10 Points in Dimension 4 not Projectively Equivalent to the Vertices of a Convex Polytope

AbstractUsing oriented matroids, and with the help of a computer, we have found a set of 10 points inR4 not projectively equivalent to the vertices of a convex polytope. This result confirms a conjecture of Larman [6] in dimension 4

A combinatorial perspective on the non-Radon partitions

AbstractLet E be a finite set of points in Rd. Then {A, E − A} is a non-Radon partition of E iff there is a hyperplane H separating A strictly from E−A. Or equivalently iff AO is an acyclic reorientation of (MAff(E), O), the oriented matroid canonically determined by E. If (M(E), O) is an oriented matroid without loops then the set NR(E, O) = {(A, E − A): AO is acyclic} determines (M(E), O). In particular the matroidal properties of a finite set of points in Rd are precisely the properties which can be formulated in non-Radon partitions terms. The Möbius function of the poset A = {A: A ⊆ E, AO is acyclic} and in a special case its homotopy type are computed. This paper generalizes recent results of P. Edelman (A partial order on the regions of Rn dissected by hyperplane

Neighborly and almost neighborly configurations, and their duals

This thesis presents new applications of Gale duality to the study of polytopes with extremal combinatorial properties. It consists in two parts. The first one is devoted to the construction of neighborly polytopes and oriented matroids. The second part concerns the degree of point configurations, a combinatorial invariant closely related to neighborliness. A d-dimensional polytope P is called neighborly if every subset of at most d/2 vertices of P forms a face. In 1982, Ido Shemer presented a technique to construct neighborly polytopes, which he named the "Sewing construction". With it he could prove that the number of neighborly polytopes in dimension d with n vertices grows superexponentially with n. One of the contributions of this thesis is the analysis of the sewing construction from the point of view of lexicographic extensions. This allows us to present a technique that we call the "Extended Sewing construction", that generalizes it in several aspects and simplifies its proof. We also present a second generalization that we call the "Gale Sewing construction". This construction exploits Gale duality an is based on lexicographic extensions of the duals of neighborly polytopes and oriented matroids. Thanks to this technique we obtain one of the main results of this thesis: a lower bound of ((r+d)^(((r+d)/2)^2)/(r^((r/2)^2)d^((d/2)^2)e^(3rd/4)) for the number of combinatorial types of neighborly polytopes of even dimension d and r+d+1 vertices. This result not only improves Shemer's bound, but it also improves the current best bounds for the number of polytopes. The combination of both new techniques also allows us to construct many non-realizable neighborly oriented matroids. The degree of a point configuration is the maximal codimension of its interior faces. In particular, a simplicial polytope is neighborly if and only if the degree of its set of vertices is [(d+1)/2]. For this reason, d-dimensional configurations of degree k are also known as "(d-k)-almost neighborly". The second part of the thesis presents various results on the combinatorial structure of point configurations whose degree is small compared to their dimension; specifically, those whose degree is smaller than [(d+1)/2], the degree of neighborly polytopes. The study of this problem comes motivated by Ehrhart theory, where a notion equivalent to the degree - for lattice polytopes - has been widely studied during the last years. In addition, the study of the degree is also related to the "generalized lower bound theorem" for simplicial polytopes, with Cayley polytopes and with Tverberg theory. Among other results, we present a complete combinatorial classification for point configurations of degree 1. Moreover, we show combinatorial restrictions in terms of the novel concept of "weak Cayley configuration" for configurations whose degree is smaller than a third of the dimension. We also introduce the notion of "codegree decomposition" and conjecture that any configuration whose degree is smaller than half the dimension admits a non-trivial codegree decomposition. For this conjecture, we show various motivations and we prove some particular cases

Helly-type problems

In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals

Combinatorial Invariants of Toric Arrangements

An arrangement is a collection of subspaces of a topological space. For example, a set of codimension one affine subspaces in a finite dimensional vector space is an arrangement of hyperplanes. A general question in arrangement theory is to determine to what extent the combinatorial data of an arrangement determines the topology of the complement of the arrangement. Established combinatorial structures in this context are matroids and -for hyperplane arrangements in the real vector space- oriented matroids. Let X be the punctured plane C- 0 or the unit circle S 1, and a(1),...,a(n) integer vectors in Z d. By interpreting the a(i) as characters of the torus T=Hom(Z d,X) isomorphic to X d we obtain a toric arrangement in T by considering the set of kernels of the characters. A toric arrangement is covered naturally by a periodic affine hyperplane arrangement in the d-dimensional complex or real vector space V=C d or R d (according to whether X = C- 0 or S 1). Moreover, if V is the real vector space R d the stratification of V given by a finite hyperplane arrangement can be combinatorially characterized by an affine oriented matroid. Our main objective is to find an abstract combinatorial description for the stratification of T given by the toric arrangement in the case X=S 1 - and to develop a concept of toric oriented matroids as an abstract characterization of arrangements of topological subtori in the compact torus (S 1) d. Part of our motivation comes from the possible generalization of known topological results about the complement of complexified toric arrangements to such toric pseudoarrangements. Towards this goal, we study abstract combinatorial descriptions of locally finite hyperplane arrangements and group actions thereon. First, we generalize the theory of semimatroids and geometric semilattices to the case of an infinite ground set, and study their quotients under group actions from an enumerative and structural point of view. As a second step, we consider corresponding generalizations of affine oriented matroids in order to characterize the stratification of R d given by a locally finite non-central arrangement in R d in terms of sign vectors