line segment]].

In other contexts, such as in [[Euclidean geometry]] and informal use, ''sphere'' is sometimes used to mean ''ball''.

In general practice, the volume of a ball is calculated as:
: <math>V = \frac{4}{3}\pi r^3 \approx 0.523 \cdot d^3</math>
where {{mvar|r}} is the radius and {{mvar|d}} is the diameter of the ball.

==In Euclidean space==
In Euclidean {{mvar|n}}-space, an (open) {{mvar|n}}-ball of radius {{mvar|r}} and center {{mvar|x}} is the set of all points of distance less than {{mvar|r}} from {{mvar|x}}. A closed {{mvar|n}}-ball of radius {{mvar|r}} is the set of all points of distance less than or equal to {{mvar|r}} away from {{mvar|x}}.

In Euclidean {{mvar|n}}-space, every ball is bounded by a [[hypersphere]].  The ball is a bounded [[Interval (mathematics)|interval]] when {{math|1=''n'' = 1}}, is a '''[[Disk (mathematics)|disk]]''' bounded by a [[circle]] when {{math|1=''n'' = 2}}, and is bounded by a [[sphere]] when {{math|1=''n'' = 3}}.

=== Volume ===
{{main article|Volume of an n-ball|l1=Volume of an {{mvar|n}}-ball}}
The {{mvar|n}}-dimensional volume of a Euclidean ball of radius {{mvar|R}} in {{mvar|n}}-dimensional Euclidean space is:<ref>Equation 5.19.4, ''NIST Digital Library of Mathematical Functions.'' http://dlmf.nist.gov/,{{dead link|date=July 2016 |bot=InternetArchiveBot |fix-attempted=yes }} Release 1.0.6 of 2013-05-06.</ref>
:<math>V_n(R) = \frac{\pi^\frac{n}{2}}{\Gamma\left(\frac{n}{2} + 1\right)}R^n,</math>
where&nbsp;{{math|Γ}} is [[Leonhard Euler]]'s [[gamma function]] (which can be thought of as an extension of the [[factorial]] function to fractional arguments). Using explicit formulas for [[particular values of the gamma function]] at the integers and half integers gives formulas for the volume of a Euclidean ball that do not require an evaluation of the gamma function. These are:
:<math>\begin{align}V_{2k}(R) &= \frac{\pi^k}{k!}R^{2k}\,,\\
V_{2k+1}(R) &= \frac{2^{k+1}\pi^k}{(2k+1)!!}R^{2k+1} = \frac{2(k!)(4\pi)^k}{(2k+1)!}R^{2k+1}\,.\end{align}</math>
In the formula for odd-dimensional volumes, the [[double factorial]] {{math|(2''k'' + 1)!!}} is defined for odd integers {{math|2''k'' + 1}} as {{math|1=(2''k'' + 1)!! = 1 · 3 · 5 · … · (2''k'' − 1) · (2''k'' + 1)}}.

==In general metric spaces==
Let {{math|(''M'', ''d'')}} be a [[metric space]], namely a set {{mvar|M}} with a [[Metric (mathematics)|metric]] (distance function) {{mvar|d}}. The open (metric) '''ball of radius''' {{math|''r'' > 0}} centered at a point {{mvar|p}} in {{mvar|M}}, usually denoted by {{math|''B<sub>r</sub>''(''p'')}} or {{math|''B''(''p''; ''r'')}}, is defined by

:<math>B_r(p) = \{ x \in M \mid d(x,p) < r \},</math>

The closed (metric) ball, which may be denoted by {{math|''B<sub>r</sub>''[''p'']}} or {{math|''B''[''p''; ''r'']}}, is defined by

:<math>B_r[p] = \{ x \in M \mid d(x,p) \le r \}.</math>

Note in particular that a ball (open or closed) always includes {{mvar|p}} itself, since the definition requires {{math|''r'' > 0}}.

The [[closure (mathematics)|closure]] of the open ball {{math|''B<sub>r</sub>''(''p'')}} is usually denoted {{math|{{overline|''B<sub>r</sub>''(''p'')}}}}. While it is always the case that {{math|''B<sub>r</sub>''(''p'') ⊆ {{overline|''B<sub>r</sub>''(''p'')}} ⊆ ''B<sub>r</sub>''[''p'']}}, it is {{em|not}} always the case that {{math|{{overline|''B<sub>r</sub>''(''p'')}} {{=}} ''B<sub>r</sub>''[''p'']}}. For example, in a metric space {{mvar|X}} with the [[discrete metric]], one has {{math|{{overline|''B''<sub>1</sub>(''p'')}} {{=}} {p<nowiki>}</nowiki>}} and {{math|''B''<sub>1</sub>[''p''] {{=}} ''X''}}, for any {{math|''p'' ∈ ''X''}}.

A '''[[unit ball]]''' (open or closed) is a ball of radius 1.

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.

The open balls of a [[metric space]] can serve as a [[base (topology)|base]], giving this space a [[topological space|topology]], the open sets of which are all possible [[union (set theory)|union]]s of open balls. This topology on a metric space is called the '''topology induced by''' the metric {{mvar|d}}.

==In normed vector spaces==
Any [[normed vector space]] {{mvar|V}} with norm <math>\|\cdot\|</math> is also a metric space with the metric <math>d (x,y)= \|x-y\|.</math>  In such spaces, an arbitrary ball <math>B_r(y)</math> of points <math>x</math> around a point <math>y</math> with a distance of less than <math>r</math> may be viewed as a scaled (by <math>r</math>) and translated (by <math>y</math>) copy of a ''unit ball'' <math>B_1(0).</math> Such "centered" balls with <math>y=0</math> are denoted with <math>B(r).</math>

The Euclidean balls discussed earlier are an example of balls in a normed vector space.

==={{mvar|p}}-norm===
In a [[Cartesian space]] {{math|ℝ<sup>''n''</sup>}} with the [[p-norm|{{mvar|p}}-norm]] {{mvar|L<sub>p</sub>}}, that is

:<math>\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dotsb + |x_n|^p \right) ^{1/p},</math>

an open ball around the origin with radius <math>r</math> is given by the set

:<math> B(r) = \left\{ x \in \R^n \,:\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dotsb + |x_n|^p \right) ^{1/p} < r \right\}.</math>

For {{math|1=''n'' = 2}}, in a 2-dimensional plane <math>\mathbb R^2</math>, "balls" according to the {{math|''L''<sub>1</sub>}}-norm (often called the ''[[Taxicab geometry|taxicab]]'' or ''Manhattan'' metric) are bounded by squares with their ''diagonals'' parallel to the coordinate axes; those according to the {{math|''L''<sub>∞</sub>}}-norm, also called the [[Chebyshev distance|Chebyshev]] metric, have squares with their ''sides'' parallel to the coordinate axes as their boundaries. The {{math|''L''<sub>2</sub>}}-norm, known as the Euclidean metric, generates the well known discs within circles, and for other values of {{mvar|p}}, the corresponding balls are areas bounded by [[Lamé curve]]s (hypoellipses or hyperellipses).

For {{math|1=''n'' = 3}}, the {{math|''L''<sub>1</sub>}}- balls are within octahedra with axes-aligned ''body diagonals'', the {{math|''L''<sub>∞</sub>}}-balls are within cubes with axes-aligned ''edges'', and the boundaries of balls for {{mvar|L<sub>p</sub>}} with {{math|''p'' > 2}} are [[superegg|superellipsoids]]. Obviously, {{math|''p'' {{=}} 2}} generates the inner of usual spheres.

===General convex norm===
More generally, given any [[central symmetry|centrally symmetric]], [[bounded set|bounded]], [[open set|open]], and [[convex set|convex]] subset {{mvar|X}} of {{math|ℝ<sup>''n''</sup>}}, one can define a [[Norm (mathematics)|norm]] on {{math|ℝ<sup>''n''</sup>}} where the balls are all translated and uniformly scaled copies of&nbsp;{{mvar|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&nbsp;{{math|ℝ<sup>''n''</sup>}}.

==In topological spaces==

One may talk about balls in any [[topological space]] {{mvar|X}}, not necessarily induced by a metric. An (open or closed) {{mvar|n}}-dimensional '''topological ball''' of {{mvar|X}} is any subset of {{mvar|X}} which is [[homeomorphic]] to an (open or closed) Euclidean {{mvar|n}}-ball. Topological {{mvar|n}}-balls are important in [[combinatorial topology]], as the building blocks of [[cell complex]]es.

Any open topological {{mvar|n}}-ball is homeomorphic to the Cartesian space {{math|ℝ<sup>''n''</sup>}} and to the open [[hypercube|unit {{mvar|n}}-cube]] (hypercube) {{math|(0, 1)<sup>''n''</sup> ⊆ ℝ<sup>''n''</sup>}}. Any closed topological {{mvar|n}}-ball is homeomorphic to the closed {{mvar|n}}-cube {{math|[0, 1]<sup>''n''</sup>}}.

An {{mvar|n}}-ball is homeomorphic to an {{mvar|m}}-ball if and only if {{math|1=''n'' = ''m''}}. The homeomorphisms between an open {{mvar|n}}-ball {{mvar|B}} and {{math|ℝ<sup>''n''</sup>}} can be classified in two classes, that can be identified with the two possible [[orientation (mathematics)|topological orientation]]s of&nbsp;{{mvar|B}}.

A topological {{mvar|n}}-ball need not be [[differentiable manifold|smooth]]; if it is smooth, it need not be [[diffeomorphic]] to a Euclidean {{mvar|n}}-ball.

==See also==
{{Div col|colwidth=20em}}
*[[Ball]] – ordinary meaning
*[[Disk (mathematics)]]
*[[Formal ball]], an extension to negative radii
*[[Neighbourhood (mathematics)]]
*[[3-sphere]]
*[[n-sphere|{{mvar|n}}-sphere]], or hypersphere
*[[Alexander horned sphere]]
*[[Manifold]]
*[[Volume of an n-ball|Volume of an {{mvar|n}}-ball]]
*[[Octahedron]] – a 3-ball in the {{math|''l''<sub>1</sub>}} metric.
*[[Spherical shell]]
{{div col end}}

==References==
{{Expand section|date=December 2009}}
{{Reflist}}
* {{cite journal |first=D. J. |last=Smith |first2=M. K. |last2=Vamanamurthy |title=How small is a unit ball? |journal=[[Mathematics Magazine]] |volume=62 |year=1989 |issue=2 |pages=101–107 |jstor=2690391 |doi=10.1080/0025570x.1989.11977419}}
* {{cite journal |title=Robin Conditions on the Euclidean ball |first=J. S. |last=Dowker |journal=[[Classical and Quantum Gravity]] |year=1996 |volume=13 |issue=4 |pages=585–610 |doi=10.1088/0264-9381/13/4/003 |arxiv=hep-th/9506042 |bibcode=1996CQGra..13..585D }}
* {{cite journal |first=Peter M. |last=Gruber |title=Isometries of the space of convex bodies contained in a Euclidean ball |journal=[[Israel Journal of Mathematics]] |year=1982 |volume=42 |issue=4 |pages=277–283 |doi=10.1007/BF02761407 }}

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[[Category:Balls]]
[[Category:Metric geometry]]
[[Category:Spheres]]
[[Category:Topology]]