Ball (mathematics)

In mathematics, in a metric space, a ball is a set containing all points within a specified distance of a given point.

The set is often written as B_r(p), meaning the ball of radius r centred on a point p. This is defined as:

B_r(p) = { x | d(x,p) < r }

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. This is usually notated by adding an overscore[?] to the B.

If r = 1, then it is called a unit ball.

A set is bounded if it is contained in a ball.