Ball (mathematics)

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In Euclidean space, A ball is the inside of a sphere

In mathematics, a ball is the space inside a sphere. It may be a closed ball (including the boundary points of 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 in Template:Var dimensions is called an [[n-ball|Template:Var-ball]] and is bounded by an [[N-sphere|(Template:Var-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 boundary.

In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball.

Balls in Euclidean space

In Euclidean Template:Var-space, an (open) Template:Var-ball of radius Template:Var and center Template:Var is the set of all points of distance < Template:Var from Template:Var. A closed Template:Var-ball of radius Template:Var is the set of all points of distance ≤ Template:Var away from Template:Var.

In Euclidean Template:Var-space, every ball is the interior of a hypersphere (a hyperball), that is a bounded interval when Template:Var = 1, the interior of a circle (a disk) when Template:Var = 2, and the interior of a sphere when Template:Var = 3.

The volume

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The Template:Var-dimensional volume of a Euclidean ball of radius Template:Var in Template:Var-dimensional Euclidean space is:

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

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:

$V_{2k}(R)={\frac {\pi ^{k}}{k!}}R^{2k},$

$V_{2k+1}(R)={\frac {2^{k+1}\pi ^{k}}{(2k+1)!!}}R^{2k+1}={\frac {2(k!)(4\pi )^{k}}{(2k+1)!}}R^{2k+1}.$

In the formula for odd-dimensional volumes, the double factorial (2Template:Var + 1)!! is defined for odd integers 2Template:Var + 1 as (2k + 1)!! = 1 · 3 · 5 ··· (2Template:Var − 1) · (2Template:Var + 1).

Balls in general metric spaces

Let (Template:Var,Template:Var) be a metric space, namely a set Template:Var with a metric (distance function) Template:Var. The open (metric) ball of radius Template:Var > 0 centered at a point Template:Var in Template:Var, usually denoted by Template:Var_Template:Var(Template:Var) or Template:Var(Template:Var; Template:Var), is defined by

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

The closed (metric) ball, which may be denoted by Template:Var_Template:Var[[[:Template:Var]]] or Template:Var[[[:Template:Var]]; Template:Var], is defined by

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

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

The closure of the open ball Template:Var_Template:Var(Template:Var) is usually denoted ${\overline {B_{r}(p)}}$. While it is always the case that $B_{r}(p)\subseteq {\overline {B_{r}(p)}}$ and ${\overline {B_{r}(p)}}\subseteq B_{r}[p]$, it is Template:Em always the case that ${\overline {B_{r}(p)}}=B_{r}[p]$. For example, in a metric space Template:Var with the discrete metric, one has ${\overline {B_{1}(p)}}=\{p\}$ and $B_{1}[p]=X$, for any $p\in X$.

A (open or closed) unit ball 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 Template:Var.

Balls in normed vector spaces

Any normed vector space Template:Var with norm |·| is also a metric space, with the metric Template:Var(Template:Var, Template:Var) = |Template:Var − Template:Var|. In such spaces, every ball Template:Var_Template:Var(Template:Var) is a copy of the unit ball Template:Var₁(0), scaled by Template:Var and translated by Template:Var.

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

Template:Var-norm

In Cartesian space $\mathbb {R} ^{n}$ with the [[p-norm|Template:Var-norm]] Template:Var_Template:Var, 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 Template:Var=2, in particular, the balls of Template:Var₁ (often called the taxicab or Manhattan metric) are squares with the diagonals parallel to the coordinate axes; those of Template:Var_∞ (the Chebyshev metric) are squares with the sides parallel to the coordinate axes. For other values of Template:Var, the balls are the interiors of Lamé curves (hypoellipses or hyperellipses).

For Template:Var = 3, the balls of Template:Var₁ are octahedra with axis-aligned body diagonals, those of Template:Var_∞ are cubes with axis-aligned edges, and those of Template:Var_Template:Var with Template:Var > 2 are superellipsoids.

General convex norm

More generally, given any centrally symmetric, bounded, open, and convex subset Template:Var of Template:Var^Template:Var, one can define a norm on R^Template:Var where the balls are all translated and uniformly scaled copies of Template:Var. 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^Template:Var.

Topological balls

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

Any open topological Template:Var-ball is homeomorphic to the Cartesian space R^Template:Var and to the open [[hypercube|unit Template:Var-cube]] (hypercube) $(0,1)^{n}\subseteq \mathbb {R} ^{n}$. Any closed topological Template:Var-ball is homeomorphic to the closed Template:Var-cube [0, 1]^Template:Var.

An Template:Var-ball is homeomorphic to an Template:Var-ball if and only if Template:Var = Template:Var. The homeomorphisms between an open Template:Var-ball Template:Var and R^Template:Var can be classified in two classes, that can be identified with the two possible topological orientations of Template:Var.

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

See also

• Ball - ordinary meaning
• Disk (mathematics)
• Formal ball, an extension to negative radii
• Neighborhood (mathematics)
• 3-sphere
• n-sphere, or hypersphere
• Alexander horned sphere
• Manifold
• Volume of an n-ball

References

• D. J. Smith and M. K. Vamanamurthy, "How small is a unit ball?", Mathematics Magazine, 62 (1989) 101–107.
• "Robin conditions on the Euclidean ball", J. S. Dowker [1]
• "Isometries of the space of convex bodies contained in a Euclidean ball", Peter M. Gruber[2]