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.
Read more about this topic: Helly Family
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