Empty Monochromatic Simplices

Empty Monochromatic Simplices

Oswin Aichholzer Institute for Software Technology, University of Technology, Graz, Austria, oaich@ist.tugraz.at    Ruy Fabila-Monroy Departamento de Matemáticas, Cinvestav, D.F. México, México, ruyfabila@math.cinsvestav.edu.mx    Thomas Hackl Institute for Software Technology, University of Technology, Graz, Austria, thackl@ist.tugraz.at, Inffeldgasse 16b/II, 8010 Graz, Austria; Tel: +43 316 873 5702; FAX: +43 316 873 5706    Clemens Huemer Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Barcelona, Spain, clemens.huemer@upc.edu    Jorge Urrutia Instituto de Matemáticas, Universidad Nacional Autónoma de México, D.F. México, México, urrutia@matem.unam.mx
July 6, 2019
Abstract

Let be a -colored (finite) set of points in , , in general position, that is, no points of lie in a common -dimensional hyperplane. We count the number of empty monochromatic -simplices determined by , that is, simplices which have only points from one color class of as vertices and no points of in their interior. For we provide a lower bound of and strengthen this to for .

On the way we provide various results on triangulations of point sets in . In particular, for any constant dimension , we prove that every set of points ( sufficiently large), in general position in , admits a triangulation with at least simplices.

1 Introduction

Let be a finite set of points in . Throughout this paper we assume that is in general position, that is, no points of lie in a common -dimensional hyperplane. A more formal definition of “general position” can be found in Section 2.1. A subset of is said to be empty if , where denotes the convex hull of (please see Section 2.1 for a detailed definition). A -coloring of is a partition of into non-empty sets called color classes. A subset of is said to be monochromatic if all its elements belong to the same color class. A -simplex is the -dimensional version of a triangle.

The problem of determining the minimum number of empty triangles any set of points in general position in the plane contains, has been widely studied [12, 3, 9, 19] and also the higher dimensional version of the problem has been considered [2]. In [12] it is noted that every set of points in general position in determines at least empty simplices. In [2] it is shown that in a random set of points in —chosen uniformly at random on a convex, bounded set with nonempty interior—the expected number of empty simplices is at most (where is a constant depending only on ).

The colored version of the problem has been introduced in [7] and was studied in [1], where empty monochromatic triangles were shown to exist in every two colored set of points in general position in the plane. This has later been improved to in [15]. Further, arbitrarily large -colored sets without empty monochromatic triangles were shown to exist in the plane in [7].

In this paper we study the higher dimensional version of this colored variant. We generalize both, the dimension and the number of colors. Specifically, we consider the problem of counting the number of empty monochromatic -simplices in a -colored set of points in .

It is shown in [18] that every sufficiently large -colored set of points in general position in contains an empty monochromatic tetrahedron. This is done by showing that any set of points in general position in can be triangulated with more than tetrahedra.

The problem of triangulating a set of points with many simplices is intimately related to the problem of determining the minimum number of empty simplices in -colored sets of points in . Remarkably this problem has received little attention. For the special case of , it even has been pronounced “the least significant” among the four extremal (maxmax, maxmin, minmax, minmin) problems in [10]. Consequently, only a trivial lower bound and an upper bound of has been shown there. Nevertheless, in [5] sets of points in in general position are shown such that every triangulation of them has tetrahedra, for points in , and in general simplices for points in . Furthermore, in [6] this minmax problem is stated as Open Problem 11 in the section “Extremal Number of Special Subconfigurations”.

In this direction we give the first, although not asymptotically improving, non-trivial lower bound and show that for every set of points in general position in admits a triangulation of at least simplices, for sufficiently large and constant.

Figure 1: Roadmap through the paper.

The paper is organized as follows: in Section 2 known results on simplicial complexes and triangulations are reviewed; in Section 3 new results on simplicial complexes and triangulations are presented; using these results in Section 4, high dimensional versions of the Order and Discrepancy Lemmas used in [1] are shown; in Section 5 the lemmas of Section 4 are put together to prove various results on the minimum number of empty monochromatic simplices in sets of points in . Our results are summarized in Table 1.

([15] and Thm 33) (Thm 33)
(Thm 29)
none ([7]) at least linear (Cor 25)
none ([7]) unknown
Table 1: Number of empty monochromatic -simplices in -colored sets of (sufficiently large) points in . The linear lower bound for and has been proved already in [18].

To provide a better general view on the paper, and especially to visualize the interrelation between the many lemmas, we present a “roadmap” through the paper in Figure 1. The lemmas (and theorems and corollaries) are shown in boxes, given with their number, if applicable a special name, and the necessary preconditions. Main results have a bold frame. The lemmas are grouped to reflect their topical and section correlation. An arrow from a Lemma A to a Lemma B depicts, that the proof of Lemma B uses the result of Lemma A. Hence, the preconditions for Lemma A have to be fulfilled in Lemma B. Theorem 35 (stated and proven in the ”Conclusions”) is not depicted in Figure 1, as there is no interrelation with other lemmas.

2 Preliminaries

In this section, following the notation of Matoušek [13], we state the definitions and known results regarding simplicial complexes and triangulations, that will be needed throughout the paper. Note that in this paper we consider the number, , of dimensions and also the number, , of different colors as constants. This means, that and do not depend on the size, , of the considered finite set of points. But of course the required minimum size of the point set might depend on and .

2.1 Simplicial Complexes

Let be a finite set of points in . The convex hull of , denoted with , is the intersection of all convex sets containing . Alternatively it may be defined as the set of points that can be written as a convex combination of elements of :

We denote the boundary of with . A point of is said to be a convex hull point if it lies in , otherwise it is called an interior point. A point set is said to be in convex position if every point of is a convex hull point.

Let denote the -dimensional zero vector. A set of points in is said to be affinely dependent if there exist real numbers , not all zero, such that and . Otherwise is said to be affinely independent. A set of points in is in general position if each subset of with at most elements is affinely independent.

A simplex is the convex hull of a finite affinely independent set in . The elements of are called the vertices of . If consists of elements, we say that is of dimension or that is an -simplex. The convex hull of any subset of vertices of a simplex is called a face of . A face of a simplex is again a simplex.

A simplicial complex is a family of simplices satisfying the following properties:

  • Each face of every simplex in is also a simplex of .

  • The intersection of two simplices is either empty or a face of both, and .

The vertex set of is the union of the vertex sets of all simplices in . We say that is of dimension , if is the highest dimension of any of its simplices. The size of a simplicial complex of dimension is the number of its simplices of dimension . The -skeleton of is the simplicial complex consisting of all simplices of of dimension at most . Hence the -skeleton is the vertex set of .

We now turn to finite sets of points in general position in . Let be such a set of elements. Note that since is in general position we may regard as a simplicial complex in a natural way. Such simplicial complexes are called simplicial polytopes. It is known that every simplicial polytope satisfies:

Theorem 1 ([4] Lower Bound Theorem).

For a simplicial polytope of dimension let be the number of its -dimensional faces. Then:

  •  for all    and

  •  .

Note that in the Lower Bound Theorem, the word dimension refers to the dimension of the simplicial polytope as a polytope. Hence, a three dimensional simplicial polytope would be a two dimensional simplicial complex.

2.2 Triangulations

A triangulation of is a simplicial complex such that its vertex set is and the union of all simplices of is . This definition generalizes the usual definition of triangulations of planar point sets. The size of a triangulation is the number of its -simplices. The minimum size of any triangulation of is known to be . We explicitly mention this result for further use:

Theorem 2 ([16]).

Every triangulation of a set of points in general position in has size at least .

Figure 2: Example in for inserting a point into a triangulation with (left) inside the convex hull and (right) outside the convex hull.

We will use the following operation of inserting a point into a triangulation frequently: Let be a point not in but such that is also in general position, and let be a triangulation of . If lies in then is contained in a unique -simplex of . We remove from and replace it with the -simplices formed by taking the convex hull of and each of the  -dimensional faces of . If, on the other hand, lies outside then a set of -dimensional faces of is visible from . We get a set of -simplices formed by taking the convex hull of and each face of , and add these simplices to . In either case the resulting family of simplices is a triangulation of (see Figure 2).

We distinguish two different types of triangulations of a set of points in general position in by their construction: A shelling triangulation of is constructed as follows. Choose any ordering of the elements of and let . Start by triangulating with only one simplex. Afterwards, for every create the triangulation of by inserting into the triangulation of . The final triangulation of this process, that of , is a shelling triangulation. A pulling triangulation of is constructed by choosing (if it exists) a point of , such that . Then is in convex position. Construct a -simplex with and each -dimensional face of that does not contain .

3 Results on Triangulations and Simplicial Complexes

In this section we present some results on triangulations and simplicial complexes that will be needed later, but are also of independent interest. We begin by showing that every point set can be triangulated with a “large number” of simplices. We use the same strategy as in [18].

3.1 Large Sized Triangulations

First we prove an at least possible size for a triangulation of a convex set of points, by building a shelling triangulation for a special sequence of points.

Lemma 3.

Every set of points in convex and general position in has a triangulation of size at least , with .

Proof.

The -skeleton of is a graph of vertices and, by the Lower Bound Theorem (Theorem 1, for ), of at least edges. Therefore, as long as there will be a vertex of degree at least in this graph.

Set and let be the -skeleton (as a graph) of . In general once is defined, let be the -skeleton (as a graph) of . Let be a vertex of degree at least in , with . We construct a shelling triangulation of , with size as claimed in the lemma.

Starting with , iteratively remove a vertex from , i.e., . Observe that . The iteration stops with as . Construct an arbitrary shelling triangulation of . By Theorem 2, has size at least . Complete to a shelling triangulation by inserting the points in reversed order of their removal ( from to ).

We prove that with each inserted point at least -simplices are added to the triangulation. Let be the degree of in and recall that . Consider the neighbors of in . Let be a -dimensional hyperplane separating and , and let be the set of intersections of with the lines spanned by and each of .

Note that are a set of points in convex position in and that the -dimensional faces of , which are visible to , project to a triangulation of in . By Theorem 2, every triangulation of points in has size at least . Thus, at least -simplices are added when inserting . Hence, the constructed shelling triangulation has size at least , which is the claimed bound of , with . ∎

Using this result it is easy to give a lower bound on the triangulation size for general point sets in dependence of a certain subset property.

Lemma 4.

Let be a set of points in general position in . Let and be two disjoint sets, such that and is in convex position. If then there exists a triangulation of of size at least , with defined as in Lemma 3.

Proof.

By Lemma 3, has a triangulation of size at least , if . Inserting each point of into adds at least one -simplex to per point in . This results in a triangulation of with size at least . ∎

Combining the previous two lemmas we prove a new non-trivial lower bound for the size of triangulations with an additive logarithmic term.

Theorem 5.

Every set of points in general position in , with convex hull points, has a triangulation of size at least , with as defined in Lemma 3.

Proof.

Let be the set of convex hull points of . We distinguish two cases:

  • . By Lemma 3, there exists a triangulation of of size at least , as . Insert the remaining points of into this triangulation. Since these points are inside , each of them contributes with additional -simplices to the final triangulation. Therefore, the resulting triangulation has size at least .

  • . By the Erdős-Szekeres Theorem (see [11]) and its best known upper bound (see [17]), contains a subset of at least points in convex position. Let . Apply Lemma 4 to obtain a triangulation of of size at least . Insert the remaining points of into . Since these inserted points are in the interior of , each of them contributes with additional -simplices to the final triangulation. Therefore, this triangulation has size at least , which is .

Note that in Lemma 3 can be improved to . Instead of stopping the process at , we continue the iteration using a vertex degree of for with , a vertex degree of for with , and so on. This way, instead of a triangulation of size at least , we can guarantee a triangulation of size at least , which results in the claimed improvement of for . Thus, for Theorem 5 can be improved to . Note that this corresponds to the bound from [10], that every set of points in general position in , with convex hull points, has a tetrahedrization of size at least for .

3.2 Pulling Complexes

Let be a set of points in general position in . In this section we present lemmas that allow us to construct -simplicial complexes of large size on , such that their -simplices contain a pre-specified subset of in their vertex set. We begin with a result for point sets, whose convex hull is a simplex.

Lemma 6.

Let be a set of points in general position in , such that is a -simplex. For every convex hull point of , there exists a triangulation of such that of its -simplices have as a vertex.

Figure 3: Illustration of the proof of Lemma 6 for and .
Proof.

We use induction on , see Figure 3 for an illustration. Start with a triangulation consisting only of the -simplex . If , is a triangulation with empty simplex containing as vertex.

Assume . Let be the interior point of closest to the only face of not incident to . (If there exist more then one such closest points, then choose an arbitrary one of them as .) Insert into . This results in a triangulation of size in which of its -simplices, , have as a vertex. Note that the remaining -simplex does not contain any point of in its interior. We apply induction on . Let be the number of points of interior to , . For each we obtain a triangulation such that of its -simplices have as a vertex. The union of the triangulations of each is a triangulation of , and of its -simplices have as a vertex. ∎

The next three lemmas give, for every point of a general point set in , a lower bound on the number of interior disjoint -simplices incident to , for the cases , , and , respectively.

Lemma 7.

Let be a set of points in general position in . For every point of there exists a -dimensional simplicial complex of size at least and such that all of its triangles have as a vertex.

Proof.

Do a cyclic ordering around of the points of . Construct a -dimensional simplicial complex by forming a triangle with and every two consecutive elements determining an angle less than . This simplicial complex has at least triangles and they all contain as a vertex. ∎

Lemma 8.

Let be a set of points in general position in . For every point of there exists a triangulation of such that at least:

  • of its -simplices have as a vertex, if is an interior point of .

  • of its -simplices contain as a vertex, if is a convex hull point of and is its degree in the -skeleton of .

Proof.

Let be the set of convex hull points of and . Construct a pulling triangulation w.r.t. of . By definition all -simplices of contain as a vertex. For every -simplex of , let be the number of points of interior to . By applying Lemma 6 we can triangulate , such that of its -simplices have as a vertex. Repeat this for every -simplex of , to obtain a triangulation of .

By Theorem 1, has (at least) faces (for this lower bound is tight).

  • If is an interior point of , contains a -simplex for every face of . Therefore, summing over all these faces we get of the -simplices in have as a vertex.

  • If is a convex hull point of , contains a -simplex for every face of not having as a vertex. This is equal to , where is the degree of in the -skeleton of . Therefore, of the -simplices in have as a vertex.

Lemma 9.

Let be a set of points in general position in . For every point of , there exists a -dimensional simplicial complex with vertex set , such that has size strictly larger than and all its -simplices have as a vertex, with defined as in Lemma 3.

Proof.

For every point distinct from let be the infinite ray with origin and passing through . Let be a halving -dimensional hyperplane of passing through , not containing any other point of . Further, let and be two -dimensional hyperplanes parallel to containing between them and not parallel to any of the rays .

Project from every point in to or , in the following way. Every ray intersects either or in a point . Take to be the projection of from . Let and be these projected points in and , respectively. Both, and , are sets of points in general position in , with and , where both, and , are strictly larger than .

By Theorem 5, there exist triangulations of and of of size at least and , respectively. Consider the simplicial complexes and that arise from replacing every point in a simplex of or with its preimage in . The -simplices of and are all visible from . Hence, we obtain a simplicial complex of dimension , by taking the convex hull of and each -simplex of and . Obviously, all -simplices of contain as a vertex. The size of is at least . This is strictly larger than . ∎

We now consider not only one point, but subsets of point sets in . The next three lemmas, applicable for , , and , respectively, provide lower bounds on the number of interior disjoint -simplices which all share the points in . Note that the second lemma in the row, Lemma 11, is true for .

Lemma 10.

Let be a set of points in general position in . For every set of points , there exists a -dimensional simplicial complex with vertex set , such that has size strictly larger than and all its -simplices have in their vertex set, with defined as in Lemma 3.

Proof.

The case is shown in Lemma 9. Thus assume that . Let be the -dimensional hyperplane containing and let be a -dimensional hyperplane orthogonal to . Project orthogonally to , and let be the resulting image. The set is projected to a single point in . Obviously . Apply Lemma 9 to , and obtain a -dimensional simplicial complex with vertex set of size at least , such that all the -simplices of have as a vertex.

To get from , lift each simplex of to the convex hull of the preimage of its vertex set. Thus is a -dimensional simplicial complex with vertex set and size larger than . As all -simplices of have as a vertex, each -simplex of has as a vertex subset. ∎

Lemma 11.

Let be a set of points in general position in . For every set of points, there exists a -dimensional simplicial complex with vertex set , such that has size at least , and all -simplices of have as a vertex.

Figure 4: Illustration of the proof of Lemma 11 for and .
Proof.

The proof is similar to that of Lemma 10, with the difference that we cannot apply Lemma 9.

Let be the -dimensional hyperplane containing and let be a -dimensional hyperplane orthogonal to . Project orthogonally to , and let be its image. The set is projected to a single point of (see Figure 4). Obviously . Apply Lemma 7 to , and obtain a -dimensional simplicial complex with vertex set of size at least , such that all triangles of have as a vertex.

To get from , lift each triangle of to the convex hull of the preimage of its vertex set. Thus is a -dimensional simplicial complex with vertex set and size . Since all triangles of have as a vertex, all -simplices of have as a vertex subset. ∎

Note that Lemma 10 and Lemma 11 leave a gap for . In this case, the point set is projected to a 3-dimensional hyperplane, where the guaranteed bounds on incident 3-simplices vary significantly for extremal and interior points, see Lemma 8. Thus we make a weaker statement for this case, which will turn out to be sufficient anyhow.

Lemma 12.

Let be a set of points in general position in . Let be a subset of points. Denote with the -dimensional hyperplane containing and with a -dimensional hyperplane orthogonal to . Project orthogonally to , and let be the resulting image. The set is projected to a single point in .

If is an interior point of , then there exists a -dimensional simplicial complex with vertex set , such that is of size at least and all -simplices of have in their vertex set.

Proof.

Obviously . As is assumed to be an interior point of , apply Lemma 8 to , and obtain a -dimensional simplicial complex with vertex set of size at least , such that all the -simplices of have as a vertex.

To get from , lift each -simplex of to the convex hull of the preimage of its vertex set. Thus is a -dimensional simplicial complex with vertex set and size at least . As all -simplices of have as a vertex, each -simplex of has as a vertex subset. ∎

In the light of the previous lemma it is of interest to know the conditions for a subset of in to project to an interior point of . We make the following statement.

Lemma 13.

Let be a set of points in and let be a subset of  points. With denote the -dimensional hyperplane spanned by and with a -dimensional hyperplane orthogonal to . Project orthogonally to and denote with the resulting image of and with the image of , respectively. Then is an extremal point of if and only if is a -dimensional facet of .

Proof.

If is a -dimensional facet of , then there exists a -dimensional hyperplane “tangential” to , containing only and having all other points of on one side. Thus, there exists a “tangential” plane at , such that all points of are on one side of . Hence, is extremal.

If is not a -dimensional facet of , then all -dimensional hyperplanes containing have points of on both sides, and therefore is not extremal in . Assume the contrary: at least one -dimensional hyperplane, , containing exists, such that all points of are on one side of . Then we could tilt keeping all of its contained points and consuming the ones it hits while tilting, until contains points; i.e., until consumed two more points, and . Still all points of , except the ones contained in , are on one side of . Observe that a hyperplane spanned by points (in a point set in general position) is a -dimensional hyperplane. Hence, has become a supporting hyperplane of a -dimensional facet, , of . As the convex hull of every subset of is a facet of , this is a contradiction to the assumption that is not a -dimensional facet of . ∎

4 Higher Dimensional Versions of The Order and Discrepancy Lemmas

We prove the higher dimensional versions of the Order and Discrepancy Lemmas from [1]. The proofs are essentially the same as in the planar case, with the difference that some facts we used in the plane are now provided by the lemmas in the previous sections.

Recall that in a partial order a chain is a set of pairwise comparable elements, whereas an antichain is a set of pairwise incomparable elements.

4.1 Order Lemma

Lemma 14.

Let be a set of points in general position in , such that is a -simplex. Then there exists a triangulation of , such that at least of its -simplices contain a convex hull point of .

Proof.

Let be the set of the interior points of . Let be the set of the -dimensional faces of . For each we define a partial order on . We say that if is in the interior of the -simplex . Our goal is to obtain a “long” chain with respect to some such that .

By Dilworth’s Theorem [8] w.r.t. , there exists a chain or an antichain in of size at least . If is a chain then we obtain , , and . Otherwise, we iteratively apply Dilworth’s Theorem w.r.t. to the points of the antichain , from downto , to obtain a chain or antichain of size at least . As soon as is a chain, terminate with , , and . Otherwise, the process ends with the antichain of size at least . But, similar to the planar case, an antichain with respect to all but two faces is a chain with respect to the remaining two faces. Hence, , , with .

Let be the points of . Construct a triangulation of , starting with consisting only of the -simplex . Then insert the points of into in the order . With each step one -simplex is replaced by new ones. This results in an intermediate triangulation of consisting of many -simplices, each of which having at least one point in as a vertex.

Let , , be the -simplices of , let be the number of interior points of , and let be a vertex of that is also in . By Lemma 6 there exists a triangulation of such that of its -simplices have as a vertex. Therefore, the remaining points can be inserted into , such that at least of the -simplices of have at least one point in