Skip to main content Accessibility help
×
Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-29T04:41:45.427Z Has data issue: false hasContentIssue false

9 - Convex Polytopes

Published online by Cambridge University Press:  18 December 2009

Anders Björner
Affiliation:
Royal Institute of Technology, Stockholm
Michel Las Vergnas
Affiliation:
Laboratoire de Probabilités, Université Pierre et Marie Curie
Bernd Sturmfels
Affiliation:
University of California, Berkeley
Neil White
Affiliation:
University of Florida
Gunter M. Ziegler
Affiliation:
Technische Universität Berlin
Get access

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).

Type
Chapter
Information
Oriented Matroids , pp. 376 - 416
Publisher: Cambridge University Press
Print publication year: 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×