Ball (mathematics)

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

In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere.<ref>Template:Cite book</ref> 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 in Template:Mvar dimensions is called a hyperball or Template:Mvar-ball and is bounded by a hypersphere or [[N-sphere|(Template:Math)-sphere]]. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be the region of space 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 the field of topology the closed <math>n</math>-dimensional ball is often denoted as <math>B^n</math> or <math>D^n</math> while the open <math>n</math>-dimensional ball is <math>\operatorname{int} B^n</math> or <math>\operatorname{int} D^n</math>.

In Euclidean spaceEdit

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.

VolumeEdit

Template:Main article The Template:Mvar-dimensional volume of a Euclidean ball of radius Template:Math in Template:Math-dimensional Euclidean space is given by <ref>Equation 5.19.4, NIST Digital Library of Mathematical Functions. [1] Release 1.0.6 of 2013-05-06.</ref> <math display="block">V_n(r) = \frac{\pi^\frac{n}{2}}{\Gamma{\left(\frac{n}{2} + 1\right)}} r^n,</math> 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: <math display="block">\begin{align}

   V_{2k}(r) &= \frac{\pi^k}{k!} r^{2k}\,,\\[2pt]
 V_{2k+1}(r) &= \frac{2^{k+1}\pi^k}{\left(2k+1\right)!!} r^{2k+1} = \frac{2\left(k!\right) \left(4\pi\right)^k}{\left(2k+1\right)!}r^{2k+1}\,.

\end{align}</math>

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 spacesEdit

Let Template:Math be a metric space, namely a set Template:Mvar with a metric (distance function) Template:Mvar, and let Template:Tmath be a positive real number. The open (metric) ball of radius Template:Mvar centered at a point Template:Mvar in Template:Mvar, usually denoted by Template:Math or Template:Math, is defined the same way as a Euclidean ball, as the set of points in Template:Mvar of distance less than Template:Mvar away from Template:Mvar, <math display="block">B_r(p) = \{ x \in M \mid d(x,p) < r \}.</math>

The closed (metric) ball, sometimes denoted Template:Math or Template:Math, is likewise defined as the set of points of distance less than or equal to Template:Mvar away from Template:Mvar, <math display="block">B_r[p] = \{ x \in M \mid d(x,p) \le r \}.</math>

In particular, a ball (open or closed) always includes Template:Mvar itself, since the definition requires Template:Math. A unit ball (open or closed) is a ball of radius 1.

A ball in a general metric space need not be round. For example, a ball in real coordinate space under the Chebyshev distance is a hypercube, and a ball under the taxicab distance is a cross-polytope. A closed ball also need not be compact. For example, a closed ball in any infinite-dimensional normed vector space is never compact. However, a ball in a vector space will always be convex as a consequence of the triangle inequality.

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.

Let <math>\overline{B_r(p)}</math> denote the closure of the open ball <math>B_r(p)</math> in this topology. While it is always the case that <math>B_r(p) \subseteq \overline{B_r(p)} \subseteq B_r[p],</math> it is Template:Em always the case that <math>\overline{B_r(p)} = B_r[p].</math> For example, in a metric space <math>X</math> with the discrete metric, one has <math>\overline{B_1(p)} = \{p\}</math> but <math>B_1[p] = X</math> for any <math>p \in X.</math>

In normed vector spacesEdit

Any normed vector space Template:Mvar with norm <math>\|\cdot\|</math> is also a metric space with the metric <math>d (x,y)= \|x - y\|.</math> In such spaces, an arbitrary ball <math>B_r(y)</math> of points <math>x</math> around a point <math>y</math> with a distance of less than <math>r</math> may be viewed as a scaled (by <math>r</math>) and translated (by <math>y</math>) copy of a unit ball <math>B_1(0).</math> Such "centered" balls with <math>y=0</math> are denoted with <math>B(r).</math>

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

Template:Mvar-normEdit

In a Cartesian space Template:Math with the [[p-norm|Template:Mvar-norm]] Template:Mvar, that is one chooses some <math>p \geq 1</math> and defines<math display="block">\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dots + |x_n|^p \right) ^{1/p},</math>Then an open ball around the origin with radius <math>r</math> is given by the set <math display="block"> B(r) = \left\{ x \in \R^n \,:\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dots + |x_n|^p \right) ^{1/p} < r \right\}.</math>For Template:Math, in a 2-dimensional plane <math>\R^2</math>, "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 disks 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. Template:Math generates the inner of usual spheres.

Often can also consider the case of <math>p = \infty</math> in which case we define <math display="block">\lVert x \rVert_\infty = \max\{\left|x_1\right|, \dots, \left|x_n\right|\}</math>

General convex normEdit

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 spacesEdit

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.

RegionsEdit

Template:See also 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

ReferencesEdit

Template:Reflist

Template:Metric spaces Template:Authority control