9 - Convex Polytopes

Summary

The combinatorial theory of convex polytopes is an important area of application for oriented matroid theory. Several new results on polytopes as well as new simplified proofs for known results have been found, and it is fair to say that oriented matroids have significantly contributed to the progress of combinatorial convexity during the past decade. This chapter aims to be both an introduction to the basics and a survey on current research topics in this branch of discrete mathematics.

Section 9.1 is concerned with basic properties of matroid polytopes. We show that oriented matroid duality is essentially equivalent to the technique of Gale transforms. In Section 9.2 we discuss matroidal analogues to polytope constructions and some applications. Section 9.3 deals with the Lawrence construction, an important general method for encoding oriented matroid properties into polytopes. Cyclic and neighborly polytopes will be studied in Section 9.4, and triangulations of matroid polytopes in Section 9.6. In Section 9.5 we discuss an oriented matroid perspective on the Steinitz problem of characterizing face lattices of convex polytopes.

Introduction to matroid polytopes

Throughout this chapter we will interpret a rank Υ oriented matroid as a generalized point configuration in affine (Υ – l)-space. Using the language of oriented matroids, we can define the convex hull of such a configuration, and this allows us to study properties of convex polytopes in this purely combinatorial setting. The following basic definitions, due to Las Vergnas (1975a, 1980a), were already discussed in Exercise 3.9. In Exercise 3.11 we gave an axiomatization of oriented matroids in terms of their convex closure operators. A weaker notion of abstract convexity was developed independently by Edelman (1980, 1982) and Jamison (1982).