Neighborly and almost neighborly configurations, and their duals
Authors:   , Padrol Sureda Arnau 
Published: October 2023

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.