Ball
Ball
Ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be either a closed ball (including the boundary points) 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 the Euclidean plane, for example, is the same thing as a disk, the area bounded by a circle. In mathematical contexts where ball is used, a sphere is usually assumed to be the boundary points only (namely, a spherical surface in three-dimensional space). In other contexts, such as in Euclidean geometry and informal use, sphere sometimes means ball.
A ball is the inside of a sphere
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.
Euclidean norm
In particular, if V is n-dimensional Euclidean space with the ordinary (Euclidean) metric, every ball is the interior of an hypersphere (a hyperball). That is a bounded interval when n=1, the interior of a circle (a disk) when n=2, and the interior of a sphere when n=3.
Ball (mathematics)
P-norm
In Cartesian space with the p-norm Lp, an open ball is the set
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 the interiors of Lam 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.
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 Rn and to the open unit n-cube . 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 ofB. A topological n-ball need not be smooth; if it is smooth, it need not be diffeomorphic to a Euclidean n-ball.
References
D. J. Smith and M. K. Vamanamurthy, "How small is a unit ball?", Mathematics Magazine, 62 (1989) 101107. "Robin conditions on the Euclidean ball", J. S. Dowker (http://www.citebase.org/fulltext?format=application/ pdf&identifier=oai:arXiv.org:hep-th/9506042) "Isometries of the space of convex bodies contained in a Euclidean ball", Peter M. Gruber (http://www. springerlink.com/content/0v74h15104232532/)
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