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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a ball is the solid figure bounded by a ''
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
''; it is also called a solid 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
but also for lower and higher dimensions, and for
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
s in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
is the same thing as a disk, the area bounded by a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
. In Euclidean 3-space, a ball is taken to be the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
bounded by a 2-dimensional sphere. In a
one-dimensional space In physics and mathematics, a sequence of ''n'' numbers can specify a location in ''n''-dimensional space. When , the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where t ...
, a ball is a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
. In other contexts, such as in
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
the closed $n$-dimensional ball is often denoted as $B^n$ or $D^n$ while the open $n$-dimensional ball is $\operatorname B^n$ or $\operatorname D^n$.

# In Euclidean space

In Euclidean -space, an (open) -ball of radius and center is the set of all points of distance less than from . A closed -ball of radius is the set of all points of distance less than or equal to away from . In Euclidean -space, every ball is bounded by a
hypersphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, call ...
. The ball is a bounded interval when , is a disk bounded by a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
when , and is bounded by a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
when .

## Volume

The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:Equation 5.19.4, ''NIST Digital Library of Mathematical Functions.'

Release 1.0.6 of 2013-05-06.
$V_n(R) = \fracR^n,$ where  is
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
's
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
(which can be thought of as an extension of the
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \ ...
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: In the formula for odd-dimensional volumes, the
double factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the ...
is defined for odd integers as .

# In general metric spaces

Let be a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, namely a set with a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
(distance function) . The open (metric) ball of radius centered at a point in , usually denoted by or , is defined by $B_r(p) = \,$ The closed (metric) ball, which may be denoted by or , is defined by Note in particular that a ball (open or closed) always includes itself, since the definition requires . A
unit ball Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
(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 In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
can serve as a base, giving this space a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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 . Let denote the closure of the open ball in this topology. While it is always the case that , it is always the case that . For example, in a metric space with the
discrete metric Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
, one has and , for any .

# In normed vector spaces

Any normed vector space with norm $\, \cdot\,$ is also a metric space with the metric $d \left(x,y\right)= \, x - y\, .$ In such spaces, an arbitrary ball $B_r\left(y\right)$ of points $x$ around a point $y$ with a distance of less than $r$ may be viewed as a scaled (by $r$) and translated (by $y$) copy of a ''unit ball'' $B_1\left(0\right).$ Such "centered" balls with $y=0$ are denoted with $B\left(r\right).$ The Euclidean balls discussed earlier are an example of balls in a normed vector space.

## -norm

In a Cartesian space with the -norm , that is $\left\, x \right\, _p = \left( , x_1, ^p + , x_2, ^p + \dots + , x_n, ^p \right) ^,$ an open ball around the origin with radius $r$ is given by the set $B(r) = \left\.$ For , in a 2-dimensional plane $\R^2$, "balls" according to the -norm (often called the ''
taxicab A taxi, also known as a taxicab or simply a cab, is a type of vehicle for hire with a driver, used by a single passenger or small group of passengers, often for a non-shared ride. A taxicab conveys passengers between locations of their choi ...
'' or ''Manhattan'' metric) are bounded by squares with their ''diagonals'' parallel to the coordinate axes; those according to the -norm, also called the
Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshe ...
metric, have squares with their ''sides'' parallel to the coordinate axes as their boundaries. The -norm, known as the Euclidean metric, generates the well known disks within circles, and for other values of , the corresponding balls are areas bounded by
Lamé curve A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape. In the C ...
s (hypoellipses or hyperellipses). For , the - balls are within octahedra with axes-aligned ''body diagonals'', the -balls are within cubes with axes-aligned ''edges'', and the boundaries of balls for with are superellipsoids. Obviously, generates the inner of usual spheres.

## General convex norm

More generally, given any
centrally symmetric In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is inv ...
, bounded,
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
, and
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
subset of , one can define a norm on where the balls are all translated and uniformly scaled copies of . 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 .

# In topological spaces

One may talk about balls in any
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, not necessarily induced by a metric. An (open or closed) -dimensional topological ball of is any subset of which is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to an (open or closed) Euclidean -ball. Topological -balls are important in
combinatorial topology In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such ...
, as the building blocks of cell complexes. Any open topological -ball is homeomorphic to the Cartesian space and to the open unit -cube (hypercube) . Any closed topological -ball is homeomorphic to the closed -cube . An -ball is homeomorphic to an -ball if and only if . The homeomorphisms between an open -ball and can be classified in two classes, that can be identified with the two possible topological orientations of . A topological -ball need not be smooth; if it is smooth, it need not be diffeomorphic to a Euclidean -ball.

# Regions

A number of special regions can be defined for a ball: *'' cap'', bounded by one plane *''
sector Sector may refer to: Places * Sector, West Virginia, U.S. Geometry * Circular sector, the portion of a disc enclosed by two radii and a circular arc * Hyperbolic sector, a region enclosed by two radii and a hyperbolic arc * Spherical sector, a po ...
'', bounded by a conical boundary with apex at the center of the sphere *'' segment'', bounded by a pair of parallel planes *'' shell'', bounded by two concentric spheres of differing radii *'' wedge'', bounded by two planes passing through a sphere center and the surface of the sphere

*
Ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
– ordinary meaning *
Disk (mathematics) In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usu ...
* Formal ball, an extension to negative radii *
Neighbourhood (mathematics) In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a po ...
*
Sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
, a similar geometric shape *
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
* -sphere, or hypersphere *
Alexander horned sphere The Alexander horned sphere is a pathological object in topology discovered by . Construction The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting ...
*
Manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
* Volume of an -ball *
Octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
– a 3-ball in the metric.

# References

* * * {{DEFAULTSORT:Ball (Mathematics) Balls Metric geometry Spheres Topology