Ball (mathematics)

In Euclidean space, a ball is the volume bounded by a sphere

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 Template:Mvar dimensions is called an Template:Mvar-ball and is bounded by an [[N-sphere|(Template:Math)-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 general practice, the volume of a ball is calculated as:

$V={\frac {4}{3}}\pi r^{3}\approx 0.523\cdot d^{3}$

where Template:Mvar is the radius and Template:Mvar is the diameter of the ball.

In Euclidean space

In Euclidean Template:Mvar-space, an (open) Template:Mvar-ball of radius Template:Mvar and center Template:Mvar is the set of all points of distance less than Template:Mvar from Template:Mvar. A closed Template:Mvar-ball of radius Template:Mvar is the set of all points of distance less than or equal to Template:Mvar away from Template:Mvar.

In Euclidean Template:Mvar-space, every ball is bounded by a hypersphere. The ball is a bounded interval when Template:Math, is a disk bounded by a circle when Template:Math, and is bounded by a sphere when Template:Math.

Volume

Template:Main article The Template:Mvar-dimensional volume of a Euclidean ball of radius Template:Mvar in Template:Mvar-dimensional Euclidean space is:

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

where Template:Math 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 Template:Math is defined for odd integers Template:Math as Template:Math.

In general metric spaces

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

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

The closed (metric) ball, which may be denoted by Template:Math or Template:Math, 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 Template:Mvar itself, since the definition requires Template:Math.

The closure of the open ball Template:Math is usually denoted Template:Math. While it is always the case that Template:Math, it is Template:Em always the case that Template:Math. For example, in a metric space Template:Mvar with the discrete metric, one has Template:Math and Template:Math, for any Template:Math.

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 Template:Mvar.

In normed vector spaces

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

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

Template:Mvar-norm

In a Cartesian space Template:Math with the [[p-norm|Template:Mvar-norm]] Template:Mvar, that is

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

an open ball around the origin with radius $r$ is given by the set

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

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

For Template:Math, the Template:Math- balls are within octahedra with axes-aligned body diagonals, the Template:Math-balls are within cubes with axes-aligned edges, and the boundaries of balls for Template:Mvar with Template:Math are superellipsoids. Obviously, Template:Math generates the inner of usual spheres.

General convex norm

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

In topological spaces

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

Any open topological Template:Mvar-ball is homeomorphic to the Cartesian space Template:Math and to the open [[hypercube|unit Template:Mvar-cube]] (hypercube) Template:Math. Any closed topological Template:Mvar-ball is homeomorphic to the closed Template:Mvar-cube Template:Math.

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

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

See also

• Ball – ordinary meaning
• Disk (mathematics)
• Formal ball, an extension to negative radii
• Neighbourhood (mathematics)
• 3-sphere
• [[n-sphere|Template:Mvar-sphere]], or hypersphere
• Alexander horned sphere
• Manifold
• [[Volume of an n-ball|Volume of an Template:Mvar-ball]]
• Octahedron – a 3-ball in the Template:Math metric.
• Spherical shell

References

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