Шардың көлемі және гиперсфералар

Volume space bounded by a sphere

In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid 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 in n dimensions is called a hyperball or n ball and is bounded by a hypersphere or (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 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 dimensional ball is often denoted as or while the open dimensional ball is or .

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. In Euclidean n space, every ball is bounded by a hypersphere. The ball is a bounded interval when 1=n = 1, is a disk bounded by a circle when 1=n = 2, and is bounded by a sphere when 1=n = 3.

More generally, given any centrally symmetric, bounded, open, and convex subset X of 'R'^(n), one can define a norm on 'R'^(n) 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 'R'^(n).

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 'R'^(n) and to the open unit n cube (hypercube) (0, 1)^(n) ⊆ 'R'^(n). 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 1=n = m. The homeomorphisms between an open n ball B and 'R'^(n) can be classified in two classes, that can be identified with the two possible topological orientations of B. A topological n ball need not be smooth; if it is smooth, it need not be diffeomorphic to a Euclidean n ball.