Editing Ball (mathematics)

{{Short description|Volume space bounded by a sphere}}
{{Distinguish|Sphere}}
[[File:Blue-sphere (crop).png|thumb|upright=1.0|In [[Euclidean space]], a '''ball''' is the volume bounded by a sphere]]{{Needs more citations|date=March 2024}}

In [[mathematics]], a '''ball''' is the [[Solid geometry|solid figure]] bounded by a ''[[sphere]]''; it is also called a '''solid sphere'''.<ref>{{Cite book|last=Sūgakkai|first=Nihon|url=https://books.google.com/books?id=WHjO9K6xEm4C&dq=great+circle+great+disk+ball&pg=PA555|title=Encyclopedic Dictionary of Mathematics|publisher=[[MIT Press]]|year=1993|isbn=9780262590204|language=}}</ref> 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'' in {{mvar|n}} dimensions is called a '''hyperball''' or '''{{mvar|n}}-ball''' and is bounded by a ''hypersphere'' or [[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 [[planar region]] bounded by a [[circle]]. In [[Euclidean space|Euclidean 3-space]], a ball is taken to be the [[region of space]] 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 the field of [[topology]] the closed <math>n</math>-dimensional ball is often denoted as <math>B^n</math> or <math>D^n</math> while the open <math>n</math>-dimensional ball is <math>\operatorname{int} B^n</math> or <math>\operatorname{int} D^n</math>.

==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 n-ball}}
The {{mvar|n}}-dimensional volume of a Euclidean ball of radius {{math|''r''}} in {{math|''n''}}-dimensional Euclidean space is given by <ref>Equation 5.19.4, ''NIST Digital Library of Mathematical Functions''. [http://dlmf.nist.gov/] Release 1.0.6 of 2013-05-06.</ref>
<math display="block">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 display="block">\begin{align}
    V_{2k}(r) &= \frac{\pi^k}{k!} r^{2k}\,,\\[2pt]
  V_{2k+1}(r) &= \frac{2^{k+1}\pi^k}{\left(2k+1\right)!!} r^{2k+1} = \frac{2\left(k!\right) \left(4\pi\right)^k}{\left(2k+1\right)!}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}}, and let {{tmath|r}} be a positive real number. The open (metric) '''ball of radius''' {{mvar|r}} 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 the same way as a Euclidean ball, as the set of points in {{mvar|M}} of distance less than {{mvar|r}} away from {{mvar|p}},
<math display="block">B_r(p) = \{ x \in M \mid d(x,p) < r \}.</math>

The ''closed'' (metric) ball, sometimes denoted {{math|''B<sub>r</sub>''[''p'']}} or {{math|''B''[''p''; ''r'']}}, is likewise defined as the set of points of distance less than or equal to {{mvar|r}} away from {{mvar|p}},
<math display="block">B_r[p] = \{ x \in M \mid d(x,p) \le r \}.</math>

In particular, a ball (open or closed) always includes {{mvar|p}} itself, since the definition requires {{math|''r'' > 0}}. A '''[[unit ball]]''' (open or closed) is a ball of radius 1.

A ball in a general metric space need not be round. For example, a ball in [[real coordinate space]] under the [[Chebyshev distance]] is a [[hypercube]], and a ball under the [[taxicab distance]] is a [[cross-polytope]]. A closed ball also need not be [[Compact space|compact]]. For example, a closed ball in any infinite-dimensional [[normed vector space]] is never compact. However, a ball in a vector space will always be [[Convex set|convex]] as a consequence of the triangle inequality.

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}}.

Let <math>\overline{B_r(p)}</math> denote the [[closure (topology)|closure]] of the open ball <math>B_r(p)</math> in this topology. While it is always the case that <math>B_r(p) \subseteq \overline{B_r(p)} \subseteq B_r[p],</math> it is {{em|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> but <math>B_1[p] = X</math> for any <math>p \in X.</math>

==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 one chooses some <math>p \geq 1</math> and defines<math display="block">\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dots + |x_n|^p \right) ^{1/p},</math>Then an open ball around the origin with radius <math>r</math> is given by the set
<math display="block"> B(r) = \left\{ x \in \R^n \,:\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dots + |x_n|^p \right) ^{1/p} < r \right\}.</math>For {{math|1=''n'' = 2}}, in a 2-dimensional plane <math>\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 disks 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 [[superellipsoid]]s. {{math|1=''p'' = 2}} generates the inner of usual spheres.

Often can also consider the case of <math>p = \infty</math> in which case we define <math display="block">\lVert x \rVert_\infty = \max\{\left|x_1\right|, \dots, \left|x_n\right|\}</math>

===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.

==Regions==
{{see also|Sphere#Regions{{!}}Spherical regions}}
A number of special regions can be defined for a ball:
*''[[spherical cap|cap]]'', bounded by one plane
*''[[spherical sector|sector]]'', bounded by a conical boundary with apex at the center of the sphere
*''[[spherical segment|segment]]'', bounded by a pair of parallel planes
*''[[spherical shell|shell]]'', bounded by two concentric spheres of differing radii
*''[[spherical wedge|wedge]]'', bounded by two planes passing through a sphere center and the surface of the sphere

==See also==
{{Div col|colwidth=20em}}
*[[Ball]] – ordinary meaning
*[[Disk (mathematics)]]
*[[Formal ball]], an extension to negative radii
*[[Neighbourhood (mathematics)]]
*[[Sphere]], a similar geometric shape
*[[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.
{{div col end}}

==References==
{{Reflist}}
* {{cite journal |first1=D. J. |last1=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 |s2cid=119438515 }}
* {{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 |s2cid=119483499 |doi-access=}}

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