Many neighborly polytopes and oriented matroids

TL;DR

This paper introduces new methods for constructing neighborly polytopes and oriented matroids, providing improved lower bounds on their combinatorial types and demonstrating the construction of many non-realizable examples.

Contribution

It presents a novel technique for constructing neighborly polytopes, generalizes sewing to oriented matroids, and establishes improved lower bounds on their combinatorial diversity.

Findings

New lower bound for the number of neighborly polytopes.

Simplified proof of the sewing technique.

Construction of many non-realizable neighborly oriented matroids.

Abstract

In this paper we present a new technique to construct neighborly polytopes, and use it to prove a lower bound of ((r+d)^((r/2+d/2)^2))/(r^((r/2)^2)d^((d/2)^2)e^(3rd/4)) for the number of combinatorial types of vertex-labeled neighborly polytopes in even dimension d with r+d+1 vertices. This improves current bounds on the number of combinatorial types of polytopes. The previous best lower bounds for the number of neighborly polytopes were found by Shemer in 1982 using a technique called the Sewing Construction. We provide a simpler proof that sewing works, and generalize it to oriented matroids in two ways: to Extended Sewing and to Gale Sewing. Our lower bound is obtained by estimating the number of polytopes that can be constructed via Gale Sewing. Combining both new techniques, we are also able to construct many non-realizable neighborly oriented matroids.