In mathematics, a ball is the space bounded by a 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 spaces in general. A ball or hyperball in n dimensions is called an n-ball and is bounded by an n. 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 general practice, the volume of a ball is calculated as:

where r is the radius and d is the diameter of the ball.

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.

The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:^[1]

where Γ 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 (2k + 1)!! is defined for odd integers 2k + 1 as (2k + 1)!! = 1 · 3 · 5 · … · (2k − 1) · (2k + 1).

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 by

The closed (metric) ball, which may be denoted by Br[p] or B[p; r], is defined by

Note in particular that a ball (open or closed) always includes p itself, since the definition requires r > 0.

The closure of the open ball Br(p) is usually denoted Br(p). While it is always the case that Br(p) ⊆ Br(p) ⊆ 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 ∈ X.

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.

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

In a Cartesian space ℝn with the p-norm Lp, that is

For n = 3, the L1- balls are within octahedra with axes-aligned body diagonals, the L∞-balls are within cubes with axes-aligned edges, and the boundaries of balls for Lp with p > 2 are superellipsoids. Obviously, p = 2 generates the inner of usual spheres.

More generally, given any centrally symmetric, bounded, open, and convex subset X of ℝn, one can define a norm on ℝn 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.

One may talk about balls in any topological space X, not necessarily induced by a metric. An (open or closed) n-dimensional topological ball of X is any subset of X which is homeomorphic to an (open or closed) Euclidean n-ball. Topological n-balls are important in combinatorial topology, as the building blocks of cell complexes.

Any open topological n-ball is homeomorphic to the Cartesian space ℝn and to the open unit n-cube (hypercube) (0, 1)n ⊆ ℝn. Any closed topological n-ball is homeomorphic to the closed n-cube [0, 1]n.

An n-ball is homeomorphic to an m-ball if and only if n = m. The homeomorphisms between an open n-ball B and ℝn can be classified in two classes, that can be identified with the two possible topological orientations of B.

A topological n-ball need not be smooth; if it is smooth, it need not be diffeomorphic to a Euclidean n-ball.