Convex Sets and Their Applications

264 Pages·1982·9.193 MB·English

by  Steven R. Lay

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CO N VEX SETS
AND THEIR AP!
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CONVEX SETS AND
TEIEIR APPLICATIONS
CONVEX SETS AN D
THEIR APPLICATION
STEVEN R. LAY
Department of Mathematics
Aurora College
A WILEY-INTERSCIENCE PUBLICATION
JOHN WILEY & SONS
New York • Chichester • Brisbane Toronto • Singapore
Copyright © 1982 by John Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of any part of this work
beyond that permitted by Sections 107 or 108 of the
1976 United States Copyright Act without the permission
of the copyright owner is unlawful. Requests for
permission or further information should be addressed to
the Permissions Department, John Wiley & Sons, Inc.
Library of Congress Cataloging in Publication Data:
Lay, Steven R., 1944-
Convex sets and their applications.
(Pure and applied mathematics, ISSN 0079-8185)
“A Wiley-Interscience publication.”
Bibliography: p.
Includes index.
1. Convex sets. I. Title. II. Series: Pure and
applied mathematics (John Wiley & Sons)
QA640.L38 511.3'2 81-19738
ISBN 0-471-09584-2 AACR2
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
To
Frederick A. Valentine,
who first introduced me to
the beauty of a convex set
PREFACE
The study of convex sets is a fairly recent development in the history of
mathematics. While a few important results date from the late 1800s, the first
systematic study was the German book Theorie der konvexen Körper by
Bonnesen and Fenchel in 1934. In the 1940s and 1950s many useful applica­
tions of convex sets were discovered, particularly in the field of optimization.
The importance of these applications has in turn sparked a renewed interest in
the theory of convex sets.
The relative youth of convexity has both good and bad implications for the
interested student. Unlike most older branches of mathematics, there is not a
vast body of background material that must be mastered before the student
can reach significant unsolved problems. Indeed, even high school students can
understand some of the basic results. Certainly upper level undergraduates
have all the tools necessary to explore the properties of convex sets. This is
good! Unfortunately, however, since the subject is so new, it has not yet
“filtered down” beneath the graduate level in any comprehensive way. To be
sure, there are several undergraduate books that devote two or three chapters
to some aspects of convex sets, but there is no text at this level which has
convex sets and their applications as its unifying theme. It is this void that the
present book seeks to fill.
The mathematical prerequisites for our study are twofold: linear algebra
and basic point-set topology. The linear algebra is important because we will
be studying convex subsets of «-dimensional Euclidean space. The reader
should be famiUar with vectors and their inner product. The topological
concepts we will encounter most frequently are open sets, closed sets, and
compactness. Occasionally we will need to refer to continuity, convergence of
sequences, or connectedness. These topics are typically covered in courses such
as Advanced Calculus or Introductory Topology.
While the intent of this book is to introduce the reader to the broad scope of
convexity, it certainly cannot hope to be exhaustive. The selection of topics has
been influenced by the following goals: The material should be accessible to
students with the aforementioned background; the material should lead the
student to open questions and unsolved problems; and the material should
highlight diverse applications. The degree to which the latter two goals are
Vll
Ylll PREFACE
attained varies from chapter to chapter. For example, Chapter 4 on
Kirchberger-type theorems includes few applications, but it does advance to
the very edge of current mathematical research. On the other hand, the
investigation of polytopes has progressed quite rapidly and so Chapter 8 can
only be considered a brief introduction to the subject. Chapter 10 on optimiza­
tion includes no unsolved problems, but has a major emphasis on applications.
For the past six years the material in this book has been used as a text in an
upper-level undergraduate course in convexity. Since there is more material
than can be covered in a three-unit semester course, the emphasis has varied
from year to year depending on which topics were included. As a result of this
flexibility, the material can be used as a geometry text for secondary teachers,
as a resource for selected topics in applications, and as a bridge to higher
mathematics for those continuing on to graduate school.
Throughout the book, the material is presented with the undergraduate
student in mind. This is not intended to imply a lack of rigor, but rather to
explain the inclusion of more examples and elementary exercises than would
be typical in a graduate text. It is recognized that a course in convexity is more
frequently found at the graduate level, but our attempt to make the material as
clear and accessible as possible should not detract from its usefulness at this
higher level as well. The exercises include not only computational problems,
but also proofs requiring various levels of sophistication. The answers to many
of the exercises are found at the back of the book, as are hints and references
to the literature.
The first three chapters of the book form the foundation for all that follows.
We begin in Chapter 1 by reviewing the fundamentals of linear algebra and
topology. This enables us to establish our notation and leads naturally into the
basic definition and properties of convex sets. Chapter 2 develops the im­
portant relationships between hyperplanes and convex sets, and this is applied
in the following chapter to the theorems of Helly and Kirchberger.
Beginning with Chapter 4, the chapters become relatively independent of
each other. Each chapter develops a particular aspect or application of convex
sets. In Chapter 4 we look at several Kirchberger-type theorems in which the
separating hyperplanes have been replaced by other geometrical figures. The
results presented in this chapter are very recent, having only appeared in the
literature in the last couple of years. Virtually any question that is posed
beyond the scope of the text opens the door to an unsolved problem. (The
answers to these will not be found in the back of the book!)
Chapter 5 investigates a number of topics which have had historical
significance in the plane. After looking at sets of constant width and universal
covers, we solve the classical isoperimetric problem under the assumption that
a set of maximal area with fixed perimeter exists. In Chapter 6 we develop
some of the tools necessary to prove the existence of such an extremal set.
There are many different ways to characterize convex sets in terms of local
properties, and these are presented in Chapter 7. Chapter 8 introduces us to
the basic properties of polytopes and gives special attention to how cubes.
PREFACE IX
pyramids, and bipyramids can be generalized to higher dimensions. In Chapter
9 we discuss two types of duaUty that occur in the study of convex sets:
polarity and dual cones.
A third type of duality appears in Chapter 10, where we show how the
theories of finite matrix games and linear programming are based on the
separation properties of convex sets. The simplex method provides a powerful
computational tool for solving a wide variety of problems. In Chapter 11 we
apply the theory of convex sets to the study of convex functions.
A final word about the exercises is in order. They have been divided into
three categories. (My students facetiously refer to them as the easy, the hard,
and the impossible.) The first few exercises at the end of each section follow
from the material in the text in a relatively straightforward manner. Some
present examples illustrating the concepts, some ask for the details omitted in a
proof, and others relate basic properties in a new way. The exercises which go
considerably beyond the scope of the text are marked by an asterisk (*),
although this distinction is, of course, sometimes arbitrary. The dagger (f) is
used to denote exercises which are either open-ended in the questions they ask
or are posing specific problems to which the answer is not currently known.
Don’t be afraid of the dagger! Only time will tell how difficult they may be.
The solution of some of them undoubtedly will be found in a new insight or a
moment of inspiration. They will lead you to the edge of current mathematical
knowledge and challenge you to jump off on your own. I encourage you to
accept this challenge. Good luck!
Steven R. Lay
Aurora, Illinois
February 1982