Ball (mathematics)
In mathematics, particularly in metric geometry, a ball is a set containing all points within a specified distance of a given point.
Examples
With the ordinary (Euclidean) metric, if the space is the line, the ball is an interval, if the space is the plane, the ball is the inside of a circle. With other metrics the shape of a ball is different; for example, in taxicab geometry a ball is diamond-shaped.
General definition
Let M be a metric space. The open ball of radius r > 0 centred at 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, a closed unit ball is also denoted Dn.
Related notion
Balls with respect to a metric d form a basis for the topology induced by d. This means, among other things, that all open sets in a metric space can be written as a union of open balls.
A set is bounded if it is contained in a ball. Conversely, a set is totally bounded if given any radius, it is covered by finitely many balls of that radius.
See also:
