Unit ball
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In mathematics, unit ball and unit sphere refer to a ball with radius equal to 1. Refer to the ball article for a more general introduction.
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Unit balls in normed vector spaces
More precisely, the open unit ball in a normed vector space <math>V<math>, with the norm <math>\|\cdot\|<math>, is
- <math> \{ x\in V: \|x\|<1 \}<math>.
It is the interior of the closed unit ball of (V,||·||),
- <math> \{ x\in V: \|x\|\le 1\}<math>.
The latter is the disjoint union of the former and their common border, the unit sphere of (V,||·||),
- <math> \{ x\in V: \|x\| = 1 \}<math>.
Comments
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.
Generalization to metric spaces
All three of the above definitions can be straightforwardly generalized to a metric space, with respect to a chosen origin. However, topological considerations (interior, closure, border) need not apply in the same way (e.g., in ultrametric spaces, all of the three are simultaneously open and closed sets), and the unit sphere may even be empty in some metric spaces.