If a family of sets is a Helly family of order k, that family is said to have Helly number k. The Helly dimension of a metric space is one less than the Helly number of the family of metric balls in that space; Helly's theorem implies that the Helly dimension of a Euclidean space equals its dimension as a real vector space.
The Helly dimension of a subset S of a Euclidean space, such as a polyhedron, is one less than the Helly number of the family of translates of S. For instance, the Helly dimension of any hypercube is 1, even though such a shape may belong to a Euclidean space of much higher dimension.
Helly dimension has also been applied to other mathematical objects. For instance M. Domokos (arXiv:math.AG/0511300) defines the Helly dimension of a group (an algebraic structure formed by an invertible and associative binary operation) to be one less than the Helly number of the family of left cosets of the group.
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