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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.
Main article: Volume of an n-ball
The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:
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:
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).
Balls in general metric spaces
The closed (metric) ball, which may be denoted by Bt[p] or B[p; r], is defined by
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 Br(p) is usually denoted . While it is always the case that , it is not always the case that . For example, in a metric space X with the discrete metric, one has and , for any .
A (open or closed) unit ball is a ball of radius 1.
Balls in normed vector spaces
Any normed vector space V with norm |·| is also a metric space, with the metric d(x,y) = | x − y |. In such spaces, every ball Br(p) is a copy of the unit ball B1(0), scaled by r and translated by p.
The Euclidean balls discussed earlier are an example of balls in a normed vector space.
For n=2, in particular, the balls of L1 (often called the taxicab or Manhattan metric) are squares with the diagonals parallel to the coordinate axes; those of L∞ (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 L1 are octahedra with axis-aligned body diagonals, those of L∞ are cubes with axis-aligned edges, and those of Lp with p > 2 are superellipsoids.
General convex norm
More generally, given any centrally symmetric, bounded, open, and convex subset X of Rn, one can define a norm on Rn 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 Rn.
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) . 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.
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- 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.