Table Of Content90
Graduate Texts in Mathematics
Editorial Board
F. W. Gehring P. R. Halmos (Managing Editor)
C. C. Moore
Arne Br0ndsted
An Introduction to
Convex Polytopes
Springer Science+Business Media, LLC
Ame Brondsled
Kobenhavns Universitets
Matematiske Institut
Universitetsparken 5
2100 Kobenha'ln (3
Danmark
Edilorial Board
c.
P. R. Halmas F. W. Gehring C. Moore
Managill9 Editar University of Miehigan University ofCalifornia
Indiana University Departmenl of al Berkeley
Department of Mathematics Department of
Mathematics Ann Arbor, MI 48104 Mathematics
Bloomington, IN 47405 U.s.A. Berkeley. CA 94720
U.5.A. U.s.A.
AMS Subjeet Classifieations (1980): 52-01. 52A25
Library ofCongress Cataloging in Publication Data
Bnmdsted. Arne.
An introduetiotl 10 convex polytopes.
(Graduate texts in mathematics: 90)
Bibliography: p.
I. Convex polylOpes. I. Tit1c. 11. Series.
QA64.0.3.B76 1982 514'.223 82·10585
With 3 I11ustrations.
© 1983 by Springer Seienee+Business Media New York
Originally published by Springer-Verlag Berlin Heidclbcrg New York in 1983
Softcover reprint of the hardcover 1s t edition 1983
All righls rcservcd. No part of this book may bc translatcd or rcproduccd in any
ronn without written pennission from Springer Scicnce+Busincss Media, LLC.
Typesct by Composilion House Ud., Salisbury. England.
9 8 7 654 3 2 1
ISBN 978-1-4612-7023-2 ISBN 978-1-4612-1148-8 (eBook)
DOI 10.1007/978-1-4612-1148-8
Preface
The aim of this book is to introduce the reader to the fascinating world of
convex polytopes.
The highlights of the book are three main theorems in the combinatorial
theory of convex polytopes, known as the Dehn-Sommerville Relations, the
Upper Bound Theorem and the Lower Bound Theorem. All the background
information on convex sets and convex polytopes which is m~eded to under
stand and appreciate these three theorems is developed in detail. This
background material also forms a basis for studying other aspects of polytope
theory.
The Dehn-Sommerville Relations are classical, whereas the proofs of
the Upper Bound Theorem and the Lower Bound Theorem are of more
recent date: they were found in the early 1970's by P. McMullen and D.
Barnette, respectively. A famous conjecture of P. McMullen on the charac
terization off-vectors of simplicial or simple polytopes dates from the same
period; the book ends with a brief discussion of this conjecture and some of
its relations to the Dehn-Sommerville Relations, the Upper Bound Theorem
and the Lower Bound Theorem. However, the recent proofs that McMullen's
conditions are both sufficient (L. J. Billera and C. W. Lee, 1980) and necessary
(R. P. Stanley, 1980) go beyond the scope of the book.
Prerequisites for reading the book are modest: standard linear algebra and
elementary point set topology in [R1d will suffice.
The author is grateful to the many people who have contributed to the
book: several colleagues, in particular Victor Klee and Erik Sparre Andersen,
supplied valuable information; Aage Bondesen suggested essential improve
ments; students at the University of Copenhagen also suggested improve
ments; and Ulla Jacobsen performed an excellent typing job.
Copenhagen ARNE BR0NDSTED
February 1982
Contents
Introduction
CHAPTER 1
Convex Sets 4
91. The Affine Structure of [Rd 4
§2. Convex Sets II
§3. The Relative Interior of a Convex Set 19
§4. Supporting Hyperplanes and Halfspaces 25
§5. The Facial Structure of a Closed Convex Set 29
§6. Polarity 37
CHAPTER 2
Convex Polytopes 44
§7. Polytopes 44
§8. Polyhedral Sets 51
§9. Polarity of Polytopes and Polyhedral Sets 56
§1O. Equivalence and Duality of Polytopes 63
§Il. Vertex-Figures 67
§12. Simple and Simplicial Polytopes 76
§13. Cyclic Polytopes 85
§14. Neighbourly Polytopes 90
§15. The Graph ofa Polytope 93
V 111 Contents
CHAPTER 3
Combinatorial Theory of Convex Polytopes 98
§16. Euler's Relation 98
§17. The Dehn-Sommerville Relations lO4
§18. The Upper Bound Theorem 112
§19. The Lower Bound Theorem 121
§20. McMullen's Conditions 129
APPENDIX 1
Lattices 135
APPENDIX 2
Graphs 137
APPENDIX 3
Combinatorial Identities 143
Bibliographical Comments 148
Bibliography 151
List of Symbols 153
Index 157
Introduction
Convex polytopes are the d-dimensional analogues of 2-dimensional convex
polygons and 3-dimensional convex polyhedra. The theme of this book is
the combinatorial theory of convex polytopes. Generally speaking, the com
binatorial theory deals with the numbers of faces of various dimensions
(vertices, edges, etc.). An example is the famous theorem of Euler, which states
that for a 3-dimensional convex polytope, the number fa of vertices, the
number II of edges and the number 12 of facets are connected by the relation
1a -II + 12 = 2.
(In contrast to the combinatorial theory, there is a metric theory, dealing
with such notions as length, angles and volume. For example, the concept
of a regular polytope belongs to the metric theory.)
The main text is divided into three chapters, followed by three appendices.
The appendices supply the necessary background information on lattices,
graphs and combinatorial identities. Following the appendices, and preceding
the bibliography, there is a section with bibliographical comments. Each of
Sections 1-15 ends with a selection of exercises.
Chapter 1 (Sections 1-6), entitled "Convex Sets," contains those parts of
the general theory of d-dimensional convex sets that are needed in what
follows. Among the basic notions are the convex hull, the relative interior
of a convex set, supporting hyperplanes, faces of closed convex sets and
polarity. (Among the basic notions of convexity theory nol touched upon
we mention convex cones and convex functions.)
The heading of Chapter 2 (Sections 7-15) is "Convex Polytopes." In
Sections 7-11 we apply the general theory of convex sets developed in
Chapter 1 to the particular case of convex polytopes. (It is the author's
belief that many properties of convex polytopes are only appreciated
2
Introduction
when seen on the background of properties of convex sets in general.) In
Sections 12-14 the important classes of simple, simplicial, cyclic and neigh
bourly polytopes are introduced. In Section 15 we study the graph determined
by the vertices and edges of a polytope.
Chapter 3 contains selected topics in the "Combinatorial Theory of
Convex Polytopes." We begin, in Section 16, with Euler's Relation in its
d-dimensional version. In Section 17 we discuss the so-called Dehn
Sommerville Relations which are "Euler-type" relations, valid for simple
or simplicial polytopes only. Sections 18 and 19 are devoted to the celebrated
Upper Bound Theorem and Lower Bound Theorem, respectively; these
theorems solve important extremum problems involving the numbers of
faces (of various dimensions) of simple or simplicial polytopes. Finally,
in Section 20 we report on a recent fundamental theorem which gives
"complete information" on the numbers of faces (of various dimensions)
of a simple or simplicial polytope.
The following flow chart outlines the organization of the book. However,
there are short cuts to the three main theorems of Chapter 3. To read the
proof of the Dehn-Sommerville Relations (Theorem 17.1) only Sections
1-12 and Euler's Relation (Theorem 16.1) are needed; Euler's Relation
also requires Theorem 15.1. To read the proof ofthe Upper Bound Theorem
(Theorem 18.1) only Sections 1-14 and Theorems 15.1-15.3 are needed.
To read the Lower Bound Theorem (Theorem 19.1) only Sections 1-12
and 15, and hence also Appendix 2, are needed. It is worth emphasizing that
none of the three short cuts requires the somewhat technical Appendix 3.
Introduction 3
{
Sections E~ndi' I
Chapter 1
1-6
1
Sections
7-12
1 1
Chapter 2
Sections Section Ep~di'l
13, 14 15
1 1
Sections Ep~ndi' I
16, 17
1 1
Section Section
Chapter 3
18 19
1 1
Section
20
