Ball (mathematics) encyclopedia topics

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.

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

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]

B[p;r]

and 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 r > 0. Finally, 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, 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 $p in X$ .

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

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.

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.

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".

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.

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.

Metric spaces

Euclidean balls

Topological balls

See also