A combinatorial version of the colorful Caratheodory theorem
aa r X i v : . [ m a t h . C O ] A ug A COMBINATORIAL VERSION OF THE COLORFUL CARATHÉODORYTHEOREM
ANDREAS F. HOLMSENA bstract . We give the following extension of Bárány’s colorful Carathéodory theo-rem: Let M be an oriented matroid, N a matroid with rank function ρ , both definedon the same ground set V and satisfying rk ( M ) < rk ( N ). If every A ⊂ V with ρ ( V − A ) < rk ( M ) contains a positive circuit of M , then there is a positive circuitof M which is independent in N .
1. I ntroduction
One of the cornerstones of convexity is Carathéodory’s theorem which states that, givena set P ⊂ R d and a point x in the convex hull of P , i.e. x ∈ conv P , there exists a subset Q ⊂ P such that | Q | ≤ d + x ∈ conv Q . In 1982, Bárány [2] gave the followinggeneralization of Carathéodory’s theorem. Theorem 1.1 (Colorful Carathéodory) . Let P , . . . , P d + be point sets in R d . If x ∈ T d + i = conv P i , then there exists p ∈ P , · · · , p d + ∈ P d + such that x ∈ conv { p , . . . , p d + } . The name originates from thinking of the P i as distinct color classes. The conclusiontells us the point x is contained in a “colorful simplex”, that is a simplex whose ver-tices are of all distinct colors. Notice that Bárány’s theorem reduces to Carathéodory’stheorem when the P i are equal.Theorem 1.1 has many applications in discrete geometry [12], and gives rise to inter-esting variations of linear programming [3]. It is easily seen that the hypothesis of The-orem 1.1 is not a necessary condition, and the following weakening of the hypothesiswas recently discovered [1, 10]. Theorem 1.2 (Strong Colorful Carathéodory) . Let P , . . . , P d + be non-empty point setsin R d . If x ∈ T d + ≤ i < j conv( P i ∪ P j ) , then there exists p ∈ P , . . . , p d + ∈ P d + such thatx ∈ conv { p , . . . , p d + } . For applications of the strengthened version, see [1]. The goal of this paper is to give atwofold generalization of Theorem 1.2.
Supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (NRF-2010-0021048). (1) Points in R d will be replaced by an oriented matroid . Every vector configura-tion in R d gives rise to an oriented matroid, but the converse does not hold. Infact, there are far more non-realizable oriented matroids than realizable ones.The question of whether the Colorful Carathéodory theorems extend to orientedmatroids is a natural one, and has been asked ever since Bárány first introducedhis result. (The rank 3 case was recently considered in [8].)(2) The color classes will be replaced by a matroid . It was noticed by Kalai andMeshulam [11] that the colorful simplices are actually playing the role of basesof the transversal matroid of the family { P , . . . , P d + } . In the closely relatedsetting of d-Leray complexes , they showed that the transversal matroid can bereplaced by an arbitrary matroid. Our main result (Theorem 1.3) can be thoughtof as dual to the result of Kalai and Meshulam, and our proof is a modificationof theirs.1.1. Matroids and oriented matroids. A matroid is a combinatorial structure de-signed to capture the notion of linear independence in vector spaces. There are nu-merous “cryptomorphic” axiom systems which define a matroid. For an introduction tomatroid theory, see e.g. [14]. The independent sets of a matroid naturally give rise to asimplicial complex whose topology will be of importance to us. For more informationin this direction we refer the reader to [4]. For a matroid N let rk ( N ) denote its rank.An oriented matroid can be thought of as a combinatorial abstraction of a finite vectorconfiguration spanning a vector space over an ordered field . As for ordinary matroids,there are several cryptomorphic axiom systems which define an oriented matroid. Ofimportance to us are the positive circuits . In the analogy of vector spaces, these corre-spond to the minimal positive linear dependencies of the configuration. For details, werefer the reader to [7, 15]. For an oriented matroid M let rk ( M ) denote the rank of itsunderlying matroid.In this paper all matroids and oriented matroids are considered to be loopless.1.2. Main result.Theorem 1.3.
Let M be an oriented matroid, N a matroid with rank function ρ , bothdefined on the same ground set V and satisfying rk ( M ) < rk ( N ) . If every A ⊂ V with ρ ( V − A ) < rk ( M ) contains a positive circuit of M , then there is a positive circuit of M which is independent in N . As an immediate corollary we obtain Theorem 1.2. To see this let M be the orientedmatroid of the vector configuration V = { p − x | p ∈ P ∪· · ·∪ P d + } and N the transversalmatroid of the family { P , . . . , P d + } .Our proof of Theorem 1.3 uses topological methods: The Folkman-Lawrence repre-sentation theorem [9] allows us to represent an oriented matroid as an arrangement of OMBINATORIAL COLORFUL CARATHÉODORY 3 open pseudohemispheres with nice intersection properties. We may therefore pass tothe nerve complex of the arrangement whose homology can be determined using theNerve theorem (see [5]). The rest of the proof (more or less) follows the arguments ofKalai and Meshulam [11]. 2. P reliminaries
Here we collect the basic facts needed for the proof of Theorem 1.3.2.1.
Simplicial complexes and homology.
First we review some standard notions fromsimplicial homology. Let X be a simplicial complex on V . For W ⊂ V let X [ W ] = { T ∈ X : T ⊂ W } denote the induced subcomplex on W . For a simplex S ∈ X let lk ( S , X ) = { T ∈ X : T ∩ S = ∅ , T ∪ S ∈ X } denote the link of S . Let e H j ( X ) denote the j -th reduced homology group of X withrational coe ffi cients, and define η ( X ) = min { j : e H j ( X ) , } + X and Y on disjoint vertex sets, the join X ∗ Y is the simplicialcomplex on the union of their vertex sets defined as X ∗ Y = { S ∪ T : S ∈ X , T ∈ Y } By the Künneth formula for the join of simplicial complexes we obtain the following.
Corollary 2.1. η ( X ∗ Y ) = η ( X ) + η ( Y )Let X be a simplicial complex on V and suppose V < X . The Alexander dual X ⋆ is thesimplicial complex on V defined as X ⋆ = { T ⊂ V : V − T < X } The homology of X and X ⋆ are related by Alexander duality, which says that if V < X then e H i ( X ⋆ ) (cid:27) e H | V |− i − ( X ) for all − ≤ i ≤ | V | − Corollary 2.2.
If V < X, S < X ⋆ , and T = V − S , then e H i ( X ⋆ [ S ]) (cid:27) e H | S |− i − ( lk ( T , X )) OMBINATORIAL COLORFUL CARATHÉODORY 4
The Nerve theorem.
Let F = { S v } v ∈ V be family of sets. The nerve N F is theabstract simplicial complex on V whose simplices consists of those T ⊂ V such that T v ∈ T S v , ∅ . We recall the following version of the Nerve theorem. (For a proof seee.g. [5]) Theorem 2.3.
Let F = { S v } v ∈ V be a family of open contractible subsets of R n such thatevery non-empty intersection T v ∈ W S v is contractible. Then F and N F are homotopyequivalent. The independence complex of a matroid.
Let N be a matroid on V . The in-dependence complex of N is the simplicial complex Y N on V whose simplices are theindependent sets of N . In other words, Y N = { S ⊂ V : S independent in N }
The following is a well-known fact. (See e.g. [4, 6].)
Lemma 2.4.
Let N be a matroid on ground set V with rank function ρ and Y = Y N itsindependence complex. Then η ( Y [ S ]) ≥ ρ ( S ) for every non-empty S ⊂ V. Topological representation of oriented matroids.
Let M be an oriented matroidof rank d on the ground set V . (Oriented matroids are assumed to be loopless.) TheFolkman-Lawrence representation theorem [9] states that M can be represented as anarrangement { S v } v ∈ V of oriented pseudospheres in S d − . Such an arrangement decom-poses S d − into a regular cell complex whose combinatorial structure encodes the ori-ented matroid. (See [7] for precise definitions and proofs.) Equivalently, M can berepresented by the collection of open pseudohemispheres { h v } v ∈ V in S d − , which havethe S v as boundaries and lie on the “positive” sides. The crucial fact for us is the fol-lowing. Corollary 2.5.
Let M be an oriented matroid on ground set V and { h v } v ∈ V a topologicalrepresentation by pseudohemispheres. The intersection h W = T w ∈ W h w is empty or con-tractible for every W ⊂ V. Moreover, h W is empty if and only if W contains a positivecircuit of M . The support complex of an oriented matroid.
Let M be an oriented matroid on V . The support complex of M is the simplicial complex X M on V whose simplices arethe subsets of V which do not contain positive circuits of M . That is, X M = { S ⊂ V : S contains no positive circuit of M} Proposition 2.6.
Let M be an oriented matroid of rank r and X = X M its supportcomplex. The following hold. (1) e H j ( X ) = for all j ≥ r. (2) e H j ( lk ( S , X )) = for all j ≥ r − and non-empty S ∈ X. OMBINATORIAL COLORFUL CARATHÉODORY 5
Proof.
Consider a topological representation of M by an arrangement of pseudohemi-spheres A = { h v } v ∈ V in S r − . Corollary 2.5 implies that X M is the nerve of A and thatevery non-empty intersection T v ∈ S h v is contractible. So by the Nerve theorem X ishomotopic to S v ∈ V h v ⊂ S r − , hence e H j ( X ) = j ≥ r .For the second part, let ∅ , S ∈ X and let h S = T v ∈ S h v . The simplices of lk ( S , X )correspond to subsets T ⊂ V \ S such that the intersection T v ∈ T h v ∩ h S is non-empty.It follows that lk ( S , X ) is the nerve of the family { h v ∩ h S } v ∈ V \ S , and the Nerve theoremimplies that lk ( S , X ) is homotopic to S v ∈ V \ S ( h v ∩ h S ) ⊂ h S . Since h S is homeomorphicto R r − it follows that e H j ( lk ( S , X )) = j ≥ r − (cid:3) Colorful simplices.
Let Z be a simplicial complex on V and S mi = V i a partition of V . A colorful simplex of Z is a simplex S ∈ Z such that | S ∩ V i | = ≤ i ≤ m . Meshulam [13] gave the following su ffi cient condition for a simplicial complex on S mi = V i to contain a colorful simplex. (For a short proof based on the Nerve theorem,see [11]) Proposition 2.7.
If for all ∅ , I ⊂ [ m ] we have η ( Z [ ∪ i ∈ I V i ]) ≥ | I | then Z contains a colorful simplex.
3. P roof of T heorem V = { v , v , . . . , v m } , r = rk ( M ), X = X M the support complex of M , and Y = Y N the independence complex of N . Make a disjoint copy V ′ = { v ′ , v ′ , . . . , v ′ m } of V and let Y ′ be an isomorphic copy of Y on V ′ . Consider the join Z = X ⋆ ∗ Y ′ and let V i = { v i , v ′ i } for 1 ≤ i ≤ m .Notice that a colorful simplex S ∪ T ′ ∈ Z implies that T = V − S is independent in N .It also implies T < X and therefore T contains a positive circuit of M . The strategy istherefore to apply Proposition 2.7 to show that Z contains a colorful simplex.For ∅ , I ⊂ [ m ] set S = { v i : i ∈ I } and S ′ = { v ′ i : i ∈ I } . By Corollary 2.1 and Lemma2.4 we have η ( Z [ ∪ i ∈ I V i ]) = η ( X ⋆ [ S ] ∗ Y ′ [ S ′ ]) = η ( X ⋆ [ S ]) + η ( Y [ S ]) ≥ η ( X ⋆ [ S ]) + ρ ( S )If S ∈ X ⋆ then X ⋆ [ S ] is contractible, which implies η ( X ⋆ [ S ]) = ∞ > | I | . We maytherefore assume S < X ⋆ and consequently T = V − S ∈ X . By hypothesis M contains OMBINATORIAL COLORFUL CARATHÉODORY 6 positive circuits, hence V < X , so by Corollary 2.2 we have e H i ( X ⋆ [ S ]) (cid:27) e H | S |− i − ( lk ( T , X ))There are two cases to consider.(1) If S = V then X ⋆ [ S ] = X ⋆ , T = ∅ , and lk ( T , X ) = X . The first case of Lemma2.6 implies that e H i ( X ⋆ ) = i ≤ | S | − r −
3, hence η ( X ⋆ ) ≥ | S | − r − ρ ( V ) = rk ( N ) > r , which implies η ( Z ) ≥ η ( X ⋆ ) + ρ ( V ) ≥ ( | S | − r − + ( r + = | V | (2) If S is a proper subset of V , then the second case of Lemma 2.6 implies that e H i ( X ⋆ [ S ]) = i ≤ | S | − r −
2, hence η ( X ⋆ [ S ]) ≥ | S | − r Since T = V − S ∈ X , T does not contain a positive circuit of M , so by hypothesis ρ ( S ) = ρ ( V − T ) ≥ r . Hence η ( Z [ ∪ i ∈ I V i ]) ≥ η ( X ⋆ [ S ]) + ρ ( S ) ≥ ( | S | − r ) + r = | I | Proposition 2.7 therefore implies that Z contains a colorful simplex. (cid:3)
4. A cknowledgments
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