Ball (mathematics)

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

The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:[2]

$${\displaystyle V_{n}(R)={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n},}$$

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:

$${\displaystyle {\begin{aligned}V_{2k}(R)&={\frac {\pi ^{k}}{k!}}R^{2k}\,,\\[2pt]V_{2k+1}(R)&={\frac {2^{k+1}\pi ^{k}}{(2k+1)!!}}R^{2k+1}={\frac {2(k!)(4\pi )^{k}}{(2k+1)!}}R^{2k+1}\,.\end{aligned}}}$$

In the formula for odd-dimensional volumes, the double factorial (2k + 1)!! is defined for odd integers 2k + 1 as (2k + 1)!! = 1 ⋅ 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 B_r(p) or B(p; r), is defined by

$${\displaystyle B_{r}(p)=\{x\in M\mid d(x,p)<r\},}$$

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

$${\displaystyle B_{r}[p]=\{x\in M\mid d(x,p)\leq r\}.}$$

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 B_r(p) denote the closure of the open ball B_r(p) in this topology. While it is always the case that B_r(p) ⊆ B_r(p) ⊆ B_r[p], it is not always the case that B_r(p) = B_r[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.

Any normed vector space V with norm ${\displaystyle \|\cdot \|}$ is also a metric space with the metric ${\displaystyle d(x,y)=\|x-y\|.}$ In such spaces, an arbitrary ball ${\displaystyle B_{r}(y)}$ of points ${\displaystyle x}$ around a point ${\displaystyle y}$ with a distance of less than ${\displaystyle r}$ may be viewed as a scaled (by ${\displaystyle r}$) and translated (by ${\displaystyle y}$) copy of a unit ball ${\displaystyle B_{1}(0).}$ Such "centered" balls with ${\displaystyle y=0}$ are denoted with ${\displaystyle B(r).}$

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

p-norm

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

$${\displaystyle \left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p},}$$

an open ball around the origin with radius ${\displaystyle r}$ is given by the set

$${\displaystyle B(r)=\left\{x\in \mathbb {R} ^{n}\,:\left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p}<r\right\}.}$$

For n = 2, in a 2-dimensional plane ${\displaystyle \mathbb {R} ^{2}}$, "balls" according to the L₁-norm (often called the taxicab or Manhattan metric) are bounded by squares with their diagonals parallel to the coordinate axes; those according to the L_∞-norm, also called the Chebyshev metric, have squares with their sides parallel to the coordinate axes as their boundaries. The L₂-norm, known as the Euclidean metric, generates the well known disks within circles, and for other values of p, the corresponding balls are areas bounded by Lamé curves (hypoellipses or hyperellipses).

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 L_p with p > 2 are superellipsoids. Obviously, p = 2 generates the inner of usual spheres.

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.

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, bounded by two concentric spheres of differing radii
• wedge, bounded by two planes passing through a sphere center and the surface of the sphere

• Smith, D. J.; Vamanamurthy, M. K. (1989). "How small is a unit ball?". Mathematics Magazine. 62 (2): 101–107. doi:10.1080/0025570x.1989.11977419. JSTOR 2690391.
• Dowker, J. S. (1996). "Robin Conditions on the Euclidean ball". Classical and Quantum Gravity. 13 (4): 585–610. arXiv:hep-th/9506042. Bibcode:1996CQGra..13..585D. doi:10.1088/0264-9381/13/4/003. S2CID 119438515.
• Gruber, Peter M. (1982). "Isometries of the space of convex bodies contained in a Euclidean ball". Israel Journal of Mathematics. 42 (4): 277–283. doi:10.1007/BF02761407.