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Ball (mathematics)

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Volume space bounded by a sphere

Not to be confused withSphere.

InEuclidean space, aball is the volume bounded by a sphere

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Inmathematics, aball is thesolid figure bounded by asphere; it is also called asolid sphere.^[1] It may be aclosed ball (including theboundary points that constitute the sphere) or anopen ball (excluding them).

These concepts are defined not only in three-dimensionalEuclidean space but also for lower and higher dimensions, and formetric spaces in general. Aball inn dimensions is called ahyperball orn-ball and is bounded by ahypersphere or(n−1)-sphere. Thus, for example, a ball in theEuclidean plane is the same thing as adisk, theplanar region bounded by acircle. InEuclidean 3-space, a ball is taken to be theregion of space bounded by a2-dimensional sphere. In aone-dimensional space, a ball is aline segment.

In other contexts, such as inEuclidean geometry and informal use,sphere is sometimes used to meanball. In the field oftopology the closed $n {\displaystyle n}$ -dimensional ball is often denoted as $B^{n}$ or $D^{n}$ while the open $n {\displaystyle n}$ -dimensional ball is $\operatorname {int} B^{n}$ or $\operatorname {int} D^{n}$ .

In Euclidean space

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In Euclideann-space, an (open)n-ball of radiusr and centerx is the set of all points of distance less thanr fromx. A closedn-ball of radiusr is the set of all points of distance less than or equal tor away fromx.

In Euclideann-space, every ball is bounded by ahypersphere. The ball is a boundedinterval whenn = 1, is adisk bounded by acircle whenn = 2, and is bounded by asphere whenn = 3.

Volume

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Main article:Volume of an n-ball

Then-dimensional volume of a Euclidean ball of radiusr inn-dimensional Euclidean space is given by^[2] $V_{n}(r)={\frac {\pi ^{\frac {n}{2}}}{\Gamma {\left({\frac {n}{2}}+1\right)}}}r^{n},$ where Γ isLeonhard Euler'sgamma function (which can be thought of as an extension of thefactorial function to fractional arguments). Using explicit formulas forparticular 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: ${\begin{aligned}V_{2k}(r)&={\frac {\pi ^{k}}{k!}}r^{2k}\,,\\[2pt]V_{2k+1}(r)&={\frac {2^{k+1}\pi ^{k}}{\left(2k+1\right)!!}}r^{2k+1}={\frac {2\left(k!\right)\left(4\pi \right)^{k}}{\left(2k+1\right)!}}r^{2k+1}\,.\end{aligned}}$

In the formula for odd-dimensional volumes, thedouble factorial(2k + 1)!! is defined for odd integers2k + 1 as(2k + 1)!! = 1 ⋅ 3 ⋅ 5 ⋅ ⋯ ⋅ (2k − 1) ⋅ (2k + 1).

The surface area of an n-ball (an (n-1)-sphere) is: $A_{n}(r)={\frac {dV_{n}}{dr}}={\frac {2\pi ^{\frac {n}{2}}}{\Gamma {\left({\frac {n}{2}}\right)}}}r^{n-1},$

In general metric spaces

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Balls of radii 1 (red) and 2 (blue) intaxicab geometry areregular octahedrons

Let(M,d) be ametric space, namely a setM with ametric (distance function)d, and let⁠ $r {\displaystyle r}$ ⁠ be a positive real number. The open (metric)ball of radiusr centered at a pointp inM, usually denoted byB_r(p) orB(p;r), is defined the same way as a Euclidean ball, as the set of points inM of distance less thanr away fromp, $B_{r}(p)=\{x\in M\mid d(x,p)<r\}.$

Theclosed (metric) ball, sometimes denotedB_r[p] orB[p;r], is likewise defined as the set of points of distance less than or equal tor away fromp, $B_{r}[p]=\{x\in M\mid d(x,p)\leq r\}.$

In particular, a ball (open or closed) always includesp itself, since the definition requiresr > 0. Aunit 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 inreal coordinate space under theChebyshev distance is ahypercube, and a ball under thetaxicab distance is across-polytope. A closed ball also need not becompact. For example, a closed ball in any infinite-dimensionalnormed vector space is never compact. However, a ball in a normed vector space will always beconvex as a consequence of the triangle inequality.

A subset of a metric space isbounded if it is contained in some ball. A set istotally bounded if, given any positive radius, it is covered by finitely many balls of that radius.

The open balls of ametric space can serve as abase, giving this space atopology, the open sets of which are all possibleunions of open balls. This topology on a metric space is called thetopology induced by the metricd.

Let ${\overline {B_{r}(p)}}$ denote theclosure of the open ball $B_{r}(p)$ in this topology. While it is always the case that $B_{r}(p)\subseteq {\overline {B_{r}(p)}}\subseteq B_{r}[p],$ it isnot always the case that ${\overline {B_{r}(p)}}=B_{r}[p].$ For example, in a metric space $X {\displaystyle X}$ with thediscrete metric, one has ${\overline {B_{1}(p)}}=\{p\}$ but $B_{1}[p]=X$ for any $p\in X.$

In normed vector spaces

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Anynormed vector spaceV with norm $\|\cdot \|$ is also a metric space with the metric $d(x,y)=\|x-y\|.$ In such spaces, an arbitrary ball $B_{r}(y)$ of points $x {\displaystyle x}$ around a point $y {\displaystyle y}$ with a distance of less than $r {\displaystyle r}$ may be viewed as a scaled (by $r {\displaystyle r}$ ) and translated (by $y {\displaystyle y}$ ) copy of aunit ball $B_{1}(0).$ Such "centered" balls with $y=0$ are denoted with $B(r).$

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

p-norm

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In aCartesian spaceRⁿ with thep-normL^p, that is one chooses some $p\geq 1$ and defines $\left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p},$ Then an open ball around the origin with radius $r {\displaystyle r}$ is given by the set $B(r)=\left\{x\in \mathbb {R} ^{n}\,:\left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p}<r\right\}.$ Forn = 2, in a 2-dimensional plane $\mathbb {R} ^{2}$ , "balls" according to theL¹-norm (often called thetaxicab orManhattan metric) are bounded by squares with theirdiagonals parallel to the coordinate axes; those according to theL^∞-norm, also called theChebyshev metric, have squares with theirsides parallel to the coordinate axes as their boundaries. TheL²-norm, known as the Euclidean metric, generates the well known disks within circles, and for other values ofp, the corresponding balls are areas bounded byLamé curves (hypoellipses or hyperellipses).

Forn = 3, theL¹-balls are within octahedra with axes-alignedbody diagonals, theL^∞-balls are within cubes with axes-alignededges, and the boundaries of balls forL^p withp > 2 aresuperellipsoids.p = 2 generates the inner of usual spheres.

Often can also consider the case of $p=\infty$ in which case we define $\lVert x\rVert _{\infty }=\max\{\left|x_{1}\right|,\dots ,\left|x_{n}\right|\}$

General convex norm

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More generally, given anycentrally symmetric,bounded,open, andconvex subsetX ofRⁿ, one can define anorm onRⁿ 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.

In topological spaces

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One may talk about balls in anytopological spaceX, not necessarily induced by a metric. An (open or closed)n-dimensionaltopological ball ofX is any subset ofX which ishomeomorphic to an (open or closed) Euclideann-ball. Topologicaln-balls are important incombinatorial topology, as the building blocks ofcell complexes.

Any open topologicaln-ball is homeomorphic to the Cartesian spaceRⁿ and to the openunitn-cube (hypercube)(0, 1)ⁿ ⊆Rⁿ. Any closed topologicaln-ball is homeomorphic to the closedn-cube[0, 1]ⁿ.

Ann-ball is homeomorphic to anm-ball if and only ifn =m. The homeomorphisms between an openn-ballB andRⁿ can be classified in two classes, that can be identified with the two possibletopological orientations of B.

A topologicaln-ball need not besmooth; if it is smooth, it need not bediffeomorphic to a Euclideann-ball.

Regions

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References

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^Sūgakkai, Nihon (1993).Encyclopedic Dictionary of Mathematics.MIT Press.ISBN 9780262590204.
^Equation 5.19.4,NIST Digital Library of Mathematical Functions.[1] Release 1.0.6 of 2013-05-06.

Smith, D. J.; Vamanamurthy, M. K. (1989). "How small is a unit ball?".Mathematics Magazine.62 (2):101–107.doi:10.1080/0025570x.1989.11977419.JSTOR 2690391.
Dowker, J. S. (1996). "Robin Conditions on the Euclidean ball".Classical and Quantum Gravity.13 (4):585–610.arXiv:hep-th/9506042.Bibcode:1996CQGra..13..585D.doi:10.1088/0264-9381/13/4/003.S2CID 119438515.
Gruber, Peter M. (1982). "Isometries of the space of convex bodies contained in a Euclidean ball".Israel Journal of Mathematics.42 (4):277–283.doi:10.1007/BF02761407.S2CID 119483499.

Metric spaces (Category)

Basic concepts

Main results

Maps

Types of
metric spaces

Sets

Examples

Manifolds	Euclidean distance Riemannian
Functional analysis andMeasure theory	Chebyshev distance Inner product space Lévy metric Lévy–Prokhorov metric Metrizable topological vector space Normed space Taxicab geometry Wasserstein metric
General topology	Discrete space Intrinsic metric Laakso space Product metric