Helly family

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In combinatorics, a Helly family of order k is a family of sets such that any minimal subfamily with an empty intersection has k or fewer sets in it. The k-Helly property is the property of being a Helly family of order k. These concepts are named after Eduard Helly (1884 - 1943); Helly's theorem on convex sets, which gave rise to this notion, states that convex sets in Euclidean space of dimension n are a Helly family of order n + 1.

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[edit] Examples

  • In the family of all subsets of the set {a,b,c,d}, the subfamily {{a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}} has an empty intersection, but removing any set from this subfamily causes it to have a nonempty intersection. Therefore, it is a minimal subfamily with an empty intersection. It has four sets in it, and is the largest possible minimal subfamily with an empty intersection, so the family of all subsets of the set {a,b,c,d} is a Helly family of order 4.
  • Let I be a set of closed intervals of the real line with an empty intersection. Then there must be two intervals A and B in I such that the left endpoint of A is larger than the right endpoint of B. {A,B} have an empty intersection, so I cannot be minimal unless I={A,B}. Therefore, all minimal families of intervals with empty intersections have 2 or fewer intervals in them, and the set of all intervals is a Helly family of order 2.

[edit] Formal definition

More formally, a Helly family of order k is a set system (F, E), with F a collection of subsets of E, such that, for any G ⊆ F with

\bigcap_{X\in G} X=\varnothing,

we can find H ⊆ G such that

\bigcap_{X\in H} X=\varnothing

and

\left|H\right|\le k.

[edit] Helly dimension

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 Domokos (arXiv:math.AG/0511300) defines the Helly dimension of a group to be one less than the Helly number of the family of left cosets of the group.

[edit] The Helly property

If a family of nonempty sets has an empty intersection, its Helly number must be at least two, so the smallest k for which the k-Helly property is nontrivial is k = 2. The 2-Helly property is also known as the Helly property. A 2-Helly family is also known as a Helly family.

A convex metric space in which the closed balls have the 2-Helly property (that is, a space with Helly dimension 1) is called injective or hyperconvex. The existence of the tight span allows any metric space to be embedded isometrically into a space with Helly dimension 1.

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