Ball (mathematics)

# Ball (mathematics)

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a ball is the space inside a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

. It may be a closed ball (including the boundary points) or an open ball (excluding them).

These concepts are defined not only in three-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

but also for lower and higher dimensions, and for metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

s in general. A ball in the Euclidean plane, for example, is the same thing as a disk
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...

, the area bounded by a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

.

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
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

and informal use, sphere sometimes means ball.

## Balls in general metric spaces

Let (M,d) be a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

, namely a set M with a metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

(distance function) d. The open (metric) ball of radius r > 0 centered at a point p in M, usually denoted by Br(p) or B(p; r), is defined by

The closed (metric) ball, which may be denoted by Br[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
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

of the open ball Br(p) is usually denoted . While it is always the case that and , it is not always the case that . For example, in a metric space with the discrete metric, one has and , for any .

An (open or closed) unit ball is a ball of radius 1.

A subset of a metric space is bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...

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
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

are a basis for a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

, whose open sets are all possible union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

s of open balls. This space is called the topology induced by the metric d.

## Balls in normed vector spaces

Any normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....

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.

### Euclidean norm

In particular, if V is n-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

with the ordinary (Euclidean) metric
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

, every ball is the interior of an hypersphere
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...

(a hyperball). That is a bounded interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

when n = 1, the interior of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

(a disk
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...

) when n = 2, and the interior of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

when n = 3.

### 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
Taxicab geometry
Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their coordinates...

or Manhattan metric) are squares with the diagonals parallel to the coordinate axes;
those of L (the Chebyshev
Chebyshev distance
In mathematics, Chebyshev distance , Maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension...

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 Lp with p > 2 are superellipsoids
Superegg
In geometry, a superegg is a solid of revolution obtained by rotating an elongated super-ellipse with exponent greater than 2 around its longest axis. It is a special case of super-ellipsoid....

.

### General convex norm

More generally, given any centrally symmetric, bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...

, open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

, and convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

subset X of Rn, one can define a norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

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.

## Topological balls

One may talk about balls in any topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

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
Combinatorial topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces were regarded as derived from combinatorial decompositions such as simplicial complexes...

, 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
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...

. 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 orientation
Orientation (mathematics)
In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

s of B.

A topological n-ball need not be smooth
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

; if it is smooth, it need not be diffeomorphic to a Euclidean n-ball.

• Ball
Ball
A ball is a round, usually spherical but sometimes ovoid, object with various uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for simpler activities, such as catch, marbles and juggling...

- ordinary meaning
• Disk (mathematics)
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...

• Neighborhood (mathematics)
• 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

• n-sphere, or hypersphere
• Alexander horned sphere
Alexander horned sphere
The Alexander horned sphere is a wild embedding of a sphere into space, discovered by . It is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:...

• Manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....