, a ball is the volume space bounded by a sphere
; 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
but also for lower and higher dimensions, and for metric space
s in general. A ''ball'' or hyperball in dimensions is called an -ball and is bounded by an ()-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 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
. The ball is a bounded interval
when , is a disk
bounded by a circle
when , and is bounded by a sphere
The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:
[Equation 5.19.4, ''NIST Digital Library of Mathematical Functions.'' http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06.]
where 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:
In the formula for odd-dimensional volumes, the double factorial
is defined for odd integers as .
In general metric spaces
Let be a metric space
, namely a set with a metric
(distance function) . The open (metric) ball of radius centered at a point in , usually denoted by or , is defined by
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 .
of the open ball is usually denoted . 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 .
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 union
s of open balls. This topology on a metric space is called the topology induced by the metric .
In normed vector spaces
Any normed vector space
is also a metric space with the metric
In such spaces, an arbitrary ball
around a point
with a distance of less than
may be viewed as a scaled (by
) and translated (by
) copy of a ''unit ball''
Such "centered" balls with
are denoted with
The Euclidean balls discussed earlier are an example of balls in a normed vector space.
In a Cartesian space
with the -norm
, that is
an open ball around the origin with radius
is given by the set
For , in a 2-dimensional plane
, "balls" according to the -norm (often called the ''taxicab
'' 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 discs within circles, and for other values of , the corresponding balls are areas bounded by Lamé curve
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
, and convex
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
, not necessarily induced by a metric. An (open or closed) -dimensional topological ball of is any subset of which is homeomorphic
to an (open or closed) Euclidean -ball. Topological -balls are important in combinatorial topology
, as the building blocks of cell complex
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 orientation
s of .
A topological -ball need not be smooth
; if it is smooth, it need not be diffeomorphic
to a Euclidean -ball.
A number of special regions
can be defined for a ball:
, bounded by one plane
, bounded by a conical boundary with apex at the center of the sphere
, bounded by a pair of parallel planes
, bounded by two concentric spheres of differing radii
, bounded by two planes passing through a sphere center and the surface of the sphere
– ordinary meaning
, an extension to negative radii
, or hypersphere
*Alexander horned sphere
*Volume of an -ball
– a 3-ball in the metric.