[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/ball-mathematics-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki19\/ball-mathematics-wikipedia\/","headline":"Ball (mathematics) – Wikipedia","name":"Ball (mathematics) – Wikipedia","description":"Volume space bounded by a sphere In mathematics, a ball is the solid figure bounded by a sphere; it is","datePublished":"2017-02-14","dateModified":"2017-02-14","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki19\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/c\/cf\/Blue_ball.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/c\/cf\/Blue_ball.png","height":"94","width":"95"},"url":"https:\/\/wiki.edu.vn\/en\/wiki19\/ball-mathematics-wikipedia\/","about":["Wiki"],"wordCount":6109,"articleBody":"Volume space bounded by a sphere  In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere.[1] 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 spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n\u22121)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 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 n{displaystyle n}-dimensional ball is often denoted as Bn{displaystyle B^{n}} or Dn{displaystyle D^{n}} while the open n{displaystyle n}-dimensional ball is Int\u2061Bn{displaystyle operatorname {Int} B^{n}} or Int\u2061Dn{displaystyle operatorname {Int} D^{n}}.Table of ContentsIn Euclidean space[edit]Volume[edit]In general metric spaces[edit]In normed vector spaces[edit]p-norm[edit]General convex norm[edit]In topological spaces[edit]Regions[edit]See also[edit]References[edit]In Euclidean space[edit]In Euclidean n-space, an (open) n-ball of radius r and center x is the set of all points of distance less than r from x. A closed n-ball of radius r is the set of all points of distance less than or equal to r away from x.In Euclidean n-space, every ball is bounded by a hypersphere.  The ball is a bounded interval when n = 1, is a disk bounded by a circle when n = 2, and is bounded by a sphere when n = 3.Volume[edit]The n-dimensional volume of a Euclidean ball of radius r in n-dimensional Euclidean space is:[2]Vn(r)=\u03c0n2\u0393(n2+1)rn,{displaystyle V_{n}(r)={frac {pi ^{frac {n}{2}}}{Gamma left({frac {n}{2}}+1right)}}r^{n},}where\u00a0\u0393 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:V2k(r)=\u03c0kk!r2k,V2k+1(r)=2k+1\u03c0k(2k+1)!!r2k+1=2(k!)(4\u03c0)k(2k+1)!r2k+1.{displaystyle {begin{aligned}V_{2k}(r)&={frac {pi ^{k}}{k!}}r^{2k},,\\[2pt]V_{2k+1}(r)&={frac {2^{k+1}pi ^{k}}{(2k+1)!!}}r^{2k+1}={frac {2(k!)(4pi )^{k}}{(2k+1)!}}r^{2k+1},.end{aligned}}}In the formula for odd-dimensional volumes, the double factorial (2k + 1)!! is defined for odd integers 2k + 1 as (2k + 1)!! = 1 \u22c5 3 \u22c5 5 \u22c5 \u22ef \u22c5 (2k \u2212 1) \u22c5 (2k + 1).In general metric spaces[edit]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 Br(p) or B(p; r), is defined byBr(p)={x\u2208M\u2223d(x,p)d(x,p)\u2264r}.{displaystyle B_{r}[p]={xin Mmid d(x,p)leq r}.}Note in particular that a ball (open or closed) always includes p itself, since the definition requires r > 0.A unit ball (open or closed) 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 can serve as a base, giving this space a topology, the open sets of which are all possible unions of open balls. This topology on a metric space is called the topology induced by the metric d.Let Br(p) denote the closure of the open ball Br(p) in this topology. While it is always the case that Br(p) \u2286 Br(p) \u2286 Br[p], it is not always the case that Br(p) = Br[p]. For example, in a metric space X with the discrete metric, one has B1(p) = {p} and B1[p] = X, for any p \u2208 X.In normed vector spaces[edit]Any normed vector space V with norm \u2016\u22c5\u2016{displaystyle |cdot |} is also a metric space with the metric d(x,y)=\u2016x\u2212y\u2016.{displaystyle d(x,y)=|x-y|.}  In such spaces, an arbitrary ball Br(y){displaystyle 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 a unit ball B1(0).{displaystyle B_{1}(0).} Such “centered” balls with y=0{displaystyle y=0} are denoted with B(r).{displaystyle B(r).}The Euclidean balls discussed earlier are an example of balls in a normed vector space.p-norm[edit]In a Cartesian space Rn with the p-norm Lp, that is\u2016x\u2016p=(|x1|p+|x2|p+\u22ef+|xn|p)1\/p,{displaystyle left|xright|_{p}=left(|x_{1}|^{p}+|x_{2}|^{p}+dots +|x_{n}|^{p}right)^{1\/p},}an open ball around the origin with radius r{displaystyle r} is given by the setB(r)={x\u2208Rn:\u2016x\u2016p=(|x1|p+|x2|p+\u22ef+|xn|p)1\/p"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/ball-mathematics-wikipedia\/#breadcrumbitem","name":"Ball (mathematics) – Wikipedia"}}]}]