Ball (mathematics)

From Wikipedia, the free encyclopedia

In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general.

In higher mathematics, a careful distinction is made between the surface of a sphere (referred to as a “sphere”), and the inside of a sphere (referred to as a “ball”). Thus, a sphere in three dimensions is considered to be a two-dimensional spherical surface embedded in three-dimensional Euclidean space, while a ball is a solid figure bounded by a sphere.

[edit] Balls in general metric spaces

Let (M,d) be a metric space, namely a set M with a metric (distance function) d. The open (metric) ball of radius r > 0 centered at a point p in M, usually denoted by $B r (p)$ or $B (p; r)$ , is defined by

$B_r(p) \triangleq \{ x \in M \mid d(x,p) < r \},$

The closed (metric) ball, which may be denoted by $B r [p]$ or $B [p; r]$ , is defined by

$B_r[p] \triangleq \{ x \in M \mid d(x,p) \le r \},$

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 $B r (p)$ is usually denoted $\overline{ B_r(p) }$ . While it is always the case that $B_r(p) \subseteq \overline{ B_r(p) }$ and $B_r(p) \subseteq B_r[p]$ , it is not always the case that $\overline{ B_r(p) } = B_r[p]$ . For example, in a metric space $X$ with the discrete metric, one has $\overline{B_1(p)} = \{p\}$ and $B 1 [p] = X$ , for any $p \in X$ .

An (open or closed) unit ball 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 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.

[edit] Balls in normed vector spaces

Any normed vector space $V$ with norm $\left|\cdot\right|$ is also a metric space, with the metric $d(x,y) = \left|x - y\right|$ . In such spaces, every ball $B r (p)$ is a copy of the unit ball $B 1 (0)$ , scaled by $r$ and translated by $p$ .

[edit] 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.

[edit] P-norm

In Cartesian space $\R^n$ with the p-norm L_p, an open ball is the set

$B(r) = \{ x \in \R^n : \sum_{i=1}^{n} \left|x_i\right|^p < r^p \}$

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 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 L_p with p> 2 are superellipsoids.

[edit] General convex norm

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 $\R^n$ .

[edit] 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 $\R^n$ and to the open unit n-cube $(0,1)^n \subseteq \R^n$ . 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 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 an Euclidean n-ball.

[edit] See also

Ball (mathematics)

From Wikipedia, the free encyclopedia

Contents

[edit] Balls in general metric spaces

[edit] Balls in normed vector spaces

[edit] Euclidean norm

[edit] P-norm

[edit] General convex norm

[edit] Topological balls

[edit] See also

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