The convex dimension of hypergraphs and the hypersimplicial Van Kampen-Flores Theorem
TTHE CONVEX DIMENSION OF HYPERGRAPHS AND THEHYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM
LEONARDO MART´INEZ-SANDOVAL AND ARNAU PADROL
Abstract.
The convex dimension of a k -uniform hypergraph is the smallestdimension d for which there is an injective mapping of its vertices into R d suchthat the set of k -barycenters of all hyperedges is in convex position.We completely determine the convex dimension of complete k -uniform hy-pergraphs, which settles an open question by Halman, Onn and Rothblum,who solved the problem for complete graphs. We also provide lower and up-per bounds for the extremal problem of estimating the maximal number ofhyperedges of k -uniform hypergraphs on n vertices with convex dimension d .To prove these results, we restate them in terms of affine projections thatpreserve the vertices of the hypersimplex. More generally, we provide a fullcharacterization of the projections that preserve its i -dimensional skeleton.In particular, we obtain a hypersimplicial generalization of the linear vanKampen-Flores theorem: for each n , k and i we determine onto which di-mensions can the ( n, k )-hypersimplex be linearly projected while preservingits i -skeleton.Our results have direct interpretations in terms of k -sets and ( i, j )-partitions,and are closely related to the problem of finding large convexly independentsubsets in Minkowski sums of k point sets. Introduction
Motivated by problems in convex combinatorial optimization [22], Halman, Onnand Rothblum introduced the concept of convex dimension of uniform hypergraphs [14].A k -uniform hypergraph is a pair H = ( V, E ) with E ⊆ (cid:0) Vk (cid:1) ; a convex embedding of H into R d is an injective map f : V → R d such that the set of k -barycenters (cid:40) k (cid:88) v ∈ e f ( v ) : e ∈ E (cid:41) is in convex position (i.e. no point is a convex combination of the others); andthe convex dimension of H , denoted cd ( H ), is the minimal d for which a convexembedding of H into R d exists.Their article focused on graphs, the k = 2 case. They studied the problem ofdetermining the convex dimension for specific families of graphs: paths, cycles,complete graphs and bipartite graphs. They also investigated the extremal prob-lem of determining the maximum number of edges that a graph on n vertices andfixed convex dimension can have. The latter problem has been studied afterwardsby several authors, in particular because of its strong relation with the problem ofdetermining large convex subsets in Minkowski sums [4, 7]. Indeed, convex embed-dings of subhypergraphs of complete k -partite k -uniform hypergraphs correspond Research supported by the grant ANR-17-CE40-0018 of the French National Research AgencyANR (project CAPPS). a r X i v : . [ m a t h . C O ] N ov LEONARDO MART´INEZ-SANDOVAL AND ARNAU PADROL to subsets in convex position inside the Minkowski sum of k sets of points. Diversevariants of the case k = 2 have been considered in the plane [4, 7, 11, 31, 33], andalso in R [32].For k >
2, the only result of which we are aware of is the upper bound cd ( H ) ≤ k for any k -uniform hypergraph H , proved by Halman et al. by mapping the verticesonto points on the moment curve in R k [14].Our first result is the complete determination of the convex dimension of K ( k ) n :=([ n ] , (cid:0) [ n ] k (cid:1) ), the complete k -uniform hypergraph on n vertices, for any k , 1 ≤ k ≤ n − Theorem 1.1.
Given positive integers n and k such that ≤ k ≤ n − , we havethat cd ( K ( k ) n ) = k if n ≥ k + 2 , n − if n ∈ { k − , k, k + 1 } , n − k if n ≤ k − .Also, cd ( K (1)2 ) = 1 and cd ( K (1) n ) = cd ( K ( n − n ) = 2 for n ≥ . This matches and extends the results for k = 2 in [14], where it is proved that cd ( K n ) = 4 for n ≥
6. Table 1 shows the explicit values of cd ( K ( k ) n ) given byTheorem 1.1 for small values of n and k . k \ n Table 1.
First values of cd ( K ( k ) n ). Green values correspond toexceptional cases with small values of n and k . Yellow values cor-respond to the cases n ∈ { k − , k, k + 1 } , when k ≥
2. Redvalues correspond to the cases n ≥ k + 2 or n ≤ k − k -barycenters of a point-set S hasalso been studied under the name of k -set polytope and denoted by P k ( S ) [3, 6] inrelation to the study of k -sets, j -facets and ( i, j )-partitions [3, 35]. In this language,Theorem 1.1 determines for which dimensions we can find point sets of cardinality n for which all k -barycenters are vertices of the k -set polytope.In general, the k -set problem consists in estimating the maximal number of ver-tices of P k ( S ) in fixed dimension d . As Sharir, Smorodinsky, and Tardos put it,this is “one of the most intriguing open problems in combinatorial geometry” [30],and there is a considerable gap between the upper and lower bounds (see [35] foran extensive survey on the subject, and [29] for the latest improvement). However,note that for a general k -uniform hypergraph H = ( V, E ), convex embeddings are
HE HYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM 3 more permissive than asking for the the k -barycenters induced by E to be verticesof the k -set polytope. Indeed, the subsets of k -barycenters given by E can be inconvex position even if they are not vertices of the whole set of k -barycenters. Weexplore further connections between these topics in Section 5.We provide a polyhedral proof of Theorem 1.1. Namely, we reformulate theexistence of a convex embedding of K ( k ) n into R d in terms of affine projectionsthat strictly preserve the vertices of the hypersimplex ∆ n,k , that is, the polytopewhose vertices are the (cid:0) nk (cid:1) incidence vectors of k -subsets of [ n ]. This polyhedralformulation is closer to the original set-up of convex combinatorial optimization [22].Hypersimplices are a widely studied family of polytopes that arise naturallyin very diverse contexts, and there has been a recent interest on hypersimplexprojections [20, 25] motivated by a result of Galashin [10] who showed that certainsubdivisions induced by hypersimplex projections are in bijection with reducedplabic graphs [19], used to describe the stratification of the totally nonnegativeGrassmannian [24].As we shall see, Theorem 1.1 is a particular case of a more general result. Let d = d ( n, k, i ) be the smallest dimension for which we can find a projection π : ∆ n,k → R d that strictly preserves the i -dimensional skeleton of ∆ n,k . We determine the valuesof d ( n, k, i ) in Theorem 1.2, which is proved in Section 3 with the framework usedby Sanyal when studying the number of vertices of Minkowski sums [27], based onZiegler’s projection lemma [38]. Theorem 1.2.
Given positive integers n , k , i such that ≤ k ≤ n − and ≤ i ≤ n − , the value of d ( n, k, i ) is determined as follows. d ( n, k, i ) = k + 2 i if n ≥ k + 2 i + 2 , n − k + 2 i if n ≤ k − i − ,n − if k − i − ≤ n ≤ k + 2 i + 1 , k ∈ A n,i ,n − if k − i − ≤ n ≤ k + 2 i + 1 , k / ∈ A n,i . Where A n,i = { , , . . . , i + 1 } ∪ { n − i − , n − i, . . . , n − } . Actually, this is a corollary of our main result, Theorem 3.6, which providesthe full characterization of the projections that attain these bounds. That is, wefully characterize which n -point configurations S verify that the k -set polytope P k ( S ) shares the i -skeleton with ∆ n,k . Our characterization shows a surprisingdichotomy: either S is neighborly enough, or it has few vertices and it is not very(almost) neighborly, see Section 3 for details.In Section 4 we exploit this characterization to solve a variant of the convexembedding problem, also posed in the work of Halman, Onn and Rothblum [22],for which we require the images of the vertices of the hypergraph to be in convexposition as well.In Theorem 1.2, for k = 1, we get d ( n, , i ) = 2 i + 2 for n ≥ i + 4 and d ( n, , i ) = n − d -dimensional polytope(from now on abbreviated as d -polytope ) that is more than (cid:4) d (cid:5) -neighborly [13,Thm. 7.1.4]; or equivalently, that no linear projection of a (2 i + 2)-simplex onto LEONARDO MART´INEZ-SANDOVAL AND ARNAU PADROL R i +1 preserves its i -skeleton. This result is sometimes referred to as the linear vanKampen-Flores Theorem (e.g. in [26, p. 95]); thus Theorem 1.2 could be called the hypersimplicial linear van Kampen-Flores Theorem .Prodsimplicial linear van Kampen-Flores Theorems for products of simpliceswere proved by Matschke, Pfeifle and Pilaud [17] and R¨orig and Sanyal [26]. Oneof their motivations was the study of dimensional ambiguity. Gr¨unbaum defineda polytope P to be dimensionally i -ambiguous if its i -skeleton is isomorphic tothat of polytope Q of a different dimension [13, Ch. 12]. Few polytopes areknown to be dimensionally ambiguous. Examples include: simplices via neigh-borly polytopes , cubes via neighborly cubical polytopes [15], products of polygonsvia projected products of polygons [28, 38], and the aforementioned products ofsimplices via prodsimplicial-neighborly polytopes [17]. Our results show that the( n, k )-hypersimplex is dimensionally i -ambiguous except, maybe, when k ∈ A n,i and 2 k − i − ≤ n ≤ k + 2 i + 1, which happens only for small values of n and k .For convenience, we present the case i = 2 in Table 2, and we invite the readerto compare it with Table 1. k \ n Table 2.
Values of d ( n, k,
2) for small values of n and k . Greenvalues correspond to exceptional cases where k ∈ A n,i and 2 k − i − ≤ n ≤ k + 2 i + 1. Yellow values correspond to the cases2 k − i − ≤ n ≤ k + 2 i + 1 with k / ∈ A n,i . Red values correspondto the cases n ≥ k + 2 i + 2 or n ≤ k − i − k = 1 , n − d ( n, k, i ) depending on the relative values of n , k and i . Thefirst possibility is that there is no dimension-reducing projection of the hypersimplexthat preserves all i -faces. This happens for some exceptional values captured by therestrictions k ∈ A n,i and 2 k − i − ≤ n ≤ k + 2 i + 1 . These are the green valueson the table. If n is very large, or very small compared to k and i , then we canproject to a space of fixed dimension, and achieve arbitrarily large codimension.These cases are shown in red in the table. Finally, there is an extra case (depictedin yellow) in which we can get a codimension 1 projection but no codimension 2projection exists. It starts happening when k ≥ i + 1 and n is of moderate size.Finally, in Section 6 we study the associated extremal problem of maximizingthe number of k -barycenters in convex position, which has been largely studied forgraphs [4, 7, 11, 14, 32]. Let g k ( n, d ) be the maximum number of hyperedges thata k -uniform hypergraph on n vertices that has a convex embedding into R d canhave. HE HYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM 5
The function g k ( n, d ) exhibits three different regimes according to the value of d .Our knowledge of the (asymptotic) growth of g k ( n, d ) for fixed d and k has differentlevels of precision for the three cases: Theorem 1.3.
For fixed values of k and d , the value of g k ( n, d ) behaves as follows: • If d ≥ k , then g k ( n, d ) = (cid:18) nk (cid:19) . • If k + 1 ≤ d ≤ k − , then g k ( n, d ) is in Θ( n k ) . More precisely, g k ( n, d ) = γ k,d · n k + o ( n k ) for some constant γ k,d satisfying (cid:18) dk (cid:19) d k ≤ γ k,d ≤ k ! (cid:32) − (cid:0) n d,k k − (cid:1) (cid:33) where n k,d = (cid:40) d + 2 if d ≥ k − , (cid:4) d (cid:5) + k if ≤ d ≤ k − . for d (cid:54) = 1 , and n , = 2 and n k, = k for k ≥ . • If d ≤ k , then g k ( n, d ) is in O ( n d ) and in Ω( n d − ) .The limits in the o , O , Ω and Θ notations are taken as n → ∞ . To put this result into perspective, we compare it with the known results for thecase of graphs ( k = 2): • For d ≥
4, every graph has a convex embedding into R d , and g ( n, d ) = (cid:0) n (cid:1) . • For d = 3, it is shown in [32] that γ , ∈ { , } , and γ , = is conjectured,which would be the case if and only if for some m there is no convexembedding of K m,m,m,m into R . Recently, Raggi and Rold´an-Pensadofound a convex embedding of K , , , into R using computational methods(personal communication). Note that for these parameters, our resultsrecover exactly these same lower and upper bounds: ≤ γ , ≤ . • For d = 2, Halman, Onn and Rothblum asked whether g ( n,
2) was linear orquadratic [14]. The answer is that g ( n, ∈ Θ( n / ), obtained as a resultof the combined effort of diverse research teams. The tight upper bound wasobtained in [7], using a generalization of the Szemer´edi-Trotter Theorem forpoints and “well-behaved” curves in the plane [23], and a matching lowerbound was given in [4] using configurations with the extremal number ofpoint-line incidences.Despite their close relation, the k -set problem and the estimation of g k ( n, d ) arefundamentally distinct problems. Indeed, the k -set problem consists in estimatingthe maximal possible number of vertices of P k ( S ), whereas we are studying thelargest possible subset of k -barycenters that are in convex position. If a k ( n, d )denotes the maximum number of k -sets that an n -point set in R d can have, thenwe trivially have a k ( n, d ) ≤ g k ( n, d ) . However, the converse is far from being true. For example, a ( n,
2) = O ( n ) (see [16,Ch. 11]), whereas g ( n, ∈ Θ( n / ) [4, 7]. LEONARDO MART´INEZ-SANDOVAL AND ARNAU PADROL
In Section 7 we provide an additional discussion of our results and we collectvarious open problems for further work.2.
Projections that strictly preserve the vertices of thehypersimplex
In this section we reformulate Theorem 1.1 in terms of polytope projections thatpreserve vertices. By projections we mean affine maps, although it suffices to focusin surjective linear maps. We assume some familiarity with the basic notions onpolytope theory and refer the reader to [37] for a detailed treatment of the subject.Our main concern is the study of faces strictly preserved under a projection, anotion introduced in Definition 3.1 in [38]. Definition 2.1.
Let P be a polytope and π : P → π ( P ) a projection. A face F ⊆ P is preserved under π if(i) π ( F ) is a face of π ( P ), and(ii) π − ( π ( F )) = F .If moreover(iii) π ( F ) is combinatorially isomorphic to F ,then we say that F is strictly preserved under π .For the restatement of Theorem 1.1 we use the following auxiliary lemma. Recallthat the ( n, k ) -hypersimplex is the polytope:∆ n,k := conv (cid:110) x ∈ { , } n (cid:12)(cid:12)(cid:12) (cid:88) ≤ j ≤ n x j = k (cid:111) . Lemma 2.1.
The existence of a convex embedding of K ( k ) n into R d is equivalent tothe existence of an affine map of the hypersimplex ∆ n,k to R d that strictly preservesits (cid:0) nk (cid:1) vertices.Proof. For n ≥ k ≥
1, let V = { v , . . . , v n } be the vertex set of K ( k ) n . To anyembedding f : V → R d we associate the linear map π : R n → R d given by π ( e i ) = k f ( v i ). Notice that π maps the vertices of ∆ n,k to the barycenters of k -subsetsof f ( V ). These are in convex position if and only if all the vertices of ∆ n,k arestrictly preserved by π . (cid:3) Said differently, the projection π : R n → R d strictly preserves the verticesof ∆ n,k if and only if the point configuration S π := { π ( e i ) | ≤ i ≤ n } has allits k -barycenters in convex position.Lemma 2.1 implies that Theorem 1.1 is a corollary of Theorem 1.2 obtained bysetting i = 0. Thus, from now on we focus on proving Theorem 1.2.For a d -polytope P ⊂ R d and a linear surjection π : R d → R e , the ProjectionLemma [38, Prop. 3.2] gives a criterion to characterize which faces of P are strictlypreserved by π in terms of an associated projection τ : R d → R d − e . More precisely, We are mainly interested in strictly preserved faces, and our definition coincides with thatin [26, 27, 38]. However, our definition of (not necessarily strictly) preserved face differs from thatin [26, 27], where they require conditions (i) and (iii) to define preserved faces. We prefer thisdefinition because it provides a bijection between faces of π ( P ) and preserved faces, and simplifiesthe notation for Section 5. HE HYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM 7 let ι : ker( π ) ∼ = R d − e (cid:44) → R d be the inclusion map of ker( π ). Then τ is the adjointmap ι ∗ : ( R d ) ∗ → ( R d − e ) ∗ after the canonical identifications ( R d ) ∗ ∼ = R d and( R d − e ) ∗ ∼ = R d − e (see [27, Sec. 3.2] for details). Lemma 2.2 (Projection Lemma [38, Prop. 3.2]) . Let P ⊂ R d be a d -polytope, π : R d → R e a linear surjection, and τ : R d → R d − e be the associated projection.Let F ⊂ P be a face of P and let { n j | j ∈ I } be the normal vectors to the facetsof P that contain F . Then F is strictly preserved if and only if { τ ( n j ) | j ∈ I } positively span R d − e ; i.e. if ∈ int conv { τ ( n j ) | j ∈ I } . As final ingredients we need the dimension and hyperplane description of ∆ n,k ,as well as its facial structure. These are well known (see for example [37, Ex. 0.11]).
Lemma 2.3.
The hypersimplex ∆ n,k is the polytope ∆ n,k = (cid:110) (cid:88) j ∈ [ n ] x j = k (cid:111) ∩ (cid:92) j ∈ [ n ] (cid:110) x j ≥ (cid:111) ∩ (cid:92) j ∈ [ n ] (cid:110) x j ≤ (cid:111) . It has (cid:0) nk (cid:1) vertices, which are the points in { , } n whose coordinate sum is k . It isa point for k ∈ { , n } , an ( n − -simplex for k ∈ { , n − } , and for ≤ k ≤ n − it is ( n − -dimensional and has n facets.For ≤ i ≤ n − , its i -faces are of the form ∆ I,Jn,k := ∆ n,k ∩ (cid:92) i ∈ I (cid:110) x i = 1 (cid:111) ∩ (cid:92) j ∈ J (cid:110) x j = 0 (cid:111) , where I, J ⊂ [ n ] are disjoint index sets with | I | ≤ k − , | J | ≤ n − k − and | I | + | J | = n − i − . The i -face ∆ I,Jn,k is isomorphic to ∆ n −| I |−| J | ,k −| I | . From here, we proceed as follows. Consider n fixed and 2 ≤ k ≤ n −
2. Towork with a full ( n − n,k with its projectiononto R n / ( R · ) ∼ = R n − , where represents the all ones vector. We want tostudy when there is an i -preserving projection π : R n − → R d , that is, one thatstrictly preserves every face of the i -skeleton of ∆ n,k . If so, Lemma 2.2 wouldensure certain positive dependencies on the vector configuration induced by theimage of the normal vectors to facets of ∆ n,k under the associated projection τ .We state explicitly these dependencies below. In Section 3 we provide an in-depthstudy of the point configurations that yield vector configurations satisfying thesedependencies and show that if d is not large enough, then they cannot all holdsimultaneously.By the description in Lemma 2.3, for 2 ≤ k ≤ n −
2, ∆ n,k ⊂ R n / ( R · ) has2 n facets whose normal vectors we may pair up as { m j , n j } for j ∈ [ n ], where m j and n j correspond to the inequalities x j ≥ x j ≤ R n / ( R · ), respectively. They satisfy(1) m j + n j = 0 for j ∈ [ n ] and (cid:88) j ∈ [ n ] m j = (cid:88) j ∈ [ n ] n j = 0 . Example . Before we continue, we provide a concrete example of our set-up.Consider Figure 1. At the top of the figure we have the hypersimplex ∆ , , whichis a 3-dimensional octahedron. By construction, its ambient space is R , but weisomorphically project it onto R / ( R · (1 , , , ∼ = R . Thus, when projecting itto the plane we get a map π : R → R . LEONARDO MART´INEZ-SANDOVAL AND ARNAU PADROL
Figure 1.
Using the Projection Lemma on a projection for thehypersimplex ∆ , , which is an octahedron.At the left side of the figure we have the image of ∆ , under π . We also showthe images of 2 e , e , e , e . Note that, as expected by Lemma 2.1, the imagesof the vertices of ∆ , are precisely the midpoints of the edges π (2 e i ) π (2 e j ), for1 ≤ i < j ≤
4, all of which lie in strictly convex position. Even though this isevident from the figure, we can also verify it using the Projection Lemma.To do so, consider the normal vectors to the faces of ∆ , . These are shown at thebottom of the figure with labels m j , n j for j = 1 , , ,
4. Since π has codimension 1,it induces a projection τ that takes these normal vectors to R , which is shown atthe right side of the figure. So, consider for example the vertex (0 , , , m , n , m , n , and 0 is strictly containedin the interior of the convex hull of τ ( { m , n , m , n } ). Thus, by Lemma 2.2 weverify that (0 , , ,
1) is strictly preserved under π .The Projection Lemma also determines whether higher dimensional faces arepreserved or not. Consider for example the edge (0 , , , , , ,
1) of ∆ , . It iscontained in the faces with normal vectors m and n . Note that 0 does not liein the interior of the convex hull of τ ( { m , n } ). By Lemma 2.2 we conclude that HE HYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM 9 π does not preserve the edge, which can be verified by inspection. We invite thereader to check the preservation remaining vertices and faces.To use these tools in general, we combine Lemma 2.2 with the facial structureof the hypersimplex to get: Lemma 2.4.
Let π : R n − → R d be a linear surjection and τ : R n − → R n − d − itsassociated projection. Let ≤ k ≤ n − , and let { n j , m j (cid:48) | j, j (cid:48) ∈ [ n ] } be the normalvectors of ∆ n,k described above. Then τ ( { n j , m j (cid:48) | j, j (cid:48) ∈ [ n ] } ) is an ( n − d − -dimensional configuration of vectors with the following strictly positive dependen-cies: a) ∈ int conv { τ ( m j ) : j ∈ [ n ] } ,b) ∈ int conv { τ ( n j ) : j ∈ [ n ] } ,c) ∈ relint conv { τ ( m j ) , τ ( n j ) } for j ∈ [ n ] ,The projection π is i -preserving if and only if: • If i = 0 , additionally for every disjoint I, J ⊂ [ n ] , | I | = k , | J | = n − k d) ∈ int conv ( { τ ( m j ) : j ∈ J } ∪ { τ ( n j ) : j ∈ I } ) ande) ∈ int conv ( { τ ( m j ) : j ∈ I } ∪ { τ ( n j ) : j ∈ J } ) . • If i ≥ , additionally for every disjoint I, J ⊂ [ n ] such that | I | ≤ k − , | J | ≤ n − k − and | I | + | J | = n − i − thatd’) ∈ int conv ( { τ ( m j ) : j ∈ I } ∪ { τ ( n j ) : j ∈ J } ) ande’) ∈ int conv ( { τ ( m j ) : j ∈ J } ∪ { τ ( n j ) : j ∈ I } ) .Proof. The positive dependencies in a ), b ) and c ) follow directly from the linearityof τ and (1). Note that (1) additionally states that the vector configuration issymmetric around the origin with pairing τ ( m j ) = − τ ( n j )For d ) we use that π preserves the vertices of ∆ n,k . Each vertex of ∆ n,k lies inexactly k hyperplanes of the form x j = 1 and n − k hyperplanes of the form x j = 0.From here we obtain, respectively, complementary index sets I and J of [ n ]. Theconclusion then follows from Lemma 2.2.The analysis for d (cid:48) ) is similar considering the description of the i -faces of ∆ n,k given in Lemma 2.3.Finally, the family of positive dependencies in e ) and e (cid:48) ) follow respectively from d ) and d (cid:48) ) and the symmetry around the origin. (cid:3) Since the configuration of vectors is symmetric around the origin, we obtain aproof of the following observation.
Corollary 2.5. An i -preserving projection π : R n − → R d exists for ∆ n,k if andonly if it exists for ∆ n,n − k . Of course, ∆ n,k and ∆ n,n − k are affinely equivalent, so Corollary 2.5 should notbe too unexpected. However, the fact that cd ( K ( k ) n ) = cd ( K ( n − k ) n ) is not entirelyobvious from the definition of cd . It has an alternative short geometric proof. Sup-pose f is a convex embedding of K ( k ) n into R d . Consider the barycenter b of f ( V ).The barycenter a of any k -subset of f ( V ), the barycenter c of the complemen-tary ( n − k )-subset and b are collinear. The segment ac is split in ratio k : n − k by b . Therefore, the set of ( n − k )-barycenters is a homothetic copy of the set of k -barycenters. Since the second is in convex position, the first one is as well. Hypersimplicial-neighborly configurations
In this section we prove Theorem 1.2. By Corollary 2.5 we may focus only onthe cases n ≥ k , and therefore it is enough to prove the following: d ( n, k, i ) = k + 2 i for n ≥ k + 2 i + 2 n − k ≤ n ≤ k + 2 i + 1, k ∈ A n,i ,n − k ≤ n ≤ k + 2 i + 1, k / ∈ A n,i . Recall that A n,i = { , , . . . , i + 1 } ∪ { n − i − , n − i, . . . , n − } is the range of someexceptional values for k .We actually prove a stronger result, as we completely characterize all projections π : R n − → R d that preserve the i -skeleton of ∆ n,k . In the remaining of thesection we assume n ≥ k ≥ π is linear and surjective and τ : R n − → R n − d − denotes the projection associated to π .Our characterization is in terms of the point configuration S π := { π ( e i ) | i ∈ [ n ] } ; more precisely, in terms of its neighborliness and almost neighborliness. Recallthat a point configuration S is called j -neighborly if every subset of at most j pointsof S is the vertex set of a face of conv( S ) (thus 1-neighborly corresponds to beingin convex position), and S is j -almost neighborly if every subset of at most j pointsof S lies in a common face of conv( S ).The relation with neighborliness was already observed by Halman, Onn andRothblum, who used cyclic 2 k -polytopes (which are k -neighborly) to provide convexembeddings of k -uniform hypergraphs into R k [14]. Their observation can beextended to higher dimensional skeleta of the hypersimplex. Lemma 3.1. If ≤ k ≤ n and S π is ( k + i ) -neighborly, then π is an i -preservingprojection of ∆ n,k .Proof. Since S π is ( k + i )-neighborly, any set of at most k + i vertices form a simplexface of S π .For a subset A ⊂ [ n ] of size k + i , consider the faces F = ∆ n,k ∩ j / ∈ A { x j = 0 } and G = conv { π ( e i ) | i ∈ A } of ∆ n,k and S π , respectively. Then π restricted to theaffine span of F is an affine isomorphism into the affine span of G . The supportinghyperplane for G in S π is also supporting for π ( F ) in π (∆ n,k ), and hence F isstrictly preserved.We conclude the proof by observing that any i -face of ∆ n,k belongs to one sucha face F . (cid:3) We will also use Gale duality (see [37, Lec. 6] or [16, Sec. 5.6] for nice introduc-tions). A short computation leads to the following observation. It can also be easilyseen by comparing our coordinate-free definition for τ with Ewald’s introductionto Gale transforms [8, Sec II.4]. We omit the details. Lemma 3.2.
The vector configuration M = { τ ( m i ) | ≤ i ≤ n } is a Gale trans-form of S π . A first consequence of this observation is that, for a point configuration S , theproperty of having all k -barycenters in convex position only depends on its un-derlying oriented matroid (in particular, it is invariant under admissible projectivetransformations, i.e. those where the hyperplane at infinity does not separate S ).Indeed, by the Projection Lemma 2.2 (and Lemma 2.4), the facial structure of HE HYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM 11 P k ( S ), the convex hull of the k -barycenters, only depends on the oriented matroidof the vector configuration { τ ( m j ) , τ ( n j ) | j ∈ [ n ] } , which is completely determinedby the oriented matroid of M by the central symmetry. This might be counter-intuitive at first, as barycenters are not preserved by projective transformations.However, the interpretation in terms of k -sets in Section 5 gives a clear explanationfor why the combinatorial type of P k ( S ) is an oriented matroid invariant.We will use the Gale dual characterization of neighborliness and almost neigh-borliness. This result is well known (see for example [18]), but we include a shortschema for the proof for completeness. Lemma 3.3. S π is j -neighborly if and only if every open linear halfspace containsat least j + 1 vectors of M , and S π is j -almost neighborly if every closed linearhalfspace contains at least j + 1 vectors of M .Proof. Using standard properties of the Gale transform characterizing faces [16,Cor. 5.6.3], j -neighborliness of S π means that every subset of j points is the vertexset of a simplex face of S π , which is equivalent to 0 being in the interior of theconvex hull of every subset of n − j points of M . By Farkas’ lemma, this is equiv-alent to every linear open halfspace containing at least one point of each subsetof M of cardinal n − j , which is equivalent to the stated condition. For almost-neighborliness, we remove the condition of being in the interior, which translatesto closed halfspaces instead. (cid:3) This provides an easy alternative proof of Lemma 3.1. We present it for conve-nience to the reader in order to provide extra insight into our later arguments.
Dual proof for Lemma 3.1. If i ≥
1, let
I, J ⊂ [ n ] be disjoint subsets such that | I | + | J | = n − i − | I | ≤ k −
1, and | J | ≤ n − k −
1. And if i = 0, let I, J ⊂ [ n ] bedisjoint subsets such | I | = k , and | J | = n − k . In both cases we have | J | ≥ n − i − k .To prove our claim, it suffices to show that 0 ∈ int conv( { τ ( m j ) | j ∈ J } ). By Farkas’Lemma, this is equivalent to showing that every linear open halfspace contains atleast one τ ( m j ) with j ∈ J . This holds because | H + ∩ M | ≥ k + i +1 (by Lemma 3.3)and J is missing at most i + k vectors from M . (cid:3) This viewpoint also allows us to provide another family of examples of i -preservingprojections. Lemma 3.4. If ≤ k ≤ n and S π is a ( n − -dimensional configuration of n pointsthat is not ( k − i − -almost neighborly, then π : R n − → R n − is an i -preservingprojection of codimension of ∆ n,k .Proof. Let M + := { j ∈ [ n ] | τ ( n j ) > } , M − := { j ∈ [ n ] | τ ( n j ) < } , and M := { j ∈ [ n ] | τ ( n j ) = 0 } .By Lemmas 3.2 and 3.3, S π not being ( k − i − {| M + ∪ M | , | M − ∪ M |} ≤ k − i − . We may assume that | M + ∪ M | ≤ | M − ∪ M | , so | M + ∪ M | ≤ k − i −
1. We pick
I, J ⊂ [ n ] such that I ∩ J = ∅ , | I | ≤ k − | J | ≤ n − k − | I | + | J | = n − i − | J | ≤ n − k −
1, we have that | I | ≥ k − i . Since n ≥ k , | J | ≥ n − k − i ≥ k − i .Therefore, we may choose j ∈ I ∩ M − and j (cid:48) ∈ J ∩ M − , which verify τ ( m j ) < τ ( m j (cid:48) ) <
0. So τ ( n j ) = − τ ( m j ) >
0. Therefore, 0 is in the desired interior ofthe convex hull. (cid:3)
We can give an explicit description of the examples provided by Lemma 3.4. Full-dimensional onfigurations of d +2 points in R d are well classified [13, Sec. 6,1]. Theyare all (up to admissible projective transformations) of the form pyr k (∆ n ⊕ ∆ m ),for k, n, m ∈ N such that k + n + m = d . Here ∆ n represents (the vertex set of) an n -dimensional simplex , the direct sum P ⊕ Q is the (dim( P ) + dim( Q ))-dimensionalconfiguration obtained by taking a copy of P and Q whose convex hulls intersectin a point in the relative interior of both, and the k -fold pyramid pyr k ( P ) is the( k + dim( P ))-dimensional configuration obtained by adding k affinely independentpoints. Note that pyr k (∆ n ⊕ ∆ m ) is min( n, m )-neighborly (but not (min( n, m )+1)-neighborly) and (min( n, m ) + k )-almost neighborly (but not (min( n, m ) + k + 1)-almost neighborly). Hence ∆ n − ⊕ ∆ , which consists of an ( n − i -preserving projection for ∆ n,k whenever k ≥ i + 2; whereas ∆ (cid:98) n / (cid:99)− ⊕ ∆ (cid:100) n / (cid:101)− corresponds to an i -preserving projection if k + i + 1 ≤ (cid:4) n (cid:5) .As it turns out, the configurations given by Lemmas 3.1 and 3.4 describe allpossible i -preserving projections. We show this first for the cases of codimension 1and then for those of larger codimension. Proposition 3.1. If ≤ k ≤ n , the surjective projection π : R n − → R n − is i -preserving for ∆ n,k if and only if either(i) S π is ( k + i ) -neighborly, or(ii) S π is not ( k − i − -almost neighborly.That is, if either(i) min {| M + | , | M − |} ≥ k + i + 1 , or(ii) min {| M + ∪ M | , | M − ∪ M |} ≤ k − i − ;where M + := { j ∈ [ n ] | τ ( n j ) > } ,M − := { j ∈ [ n ] | τ ( n j ) < } and M := { j ∈ [ n ] | τ ( n j ) = 0 } . Proof.
We have already seen that if S π verifies (i) or (ii), then π is i -preserving.Now we show that no other projection π of codimension 1 can be i -preserving.If C π is not ( k + i )-neighborly, then min {| M + | , | M − |} ≤ k + i . Without loss ofgenerality, say | M + | ≥ | M − | and | M − | ≤ k + i . If moreover C π is ( k − i − | M + ∪ M | ≥ k − i and | M − ∪ M | ≥ k − i . We also have | M + ∪ M | = n − | M − | ≥ n − k − i. Assume first i ≥
1. Now, • If | M + | ≥ n − k −
1, let J be a subset of M + of size n − k − I be asubset of M − ∪ M of size k − i . • If | M + | ≤ n − k −
2, let Z ⊂ M be a minimal (maybe empty) set suchthat n − k − i ≤ | M + ∪ Z | ≤ n − k − , define J = M + ∪ Z and let I be a subset of [ n ] \ J ⊆ M − ∪ M of size n − i − − | J | .If i = 0, HE HYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM 13 • If | M + | ≥ n − k , let J be a subset of M + of size n − k and I be a subsetof M − ∪ M of size k . • If | M + | ≤ n − k −
1, let Z ⊂ M be a set such that n − k = | M + ∪ Z | , define J = M + ∪ Z and let I = [ n ] \ J .In both cases, we obtain disjoint subsets I , J that satisfy • | I | ≤ k − | J | ≤ n − k − | I | + | J | = n − i − i ≥ • | I | = k , | J | = n − k when i = 0, and • τ ( m j ) ≥ j ∈ I and τ ( n j ) ≥ j ∈ J .Then 0 is not in the interior of the convex hull of { τ ( m j ) : j ∈ I } ∪ { τ ( n j ) : j ∈ J } , and thus π is not i -preserving by Lemma 2.4. (cid:3) Proposition 3.2. If ≤ k ≤ n , the projection π : R n − → R n − is i -preservingfor ∆ n,k if and only if S π is ( k + i ) -neighborly.That is, if for every linear open halfplane H + we have (2) | H + ∩ M | ≥ k + i + 1 . Proof.
Let M = τ ( { m j | j ∈ [ n ] } ) ⊂ R and N = τ ( { n j | j ∈ [ n ] } ) ⊂ R . Byconstruction, M ∪ N is a centrally symmetric 2-dimensional vector configuration.And π is i -preserving if and only if some prescribed positive dependencies areverified, by Lemma 2.4. We will show that these dependencies are equivalent tocondition (2).The fact that (2) implies that π is i -preserving is a direct corollary of Lemmas 3.1and 3.3. Hence, it suffices to prove the converse, that is, that if π is i -preservingthen (2) holds. Note that it is sufficient to show that we have (2) for the halfplanessupported by lines spanned by the vectors in M .Consider an oriented line (cid:96) through the origin. Let A ⊂ [ n ] and B ⊂ [ n ] bethe indices of the vectors of M strictly to the right and left of (cid:96) respectively. Let C ⊂ [ n ] index the vectors of M on the open ray from 0 with the same directionas (cid:96) and D ⊂ [ n ] the vectors of M on the opposite ray (see Figure 2). Then E = [ n ] \ ( A ∪ B ∪ C ∪ D ) are the indices of the vectors that are copies of 0. Let a = | A | , b = | B | , c = | C | , d = | D | , and e = | E | .The same arguments of the proof of Proposition 3.1 show that we must haveeither(i) min { a, b } ≥ k + i + 1, or(ii) min { a + c + d + e, b + c + d + e } ≤ k − i − (cid:96) clockwise. The sign of a − b changes after half a completerotation. This ensures that we can find a position of the line for which a + c ≥ b + d , b + c ≥ a + d and c + d (cid:54) = 0, as the switch has to take place at one of the linesspanned by a vector of M .We claim that (i) holds. Indeed, if a + c + d + e ≤ k − i −
1, then b = n − ( a + c + d + e ) ≥ k − ( k − i −
1) = k + i + 1. Hence, we have b + d ≥ k + i + 1 ≥ k − i − ≥ a + c , a contradiction. And analogously, if b + c + d + e ≤ k − i − a + d ≥ k + i + 1 ≥ k − i − ≥ b + c . Figure 2.
The partition
A, B, C, D, E induced by an oriented line.In this example, a = 2, b = 3, c = 4, d = 2, and e = 0, and wehave a + c = 6 ≥ b + d , b + c = 7 ≥ a + d and c + d = 6 (cid:54) = 0.We conclude that there is at least a line for which (i) holds. Now, assume that (cid:96) is a line for which (i) holds, and let (cid:96) (cid:48) be the next line spanned by M in clockwiseorder. It defines a new partition, with corresponding values a (cid:48) , b (cid:48) , c (cid:48) , d (cid:48) , e (cid:48) . We have a (cid:48) = a − c (cid:48) + d , b (cid:48) = b − d (cid:48) + c and e (cid:48) = e . Thus, a (cid:48) + c (cid:48) + d (cid:48) + e (cid:48) = a + d + d (cid:48) + e (cid:48) ≥ a ≥ k + i + 1 > k − i − , and b (cid:48) + c (cid:48) + d (cid:48) + e (cid:48) = b + c + c (cid:48) + e (cid:48) ≥ b ≥ k + i + 1 > k − i − . Which shows that (ii) cannot hold. And hence that (i) must hold for (cid:96) (cid:48) , and, byinduction, for all lines spanned by M . (cid:3) Corollary 3.5. If ≤ k ≤ n and (cid:96) ≥ , the projection π : R n − → R n − − (cid:96) is i -preserving for ∆ n,k if and only if S π is ( k + i ) -neighborly.That is, if for every linear open halfspace H + we have (2) | H + ∩ M | ≥ k + i + 1 . Proof.
Actually, the proof of Proposition 3.2 extends to the general case almostverbatim. The direct implication follows from Lemma 3.1. For the converse, let F be a ( d − M ∩ H . Then we can repeat theargument of the proof of Proposition 3.2 by pivoting a hyperplane H containing F clockwise around F to conclude that all hyperplanes containing F verify (2). (cid:3) Theorem 3.6. If ≤ k ≤ n and ≤ i ≤ n − , the projection π : R n − → R d is i -preserving for ∆ n,k if and only if(i) S π is ( k + i ) -neighborly,(ii) S π is ( n − -dimensional and not ( k − i − -almost neighborly, or(iii) π is an affine isomorphism.In particular, such a projection exists only if(i) k + 2 i ≤ d ≤ n − (or d = n − ≥ k + i ),(ii) d = n − and k ≥ i + 2 , or(iii) d = n − . HE HYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM 15
Proof. If π is an isomorphism, then d = n − S π is just the vertex set of a ( n − k ≥ i + 2. In this case, the configuration of n points in R n − consisting of the vertex set of an ( n − j -neighborly polytopes in all dimensions d ≥ j . For example the cyclic polytopes which are the convex hulls of n pointsin the moment curve (cid:8) ( t, t , . . . , t d ) (cid:12)(cid:12) t ∈ R (cid:9) ⊂ R d are (cid:4) d (cid:5) -neighborly [37, Cor. 0.8].It is also well known that no d -polytope other than the simplex is more than (cid:4) d (cid:5) -neighborly [37, Exercice 0.10]. (cid:3) Theorem 1.2 follows directly from this.
Proof of Theorem 1.2. If n ≥ k + 2 i + 2, then 2 k + 2 i ≤ n − k + i )-neighborly polytope in R k +2 i . If 2 k ≤ n ≤ k +2 i +1, thenthere is no neighborly embedding except for the isomorphism, and hence whether d ( n, k, i ) = n − d ( n, k, i ) = n − k ≥ i +2. That is if k / ∈ A n,i ,then the simplex with an interior point works, otherwise the only embedding is theisomorphism. The remaining cases follow by symmetry. (cid:3) Strong convex embeddings
Halman, Onn and Rothblum [14] also define the following notions. A strongconvex embedding for a hypergraph H = ( V, E ) is a convex embedding f : V → R d for which f ( V ) is also in convex position; and the strong convex dimension scd ( H )of H is the minimal d for which a strong convex embedding of H into R d exists.Strong convex embeddings have some intrinsic interest (and the associated ex-tremal problems have a different behavior, see Section 7). They also yield usefulinformation on (normal) convex embeddings. For example, Swanepoel and Valtr[32] use the non-existence of a strong convex embedding for K into R as a keystep in proving that there is no (normal) convex embedding for K , , , , into R .Here we exploit our characterization of i -preserving projections for ∆ n,k to de-termine scd ( K ( k ) n ). In fact, we determine the value of d (cid:48) ( n, k, i ), the smallest di-mension for which we can find a projection π : ∆ n,k → R d that strictly preservesthe i -dimensional skeleton of ∆ n,k and for which the associated point configuration S π is in convex position. Corollary 4.1.
Given positive integers n , k , i such that ≤ k ≤ n − and ≤ i ≤ n − , the value of d (cid:48) ( n, k, i ) is determined as follows. d (cid:48) ( n, k, i ) = k + 2 i if n ≥ k + 2 i + 2 , n − k + 2 i if n ≤ k − i − ,n − if k − i − ≤ n ≤ k + 2 i + 1 , k ∈ C n,i ,n − if k − i − ≤ n ≤ k + 2 i + 1 , k / ∈ C n,i . Where C n,i = { , , . . . , i + 2 } ∪ { n − i − , n − i − , . . . , n − } . Note that C n,i = A n,i ∪{ i +2 , n − i − } , so we are saying that d (cid:48) ( n, k, i ) = d ( n, k, i )except for the cases 2 k − i − ≤ n ≤ k + 2 i + 1 and k ∈ { i + 2 , n − i − } , inwhich d (cid:48) ( n, k, i ) = d ( n, k, i ) + 1 = n −
1. Thus, d and d (cid:48) have essentially the samebehavior, except from a very specific case. Proof. If n ≥ k + 2 i + 2, then 2 k + 2 i ≤ n − k + i )-neighborly polytope in R k +2 i , whose vertices are in convex position.If 2 k ≤ n ≤ k + 2 i + 1, then there is no ( k + i )-neighborly embedding except forthe isomorphism, and hence whether d (cid:48) ( n, k, i ) = n − d (cid:48) ( n, k, i ) = n − k − i − k = i + 2,then the embedding must be non 1-almost neighborly, so the vertices cannot bein convex position, hence in this case d (cid:48) ( n, k, i ) = n −
1. Finally, if k ≥ i + 3,then the projection with point configuration ∆ n − ⊕ ∆ is 1-almost neighborly, butnot 2-almost neighborly. Hence it has all its vertices in convex position and it is i -preserving.The remaining cases follow by symmetry. (cid:3) Relation with k -sets and ( i, j ) -partitions Let S = { s , . . . , s n } be a finite point set in R d . A subset of S of cardinality k iscalled a k -set of S if it is the intersection of S with an open halfspace. Studying themaximal possible number of k -sets is a central problem in combinatorial geometry,and only partial results are known. We refer to [16, Ch. 11] for an introduction tothe topic, to [35] for a larger survey, and to [29] for the latest improvement.In [6], the k -set polytope P k ( S ) is defined as the convex hull of all k -barycentersof S . Note that, if π : R n → R d is the linear projection with π ( e i ) = k s i , then P k ( S ) = π (∆ n,k ) ( k -set polytopes are defined without the k factor in some refer-ences like [3]). Hence, k -set polytopes for point sets of cardinality n are projectionsof the ( n, k )-hypersimplex. The importance of P k ( S ) lies in the fact that its verticesare in bijection with the k -sets of S .In [3], Andrzejak and Welzl studied further the facial structure of k -set polytopes.To this end, they define an ( i, j )-partition of S as a pair ( A, B ) of subsets of S with | A | = i and | B | = j for which there is an oriented hyperplane H such that A = S ∩ H , and B = S ∩ H > , where H > is the positive open halfspace definedby H . These generalize k -sets (which are (0 , k )-partitions) and j -facets of point setsin general position (which are ( d, j )-partitions). Denoting by D i,j ( S ) the number of( i, j )-partitions of S , Andrzejak and Welzl observed that, for S in general position, f i − ( P k ) = , if i = 0 ,D ,k , if i = 1 , (cid:80) k − j = k − ( i − D i,j otherwise;which allowed them to use Euler’s relation on P k to derive linear relations on thenumbers of ( i, j )-partitions. See also [2, Sec. 3.2], which considers configurationsthat are not in general position.In this section, we provide an interpretation in terms of preserved faces underhypersimplex projections.We define the dimension of an ( i, j )-partition ( A, B ) as the dimension of theaffine hull of A . Of course, if S is in general position, then the dimension of any( i, j )-partition is i − HE HYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM 17
Proposition 5.1.
Let P k ( S ) = π (∆ n,k ) ⊂ R d as before. Then(i) the vertices of P k ( S ) are in bijection with the k -sets of S , and(ii) the e -faces of P k ( S ) with e ≥ are in bijection with e -dimensional ( i, j ) -partitions with j + 1 ≤ k ≤ i + j − .More precisely, for disjoint subsets X, Y ⊆ [ n ] with | Y | + 1 ≤ k ≤ | X | + | Y | − ,and A = { s i | i ∈ X } and B = { s j | j ∈ Y } , we have that ( A, B ) is an e -dimensional ( i, j ) -partition if and only if the face F = ∆ I,Jn,k is preserved under π , and π ( F ) is e -dimensional; where I = Y and J = [ n ] \ ( X ∪ Y ) .In particular, the strictly preserved e -faces with e ≥ are in bijection with the ( i − -dimensional ( i, j ) -partitions with j + 1 ≤ k ≤ i + j − .Proof. Faces of P k ( S ) are in bijection with faces of ∆ n,k preserved under π . Moreprecisely, the face of P k ( S ) that maximizes the linear functional f ∈ ( R d ) ∗ is theimage of the face F of ∆ n,k maximized by π ∗ ( f ), where π ∗ denotes the adjoint of π .That is, the vertices of F are the incidence vectors of the S ∈ (cid:0) [ n ] k (cid:1) that maximizes (cid:80) i ∈ S f ( s i ), where s i = π ( e i ).Fix f ∈ ( R d ) ∗ and let e ∈ (cid:0) [ n ] k (cid:1) be one of the subsets that maximizes (cid:80) i ∈ e f ( s i ).Set c = min i ∈ e f ( s i ), X = { i ∈ [ n ] | f ( s i ) = c } and Y = { i ∈ [ n ] | f ( s i ) > c } . Thehyperplane f ( x ) = c defines an ( | X | , | Y | )-partition ( A, B ) with A = { s i | i ∈ X } and B = { s j | j ∈ Y } . Note that every i ∈ Y belongs to e , and that for e (cid:48) ∈ (cid:0) [ n ] k (cid:1) we have (cid:80) i ∈ e (cid:48) f ( s i ) = (cid:80) i ∈ e f ( s i ) if and only if Y ⊆ e (cid:48) and e (cid:48) ⊆ Y ∪ X . Therefore, the faceof ∆ n,k maximized by π ∗ ( f ) is precisely F = ∆ I,Jn,k with I = Y and J = [ n ] \ ( X ∪ Y ).If F is not a vertex, then we have | Y | ≤ k − | X ∪ Y | ≥ k + 1. Conversely,every ( i, j )-partition with j + 1 ≤ k ≤ i + j − π ( F ) is affinely equivalent to P k − j ( A ),which has the same dimension as the affine span of A provided that 1 ≤ k − j ≤ i − P k ( S ) must be a vertex of ∆ n,k . Indeed,while it is true that there might be subsets e, e (cid:48) ∈ (cid:0) [ n ] k (cid:1) for which (cid:80) i ∈ e (cid:48) s i = (cid:80) i ∈ e s i ,such a subset cannot define a vertex of P k ( S ). The reason is that, if all the s i ’s aredifferent, and if f is a generic functional maximized at (cid:80) i ∈ e f ( s i ), then there mustbe some j ∈ e (cid:48) \ e with f ( s j ) > k (cid:80) i ∈ e f ( s i ), and hence a subset e (cid:48)(cid:48) ∈ (cid:0) [ n ] k (cid:1) with (cid:80) i ∈ e (cid:48)(cid:48) f ( s i ) > (cid:80) i ∈ e f ( s i ), contradicting the maximality of e . (cid:3) Hypergraphs with many barycenters in convex position
Now we study the extremal function g k ( n, d ) that counts the maximum number ofbarycenters in convex position that a k -uniform hypergraph on n vertices embeddedin R d may have. As explained in Theorem 1.3, we distinguish three regimes: d ≥ k , k + 1 ≤ d ≤ k − d ≤ k . By Theorem 1.1, we have g k ( n, d ) = (cid:0) nk (cid:1) for d ≥ k ,which covers the first case.By Theorem 1.1, we also know that g k ( n, d ) < (cid:0) nk (cid:1) when d ≤ k −
1. Bycombining this result with de Caen’s bound on Tur´an numbers for hypergraphs [5]we can get sharper upper bounds for g k when d ≤ k −
1, as n grows.Fix k and 1 ≤ d ≤ k −
1. Using Theorem 1.1, we obtain that the maximumvalue n = n k,d so that K ( k ) n has a convex embedding into R d is for d ≥ n k,d = (cid:4) d (cid:5) + k if 1 ≤ d ≤ k − ,d + 2 if d ∈ { k − , k − , k − }∞ if d ≥ k. (3)and for d = 1, n , = 2, n k, = k for k ≥ n k,d are contained in Table 3. k \ d ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ Table 3.
Values of n k,d for small values of d and k . Yellow valuesare the cases d ∈ { k − , k − , k − } , red values are the cases1 ≤ d ≤ k −
4, and green values are the exceptional cases with d = 1 where the standard formula does not hold.We recall the following bound for Tur´an numbers for complete hypergraphs byde Caen [5]: Theorem 6.1. A k -uniform hypergraph with no complete K ( k ) (cid:96) as an induced sub-hypergraph can have at most EX ( n, k, (cid:96) ) ≤ (cid:32) − n − (cid:96) + 1 n − k + 1 · (cid:0) (cid:96) − k − (cid:1) (cid:33) (cid:18) nk (cid:19) edges. Theorem 6.2.
For ≤ d ≤ k − we have g k ( n, d ) ≤ c k,d · n k + o ( n k ) , where c k,d = 1 k ! (cid:32) − (cid:0) n k,d k − (cid:1) (cid:33) , for n k,d as defined in (3) .Proof. If a k -uniform hypergraph G has convex dimension d , then any induced sub-hypergraph must also have convex dimension d . In particular, its largest completesub-hypergraph cannot have more than n k,d vertices. Therefore, we may apply deCaen’s bound with (cid:96) = n k,d + 1 to obtain that G has at most EX ( n, k, n k,d + 1)edges.The result follows by using that (cid:0) nk (cid:1) = n k k ! + o ( n k ) and collecting the n − n k,d n − k +1 coefficient in the o ( n k ) term. (cid:3) HE HYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM 19
For d ≥ k + 1, we have an accompanying lower bound for g k of the same as-ymptotic order. Denote by K ( k ) m × n = K ( k ) n,...,n the complete m -partite k -uniformhypergraph with m parts of n vertices each. One obtains a first bound of size g k ( n, d ) ≥ ( n / k ) k + o ( n k ) by showing that K ( k ) k × n has a convex embedding into R k +1 .This can be done using a particular case of a result by Matschke, Pfeifle and Pi-laud [17]. Namely, Theorem 2.6 in [17] (with parameters r = k , k = 0 and n i = n for each i) provides k sets of n points in R k +1 whose Minkowski sum has all thepossible n k vertices. Mapping the vertices of K ( k ) k × n to these sets gives the desiredconvex embedding.For k = 2 and d = 3, this gives a lower bound of order n / + o ( n ). However,for this case a better lower bound of size (cid:4) n / (cid:5) was found by Swanepoel and Valtrin [32, Theorem 6] using a convex embedding of K n,n,n into R .We close the gap by providing below an improved lower bound of size g k ( n, d ) ≥ (cid:18) dk (cid:19) (cid:16) nd (cid:17) k + o ( n k )for any d ≥ k + 1. One can easily verify that this bound is increasing with d byusing Bernoulli’s inequality. In particular, we have (cid:18) dk (cid:19) (cid:16) nd (cid:17) k ≥ n k ( k + 1) ( k − ≥ (cid:16) nk (cid:17) k ;which shows that our bound improves the one arising from the construction in [17].To do so, we construct a convex embedding of K ( k ) d × n into R d for any d ≥ k + 1. Theorem 6.3.
For fixed d ≥ k + 1 , there is a convex embedding of the complete d -partite k -uniform hypergraph K ( k ) d × n into R d . Therefore, g k ( n, d ) ≥ (cid:0) dk (cid:1) (cid:0) nd (cid:1) k + o ( n k ) as n → ∞ .Proof. Let e , . . . , e d be the standard basis vectors of R d , and set e = − (cid:80) d − i =1 e i (notice that the sum starts at 1 and ends at d − A of n distinct positive real numbers. For any k -subset S of { , . . . , d − } we define the set X S , of cardinality n k as X S := (cid:40)(cid:88) i ∈ S ( a i · e i + a i · e d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i ∈ A for each i ∈ S (cid:41) . Let X := (cid:83) S X S be the union of these point sets for all k -subsets of { , . . . , d − } .Note that | X | = (cid:0) nd (cid:1) n k , as the subsets are disjoint (here we use the positivity of A ).We will prove that all the points in X are in convex position. Notice that for anypair of k -subsets S and T , there is a linear automorphism of X that sends X S to X T . Hence, it suffices to show that the points of X S are vertices of conv( X ) for S = { , . . . , k } .We do so by exhibiting a supporting hyperplane for each of these points. Fix apoint p = (cid:80) i ∈ S ( a i · e i + a i · e d ) ∈ X S , and consider the linear functional v ∈ ( R d ) ∗ given by v = − e ∗ d + (cid:80) i ∈ S a i · e ∗ i . Then we have that (cid:104) v , p (cid:105) = (cid:80) i ∈ S a i . We will seethat (cid:104) v , q (cid:105) < (cid:80) i ∈ S a i for any other q ∈ X . Let q = (cid:80) i ∈ T ( b i · e i + b i · e d ) ∈ X T ,and set b i = 0 for any 0 ≤ i ≤ d − T . We use k to differentiate their parameter from our variable k . Then, using that a i , b i ≥ (cid:104) v , q (cid:105) = (cid:88) i ∈ S (2 a i b i − a i b − b i ) − (cid:88) i ∈ T (cid:114) S b i ≤ (cid:88) i ∈ S ( a i − ( a i − b i ) ) ≤ (cid:88) i ∈ S a i , which can only be an equality if S = T and a i = b i for all i ∈ S ; that is, if p = q .This shows that all the points in X S are vertices of conv( X ), and, by symmetry,that all the points of X are vertices of conv( X ).To conclude the proof, let V = V ∪· · ·∪ V d be the vertex set of K ( k ) d × n , and considera map f : V → R d that maps bijectively each V i to (cid:8) a · e i − + a · e d (cid:12)(cid:12) a ∈ A (cid:9) .Any hyperedge from K ( k ) d × n is obtained by choosing a k -subset S of { , . . . , d − } and then one vertex from each V i with i ∈ S . And hence every k -barycenter isprecisely of the form 1 k (cid:88) i ∈ S ( a i · e i + a i · e d )with a i ∈ A for i ∈ S . All these barycenters lie in convex position (they form k X ),so f is indeed a convex embedding into R d . (cid:3) Combining Theorems 6.2 and 6.3, we get the following estimation for the coeffi-cient of n k in the asymptotic development of γ k,d in the range k + 1 ≤ d ≤ k − Corollary 6.4. If k + 1 ≤ d ≤ k − , then g k ( n, d ) is in Θ( n k ) . More precisely, g k ( n, d ) = γ k,d · n k + o ( n k ) for some constant γ k,d verifying (cid:18) dk (cid:19) d k ≤ γ k,d ≤ k ! (cid:32) − (cid:0) n d,k k − (cid:1) (cid:33) where n k,d = (cid:40) d + 2 if d ≥ k − , (cid:4) d (cid:5) + k if ≤ d ≤ k − . for d (cid:54) = 1 , and n , = 2 and n k, = k for k ≥ . For d ≤ k , we do not know the asymptotic order of g k ( n, d ). We will show that g k ( n, d ) ∈ O ( n d ) and that g k ( n, d ) ∈ Ω( n d − ). The proof of the upper bound isinspired from the analogous bound for k -sets. Theorem 6.5. g k ( n, d ) ≤ (cid:18)(cid:18) n − d (cid:19) + (cid:18) n − d − (cid:19) + · · · (cid:18) n − (cid:19)(cid:19) ∈ O ( n d ) . Proof.
Let H = ( V, E ) be a k -uniform hypergraph of convex dimension d . Let S = { s , . . . , s n } ⊂ R d be the images of the vertices for some convex embedding.Set P H ( S ) = 1 k conv (cid:40)(cid:88) i ∈ e s i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ∈ E (cid:41) . Since the embedding is convex, for every k -subset e ∈ E we have that (cid:80) i ∈ e f ( s i )is a vertex of P H ( S ), and hence there is a linear functional f ∈ ( R d ) ∗ such that (cid:80) i ∈ e f ( s i ) > (cid:80) i ∈ e (cid:48) f ( s i ) for every e (cid:48) ∈ E \ e . Therefore, there is a closed halfspace HE HYPERSIMPLICIAL VAN KAMPEN-FLORES THEOREM 21 H + such that e is the only hyperedge contained in H + ∩ S . The maximal numberof different subsets cut out by halfspaces is well known to be2 (cid:18)(cid:18) n − d (cid:19) + (cid:18) n − d − (cid:19) + · · · (cid:18) n − (cid:19)(cid:19) (see [3, Proof of Thm 2.2] for a proof). (cid:3) We conclude with a construction of Weibel [36] for Minkowski sums of polytopesthat will provide a lower bound of Ω( n d − ) for g k ( n, d ). (We refer to [36, Sect. 5]for the details of the construction.) Theorem 6.6 (Theorem 3 in [36]) . Let k ≥ d , and n ≥ d + 1 , then there exist d -dimensional polytopes P , . . . , P k ⊂ R d with n vertices such that P + · · · + P k has θ ( n d − ) vertices. Corollary 6.7.
For k ≥ d , there is a subhypergraph of the complete k -uniform k -partite hypergraph K ( k ) n,n,...,n with θ ( n d − ) hyperedges that has a convex embeddinginto R d . Therefore, we have that g k ( n, d ) is in Ω( n d − ) as n → ∞ .Proof. Consider the polytopes P i ⊂ R d , 1 ≤ i ≤ k , with vertices { p i , . . . , p in } from Theorem 6.6. We define the subhypergraph H of K ( k ) n,n,...,n with vertex set V = V ∪ · · · ∪ V k with V i = { v i , . . . , v in } for i ∈ [ k ]and whose hyperedges correspond to the k -tuples ( v j , . . . , v j k ) such that p j + · · · + p kj k is a vertex of P + · · · + P k . Then the map that sends v ij to p ij is a convexembedding. (cid:3) Discussion and open problems
Other hypergraphs:
After studying complete hypergraphs, it would be interestingto determine the convex dimension of other families of uniform hypergraphs. Aparticularly interesting family of uniform hypergraphs that comes to mind are (setsof bases of) matroids . The corresponding polytopes, known as matroid polytopes ,have been extensively studied and many of their properties are well understood.They are in particular a relevant family in the context of the convex combinatorialoptimization problems that originally motivated the study of the convex dimen-sion of hypergraphs [22]. The associated optimization problem is known as convexmatroid optimization [21].
Asymptotic behavior of g k ( n, d ) : It is a challenging question to understand theasymptotic growth of g k ( n, d ) when d ≤ k . For d ≤ k , we know that g k ( n, d ) ∈O ( n d ) ∩ Ω( n d − ), and we believe that g k ( n, d ) ∈ Θ( n α d,k ) for some rational d − <α d,k < d . This is what happens for d = k = 2, where it is known that α , = [4, 7].When k + 1 ≤ d ≤ k −
1, we know that g k ( n, d ) = γ d,k · n k + o ( n k ), but the exactvalue of γ d,k is unknown (see Theorem 1.3), even when k = 2 and d = 3 [32]. Large subsets of Minkowski sums:
Restricting this extremal problem to subgraphsof complete k -partite k -uniform hypergraphs is equivalent to the question of findinglarge convexly independent subsets of the Minkowski sum of k point sets. Indeed,the k -barycenters of an embedding of a complete k -partite k -uniform hypergraphare (a dilation) of the Minkowksi sum of the embeddings of each of the k parts.The planar case with k = 2 has been an active area of research during the last decade. The unconstrained version was (asymptotically) solved in [4, 7], and thecase where the point sets are themselves in convex position was (asymptotically)solved in [33, 31]. Some of these cases were considered in [11], who introducedvariants with weak convexity and sharpened the bounds when the two point setscoincide and are in convex position. The case k = 2 and d = 3 was studied in [32].However, we are not aware of any result for larger k except for the lower boundsarising from Minkowski sums with many vertices. Finding the maximal numberof vertices of a Minkowski sum has been solved [1] (although closed formulas seemrather involved and are not explicit). Extremal problem for strong convex embeddings:
The notion of strong convex di-mension of a hypergraph poses an analogous extremal problem on the maximumnumber of hyperedges h k ( n, d ) that a k -uniform hypergraph on n vertices with astrong convex embedding to R d can have. The asymptotic values of g k ( n, d ) and h k ( n, d ) may largely differ: it is shown in [14, 11] that h ( n,
2) is linear, while g ( n,
2) is in Θ( n / ) [4, 7]. What are other quantitative and qualitative differencesbetween convex and strong convex embeddings when k > Combinatorial and topological hypersimplices:
The (topological) van Kampen-FloresTheorem [9, 34] states that the i -skeleton of the (2 i +2)-simplex cannot be embeddedin R i . This begs the question whether an analogous result for all hypersimplicesalso holds: Is d ( n, k, i ) − i -skeleton of ∆ n,k canbe topologically embedded ?This would imply that Theorem 1.2 also holds for combinatorial hypersimplices.This is not immediate from our results, since there are plenty of polytopes thatare combinatorially but not affinely isomorphic to a hypersimplex (in fact, therealization spaces of hypersimplices are far from being understood [12]). Acknowledgements
We are grateful to Kolja Knauer for introducing us to this problem and manyinteresting discussions. We thank Raman Sanyal for useful comments on an earlierversion of this manuscript. We also want to thank Edgardo Rold´an-Pensado forkeeping us informed about their progress on convex embeddings of multipartitegraphs into R . References
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E-mail address , L. Mart´ınez-Sandoval: [email protected]
E-mail address , A. Padrol:, A. Padrol: