Ball (mathematics)
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Categories: Metric geometry | Topology
- A synonym for ball (in geometry or topology, and in any dimension) is disc (or disk); however, a 3-dimensional ball is generally called a ball, and a 2-dimensional ball (e.g., the interior of a circle in the plane) is generally called a disc.
In mathematics, a ball is the inside of a sphere; both concepts apply not only in 3D but also for lower and higher dimensions, and for metric spaces in general; ball even applies for topological spaces in general.
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Metric spaces
Let M be a metric space. The (open) ball of radius r > 0 centred at a point p in M is defined as
- <math>B_r(p) = \{ x \in M \mid d(x,p) < r \},</math>
where d is the distance function or metric. If the less-than symbol (<) is replaced by a less-than-or-equal-to (≤), the above definition becomes that of a closed ball:
- <math>{\bar B}_r(p) = \{ x \in M \mid d(x,p) \le r \}</math>.
Note in particular that a ball (open or closed) always includes p itself, since r > 0. A (open or closed) unit ball is a ball of radius 1.
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 outside is the n-1-sphere Sn-1, e.g. the 3-sphere S3 is the outside 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
Note that in most cases rotated balls are not balls.
Related notions
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.
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.
See also
Topological spaces
In topology, ball has two meanings, with context governing which is meant.
The term (open) ball is sometimes informally used to refer to any open set: one speaks of "a ball about the point p" when one means an open set containing p. What this set is homeomorphic to depends on the ambient space and on the open set chosen. Likewise, closed ball is sometimes used to mean the closure of such an open set. (This can be quite misleading, as e.g. in ultrametric spaces a closed ball is not the closure of the open ball with the same radius, both being simultaneously open and closed sets.) Sometimes, neighborhood (or neighbourhood) is used for this meaning of ball, although neighborhood has a more general meaning: a neighborhood of p is any set containing an open set about p, thus not in general an open set.
Also (and more formally), an (open or closed) ball is a space homeomorphic to the (open or closed) Euclidean ball described above, but perhaps lacking its metric. A ball is known by its dimension: an n-dimensional ball is called an n-ball and denoted <math>B^n</math> or <math>D^n</math>. 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.