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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. Equivalently, every finite subfamily such that every <math>k</math>-fold intersection is non-empty has non-empty total intersection.<ref name="b86">{{citation|title=Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability|first=Béla|last=Bollobás|authorlink=Béla Bollobás|publisher=Cambridge University Press|year=1986|isbn=9780521337038|page=82|url=http://books.google.com/books?id=psqFNlngZDcC&pg=PA82}}.</ref>
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| The ''k''-'''Helly property''' is the property of being a Helly family of order ''k''.<ref name="d95">{{citation
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| | last = Duchet | first = Pierre
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| | editor1-last = Graham | editor1-first = R. L.
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| | editor2-last = Grötschel | editor2-first = M.
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| | editor3-last = Lovász | editor3-first = L.
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| | contribution = Hypergraphs
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| | location = Amsterdam
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| | mr = 1373663
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| | pages = 381–432
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| | publisher = Elsevier
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| | title = Handbook of combinatorics, Vol. 1, 2
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| | year = 1995}}. See in particular Section 2.5, "Helly Property", [http://books.google.com/books?id=5Y9NCwlx63IC&pg=PA393 pp. 393–394].</ref> These concepts are named after [[Eduard Helly]] (1884 - 1943); [[Helly's theorem]] on [[convex set]]s, which gave rise to this notion, states that convex sets in [[Euclidean space]] of dimension ''n'' are a Helly family of order ''n'' + 1.<ref name="b86"/> The number ''k'' is frequently omitted from these names in the case that ''k'' = 2.
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| == Examples ==
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| * 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.
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| * Let I be a finite set of closed [[Interval (mathematics)|intervals]] of the [[real line]] with an empty intersection. Let A be the interval whose left endpoint ''a'' is as large as possible, and let B be the interval whose right endpoint ''b'' is as small as possible. Then, if ''a'' were less than or equal to ''b'', all numbers in the range [''a'',''b''] would belong to all invervals of I, violating the assumption that the intersection of I is empty, so it must be the case that ''a'' > ''b''. Thus, the two-interval subfamily {A,B} has an empty intersection, and the family I cannot be minimal unless I = {A,B}. Therefore, all minimal families of intervals with empty intersections have two or fewer intervals in them, showing that the set of all intervals is a Helly family of order 2.<ref>This is the one-dimensional case of Helly's theorem. For essentially this proof, with a colorful phrasing involving sleeping students, see {{citation|title=Mathematical Miniatures|volume=43|series=New Mathematical Library|first1=Svetoslav|last1=Savchev|first2=Titu|last2=Andreescu|publisher=Mathematical Association of America|year=2003|isbn=9780883856451|contribution=27 Helly's Theorem for One Dimension|pages=104–106|url=http://books.google.com/books?id=qnkiAW6VQUAC&pg=PA104}}.</ref>
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| * The family of infinite [[arithmetic progression]]s of [[integer]]s also has the 2-Helly property. That is, whenever a finite collection of progressions has the property that no two of them are disjoint, then there exists an integer that belongs to all of them; this is the [[Chinese remainder theorem]].<ref name="d95"/>
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| == Formal definition ==
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| More formally, a '''Helly family of order ''k''''' is a [[set system]] (''F'', ''E''), with ''F'' a collection of [[subset]]s of ''E'', such that, for every finite ''G'' ⊆ ''F'' with
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| :<math>\bigcap_{X\in G} X=\varnothing,</math>
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| we can find ''H'' ⊆ ''G'' such that
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| :<math>\bigcap_{X\in H} X=\varnothing</math>
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| and
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| :<math>\left|H\right|\le k.</math><ref name="b86"/>
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| In some cases, the same definition holds for every subcollection ''G'', regardless of finiteness. However, this is a more restrictive condition. For instance, the [[open interval]]s of the real line satisfy the Helly property for finite subcollections, but not for infinite subcollections: the intervals (0,1/''i'') (for ''i'' = 0, 1, 2, ...) have pairwise nonempty intersections, but have an empty overall intersection.
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| == Helly dimension ==
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| 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]].<ref>{{citation|title=Excursions Into Combinatorial Geometry|first=Horst|last=Martini|publisher=Springer|year=1997|isbn=9783540613411|pages=92–93|url=http://books.google.com/books?id=U0LHc-DoBHgC&pg=PA92}}.</ref>
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| 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 [[Translation (geometry)|translates]] of S.<ref>{{citation|title=Classical Topics in Discrete Geometry|first=Károly|last=Bezdek|authorlink=Károly Bezdek|publisher=Springer|year=2010|isbn=9781441906007|page=27|url=http://books.google.com/books?id=Tov0d9VMOfMC&pg=PA27}}.</ref> 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.<ref>{{citation
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| | last = Sz.-Nagy | first = Béla
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| | journal = Acta Universitatis Szegediensis
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| | mr = 0065942
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| | pages = 169–177
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| | title = Ein Satz über Parallelverschiebungen konvexer Körper
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| | url = http://acta.fyx.hu/acta/showCustomerArticle.action?id=6292&dataObjectType=article
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| | volume = 15
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| | year = 1954}}.</ref>
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| Helly dimension has also been applied to other mathematical objects. For instance {{harvtxt|Domokos|2007}} defines the Helly dimension of a [[Group (mathematics)|group]] (an algebraic structure formed by an invertible and associative binary operation) to be one less than the Helly number of the family of [[Coset|left cosets]] of the group.<ref>{{citation
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| | last = Domokos | first = M.
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| | arxiv = math/0511300
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| | doi = 10.1007/s00031-005-1131-4
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| | issue = 1
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| | journal = Transformation Groups
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| | mr = 2308028
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| | pages = 49–63
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| | title = Typical separating invariants
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| | volume = 12
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| | year = 2007}}.</ref>
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| == The Helly property ==
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| 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.
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| The 2-Helly property is also known as the '''Helly property'''. A 2-Helly family is also known as a '''Helly family'''.<ref name="b86"/><ref name="d95"/>
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| A [[convex metric|convex]] [[metric space]] in which the closed [[Ball (mathematics)|balls]] have the 2-Helly property (that is, a space with Helly dimension 1, in the stronger variant of Helly dimension for infinite subcollections) is called [[injective metric space|injective]] or hyperconvex.<ref>{{citation|title=Encyclopedia of Distances|first1=Michel Marie|last1=Deza|author1-link=Michel Deza|first2=Elena|last2=Deza|publisher=Springer|year=2012|isbn=9783642309588|page=19|url=http://books.google.com/books?id=QxX2CX5OVMsC&pg=PA19}}</ref> The existence of the [[tight span]] allows any metric space to be embedded isometrically into a space with Helly dimension 1.<ref>{{citation
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| | last = Isbell | first = J. R. | authorlink = John R. Isbell
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| | doi = 10.1007/BF02566944
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| | journal = Comment. Math. Helv.
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| | pages = 65–76
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| | title = Six theorems about injective metric spaces
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| | volume = 39
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| | year = 1964}}.</ref>
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| ==References==
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| {{reflist}}
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| [[Category:Set families]]
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