BIGpedia - Unit ball - Encyclopedia and Dictionary Online

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

$\{x\in V: \|x\|<1\}$ .

The closed unit ball on $V$ under $\|\cdot\|$ is

$\{x\in V: \|x\|\le 1\}$ .

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.