Unit ball - Definition and Overview

In mathematics, the open unit ball in a normed vector space <math>V<math> for a given norm <math>\|\cdot\|<math> is

<math>\{x\in V: \|x\|<1\}<math>.

The closed unit ball on <math>V<math> under <math>\|\cdot\|<math> is

<math>\{x\in V: \|x\|\le 1\}<math>.

The 'shape' of the unit ball is entirely dependent on the chosen norm; it may well have 'corners', and for example may look like [−1,1]ⁿ, in the case of the norm l_∞ in Rⁿ. The round ball is understood as the usual Hilbert space norm, based in the finite dimensional case on the Euclidean distance; its boundary is what is usually meant by the unit sphere.

See also:

• unit circle
• unit square
• unit disk
• sphere
• ball (mathematics)
• Table of mathematical symbols