Paper detail

Neighborly and almost neighborly configurations, and their duals

This thesis presents new applications of Gale duality to the study of polytopes, point configurations and oriented matroids with extremal combinatorial properties. The first part of the thesis explores construction techniques for neighborly polytopes and oriented matroids. First, we provide a new interpretation of Shemer's classical Sewing Construction for neighborly polytopes in terms of lexicographic extensions of oriented matroids. This allows us to provide a simplified proof and to generalize it to oriented matroids in two ways: the Extended Sewing Construction and the Gale Sewing Construction. Estimating the number of polytopes constructed with the later, we can provide new lower bounds for the number of combinatorial types of neighborly polytopes that even improve 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. 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 for configurations of small degree in terms of the novel concepts of weak Cayley configurations and codegree decompositions.