open ball

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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 In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of various solid fig ...
bounded by a ''
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
''; 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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
but also for lower and higher dimensions, and for
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
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 geometry, geometric setting in which two real number, real quantities are required to determine the position (geometry), position of each point (mathematics), ...
is the same thing as a disk, the area bounded by a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. Equivalently, it is the curve traced out by a point that moves in ...
. In
Euclidean 3-space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
, 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, a ball is a
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...
. In other contexts, such as in
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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 point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. Equivalently, it is the curve traced out by a point that moves in ...
when , and is bounded by a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
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 in ma ...
'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 except th ...
(which can be thought of as an extension of the
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), 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 ...
function to fractional arguments). Using explicit formulas for
particular values of the gamma function The 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 compl ...
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 (mathematics), parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \c ...
is defined for odd integers as .

In general metric spaces

Let be a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
, 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 mathema ...
(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 (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 (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
can serve as a base, giving this space a
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, 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, one has and , for any .

In normed vector spaces

Any
normed vector space In mathematics, a normed vector space or normed space is a vector space over the Real number, real or Complex number, complex numbers, on which a Norm (mathematics), norm is defined. A norm is the formalization and the generalization to real ve ...
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 Driving, driver, used by a single passenger or small group of passengers, often for a non-shared ride. A taxicab conveys passengers between locations of thei ...
'' or ''Manhattan'' metric) are bounded by squares with their ''diagonals'' parallel to the coordinate axes; those according to the -norm, also called the Chebyshev 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 curves (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, bounded,
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
, 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 polytope, ...
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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, not necessarily induced by a metric. An (open or closed) -dimensional topological ball of is any subset of which is
homeomorphic In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
to an (open or closed) Euclidean -ball. Topological -balls are important in combinatorial topology, 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 In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
to a Euclidean -ball.

Regions

A number of special regions can be defined for a ball: *'' cap'', bounded by one plane *'' sector'', bounded by a conical boundary with apex at the center of the sphere *'' segment'', bounded by a pair of parallel planes *''
shell Shell may refer to: Architecture and design * Shell (structure), a thin structure ** Concrete shell, a thin shell of concrete, usually with no interior columns or exterior buttresses ** Thin-shell structure Science Biology * Seashell, a hard out ...
'', bounded by two concentric spheres of differing radii *''
wedge A wedge is a triangle, triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions b ...
'', bounded by two planes passing through a sphere center and the surface of the sphere

*
Ball A ball is a round object (usually sphere, 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 ...
– ordinary meaning *
Disk (mathematics) In geometry, a disk (also Spelling of disc, spelled disc). is the region in a plane (geometry), 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. Fo ...
* 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 (topology), interior. Intuitively speaking, a n ...
*
Sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, 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 (mathematics), ba ...
* -sphere, or hypersphere * Alexander horned sphere *
Manifold 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 i ...
* Volume of an -ball *
Octahedron In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field ...
– a 3-ball in the metric.

References

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