Convex and Discrete Geometry
Convex and Discrete Geometry is an area of mathematics situated between analysis, geometry and discrete mathematics with numerous relations to other areas. The book gives an overview of major results, methods and ideas of convex and discrete geometry
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Series editors M. Berger B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin L. Hörmander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane A. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates
S.R.S. Varadhan
336
Peter M. Gruber
Convex and Discrete Geometry
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Peter M. Gruber Institute of Discrete Mathematics and Geometry Vienna University of Technology Wiedner Hauptstrasse 8-10 1040 Vienna, Austria e-mail: [email protected]
Library of Congress Control Number: 2007922936 Mathematics Subject Classification (2000): Primary: 52-XX, 11HXX Secondary: 05Bxx, 05Cxx, 11Jxx, 28Axx, 46Bxx, 49Jxx, 51Mxx, 53Axx, 65Dxx, 90Cxx, 91Bxx ISSN 0072-7830 ISBN 978-3-540-71132-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A EX macro package Typesetting by the author and SPi using a Springer LT Cover design: design & production GmbH, Heidelberg
Printed on acid-free paper
SPIN: 11966739
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Preface
In this book we give an overview of major results, methods and ideas of convex and discrete geometry and their applications. Besides being a graduate-level introduction to the field, the book is a practical source of information and orientation for convex geometers. It should also be of use to people working in other areas of mathematics and in the applied fields. We hope to convince the reader that convexity is one of those happy notions of mathematics which, like group or measure, satisfy a genuine demand, are sufficiently general to apply to numerous situations and, at the same time, sufficiently special to admit interesting, non-trivial results. It is our aim to present convexity as a branch of mathematics with a multitude of relations to other areas. Convex geometry dates back to antiquity. Results and hints to problems which are of interest even today can already be found in the works of Archimedes, Euclid and Zenodorus. We mention the Platonic solids, the isoperimetric problem, rigidity of polytopal convex surfaces and the problem of the volume of pyramids as examples. Contributions to convexity in modern times started with the ge
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