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In mathematics, a **ball** is the solid figure bounded by a *sphere*; it is also called a **solid sphere**.^{[1]} 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 *n* = 1, is a **disk** bounded by a circle when *n* = 2, and is bounded by a sphere when *n* = 3.

The n-dimensional volume of a Euclidean ball of radius *r* in *n*-dimensional Euclidean space is:^{[2]}
where Γ is Leonhard Euler's gamma function (which can be thought of as an extension of the factorial function to fractional arguments). Using explicit formulas for particular values of the gamma function at the integers and half integers gives formulas for the volume of a Euclidean ball that do not require an evaluation of the gamma function. These are:

In the formula for odd-dimensional volumes, the double factorial (2*k* + 1)!! is defined for odd integers 2*k* + 1 as (2*k* + 1)!! = 1 ⋅ 3 ⋅ 5 ⋅ ⋯ ⋅ (2*k* − 1) ⋅ (2*k* + 1).

Let (*M*, *d*) be a metric space, namely a set M with a metric (distance function) d, and let be a positive real number. The open (metric) **ball of radius** r centered at a point p in M, usually denoted by *B _{r}*(

The *closed* (metric) ball, sometimes denoted *B _{r}*[

In particular, a ball (open or closed) always includes p itself, since the definition requires *r* > 0. A **unit ball** (open or closed) is a ball of radius 1.

A ball in a general metric space need not be round. For example, a ball in real coordinate space under the Chebyshev distance is a hypercube, and a ball under the taxicab distance is a cross-polytope. A closed ball also need not be compact. For example, a closed ball in any infinite-dimensional normed vector space is never compact. However, a ball in a vector space will always be convex as a consequence of the triangle inequality.

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 can serve as a base, giving this space a topology, the open sets of which are all possible unions of open balls. This topology on a metric space is called the **topology induced by** the metric d.

Let denote the closure of the open ball in this topology. While it is always the case that it is *not* always the case that For example, in a metric space with the discrete metric, one has but for any

Any normed vector space V with norm is also a metric space with the metric In such spaces, an arbitrary ball of points around a point with a distance of less than may be viewed as a scaled (by ) and translated (by ) copy of a *unit ball* Such "centered" balls with are denoted with

The Euclidean balls discussed earlier are an example of balls in a normed vector space.

In a Cartesian space **R**^{n} with the p-norm L_{p}, that is one chooses some and defines Then an open ball around the origin with radius is given by the set
For *n* = 2, in a 2-dimensional plane , "balls" according to the *L*_{1}-norm (often called the *taxicab* or *Manhattan* metric) are bounded by squares with their *diagonals* parallel to the coordinate axes; those according to the *L*_{∞}-norm, also called the Chebyshev metric, have squares with their *sides* parallel to the coordinate axes as their boundaries. The *L*_{2}-norm, known as the Euclidean metric, generates the well known disks within circles, and for other values of p, the corresponding balls are areas bounded by Lamé curves (hypoellipses or hyperellipses).

For *n* = 3, the *L*_{1}- balls are within octahedra with axes-aligned *body diagonals*, the *L*_{∞}-balls are within cubes with axes-aligned *edges*, and the boundaries of balls for L_{p} with *p* > 2 are superellipsoids. *p* = 2 generates the inner of usual spheres.

Often can also consider the case of in which case we define

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 *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.

A number of special regions can be defined for a ball:

- Ball – ordinary meaning
- Disk (mathematics)
- Formal ball, an extension to negative radii
- Neighbourhood (mathematics)
- Sphere, a similar geometric shape
- 3-sphere
- n-sphere, or hypersphere
- Alexander horned sphere
- Manifold
- Volume of an n-ball
- Octahedron – a 3-ball in the
*l*_{1}metric.

- Smith, D. J.; Vamanamurthy, M. K. (1989). "How small is a unit ball?".
*Mathematics Magazine*.**62**(2): 101–107. doi:10.1080/0025570x.1989.11977419. JSTOR 2690391. - Dowker, J. S. (1996). "Robin Conditions on the Euclidean ball".
*Classical and Quantum Gravity*.**13**(4): 585–610. arXiv:hep-th/9506042. Bibcode:1996CQGra..13..585D. doi:10.1088/0264-9381/13/4/003. S2CID 119438515. - Gruber, Peter M. (1982). "Isometries of the space of convex bodies contained in a Euclidean ball".
*Israel Journal of Mathematics*.**42**(4): 277–283. doi:10.1007/BF02761407. S2CID 119483499.