Constructing neighborly polytopes and oriented matroids

Abstract

A d-polytope P is neighborly if every subset of b d 2 c vertices is a face of P. In 1982, Shemer introduced a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many different neighborly polytopes. The concept of neighborliness extends naturally to oriented matroids. Duals of neighborly oriented matroids also have a nice characterization: balanced oriented matroids. In this paper, we generalize Shemer’s sewing construction to oriented matroids, providing a simpler proof. Moreover we provide a new technique that allows to construct balanced oriented matroids. In the dual setting, it constructs a neighborly oriented matroid whose contraction at a particular vertex is a prescribed neighborly oriented matroid. We compare the families of polytopes that can be constructed with both methods, and show that the new construction allows to construct many new polytopes.