- Helly family
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. In other words, any subfamily such that every $(k+1)$-fold intersection is non-empty has non-empty total intersection.The "k"-

**Helly property**is the property of being a Helly family of order "k". These concepts are named afterEduard Helly (1884 - 1943);Helly's theorem onconvex set s, which gave rise to this notion, states that convex sets inEuclidean space of dimension "n" are a Helly family of order "n" + 1.**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

interval s of thereal 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.**Formal definition**More formally, a

**Helly family of order "k**" is aset system ("F", "E"), with "F" a collection ofsubset s of "E", such that, for any "G" ⊆ "F" with:$igcap\_\{Xin\; G\}\; X=varnothing,$

we can find "H" ⊆ "G" such that

:$igcap\_\{Xin\; H\}\; X=varnothing$

and

:$left|H\; ight|le\; k.$

**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 ametric 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 realvector 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 anyhypercube 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|archive=math.AG|id=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.

**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 thetight span allows any metric space to be embedded isometrically into a space with Helly dimension 1.**References***cite book

last = Balakrishnan

first = R.

coauthors = Ranganathan, K.

title = A textbook of graph theory

publisher = New York: Springer

date = 2000

pages =

isbn = 0387988599 (acid-free paper)*cite book

last = Kloks

first = Ton

title = Treewidth: computations and approximations

publisher = Berlin; New York: Springer-Verlag

date = 1994

pages =

isbn = 3540583564

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