Ball (mathematics)

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 ( $n - 1$ )-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 spherical shell. 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.

Balls in Euclidean space

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 volume

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

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:

{\begin{aligned}V_{2k}(R)&={\frac {\pi ^{k}}{k!}}R^{2k}\,,\\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 $(2 k + 1)!!$ is defined for odd integers $2 k + 1$ as $(2 k + 1)!! = 1 \cdot 3 \cdot 5 \cdot \dots \cdot (2 k - 1) \cdot (2 k + 1)$ .

Balls in general metric spaces

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

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

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

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 $r > 0$ .

The closure of the open ball $B r (p)$ is usually denoted $B r (p)$ . While it is always the case that $B r (p) \subseteq B r (p) \subseteq B r [p]$ , 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 $B 1 (p) = {p}$ and $B 1 [p] = X$ , for any $p \in X$ .

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 are a basis for a topological space, whose open sets are all possible unions of open balls. This space is called the topology induced by the metric $d$ .

Balls in normed vector spaces

Any normed vector space $V$ with norm $| \cdot |$ is also a metric space, with the metric $d (x, y) = | x - y |$ . In such spaces, every ball $B r (p)$ is a copy of the unit ball $B 1 (0)$ , scaled by $r$ and translated by $p$ .

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

$p$ -norm

In Cartesian space $ℝ n$ with the $p$ -norm $L p$ , an open ball, is the set

B(r)=\left\{x\in \mathbb {R} ^{n}\,:\,\sum _{i=1}^{n}\left|x_{i}\right|^{p}<r^{p}\right\}.

For $n = 2$ , in particular, the balls of $L 1$ (often called the taxicab or Manhattan metric) are squares with the diagonals parallel to the coordinate axes; those of $L \infty$ (the Chebyshev metric) are squares with the sides parallel to the coordinate axes. For other values of $p$ , the balls are bounded by Lamé curves (hypoellipses or hyperellipses).

For $n = 3$ , the balls of $L 1$ are octahedra with axis-aligned body diagonals, those of $L \infty$ are cubes with axis-aligned edges, and those of $L p$ with $p > 2$ are superellipsoids.

General convex norm

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

Topological balls

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 $ℝ n$ and to the open unit $n$ -cube (hypercube) $(0, 1) n \subseteq ℝ n$ . 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 $ℝ n$ 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.

References

^ Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/,^{[permanent dead link‍]} Release 1.0.6 of 2013-05-06.

Smith, D. J.; Vamanamurthy, M. K. (1989). "How small is a unit ball?". Mathematics Magazine. 62 (2): 101–107. JSTOR 2690391.
Dowker, J. S. (1996). "Robin Conditions on the Euclidean ball". Classical and Quantum Gravity. 13 (4): 585–610. doi:10.1088/0264-9381/13/4/003.
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.

[1] Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/,^{[permanent dead link‍]} Release 1.0.6 of 2013-05-06.

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