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.

Peer Reviewed