NationMaster - Encyclopedia: Ball (mathematics)

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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see Sphere (disambiguation). ...

1 Metric spaces
2 Euclidean balls
3 Topological balls
4 See also

Metric spaces

Let M be a metric space. The (open) ball of radius r > 0 centered at a point p in M is usually denoted by $B r (p)$ or $B (p; r)$ and defined by In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

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

where d is the distance function or metric. This is also called an (open) metric ball. If the less-than symbol (<) is replaced by a less-than-or-equal-to (≤), the above definition becomes that of a closed (metric) ball, which is denoted by $B r [p]$ or $B [p; r]$ and defined by: In mathematics a metric or distance function is a function which defines a distance between elements of a set. ...

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

Note in particular that a ball (open or closed) always includes p itself, since r > 0. Finally, the closure of the open ball $B r (p)$ is usually denoted $overline{ B_r(p) }$ . In mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set. ...

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, consider a nonempty metric space $X$ with the discrete metric. In this case, for any $p in X$ , $overline{B_1(p)} = {p}$ and $B 1 [p] = X$ , so clearly $overline{B_1(p)} neq B_1[p]$ for all points $p in X$ . In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ...

A (open or closed) unit ball is a ball of radius 1. some unit spheres In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used. ...

A subset of a metric space is bounded if it is contained in a ball. A set is totally bounded if given any radius, it is covered by finitely many balls of that radius. In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ...

Open balls with respect to a metric d form a basis for the topology induced by d (by definition). This means, among other things, that all open sets in a metric space can be written as a union of open balls. In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

Euclidean balls

In n-dimensional Euclidean space with the ordinary (Euclidean) metric, if the space is the line, the ball is an interval, and if the space is the plane, the ball is the disc inside a circle. A closed unit ball is denoted by Dⁿ; its boundary (or "edge") is the n-1-sphere Sⁿ⁻¹, e.g., the 3-sphere S³ is the boundary of D⁴ in 4D. These and the corresponding objects in even higher dimensions are called hyperball and hypersphere. See the latter for "volumes" and "areas". Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... This article is about the shape and mathematical concept of circle. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ...

With other metrics the shape of a ball can be different; examples:

in 2D:
- with the 1-norm (i.e., in taxicab geometry) a ball is a square with the diagonals parallel to the coordinate axes
- with the Chebyshev distance a ball is a square with the sides parallel to the coordinate axes

in 3D:
- with the 1-norm a ball is a regular octahedron with the body diagonals parallel to the coordinate axes
- with the Chebyshev distance a ball is a cube with the edges parallel to the coordinate axes.

Manhattan versus Euclidean distance: The red, blue, and yellow lines representing the Manhattan distance all have the same length (12), whereas the green line representing the Euclidian distance has length 6Ã—âˆš2 â‰ˆ 8. ... In a plane, the Chebyshev distance between the point P1 with coordinates (x1, y1) and the point P2 at (x2, y2) is This concept is named after Pafnuty Chebyshev. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ...

Topological balls

One may talk about balls in any topological space, not necessarily induced by a metric. An (open or closed) ball in such a space is a set which is homeomorphic to an (open or closed) Euclidean ball described above. A ball is known by its dimension: an n-dimensional ball is called an n-ball and denoted $B n$ or $D n$ . For distinct n and m, an n-ball is not homeomorphic to an m-ball. A ball need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean ball. This word should not be confused with homomorphism. ... 2-dimensional renderings (ie. ... Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

Encyclopedia > Ball (mathematics)

Contents

Metric spaces

Euclidean balls

Topological balls

See also