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Grundlehren Der Mathematischen Wissenschaften 336 a Series of Comprehensive Studies in Mathematics Grundlehren der mathematischen Wissenschaften 336 A Series of Comprehensive Studies in Mathematics 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 Peter M. Gruber Convex and Discrete Geometry ABC 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. Typesetting by the author and SPi using a Springer LTA EX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11966739 41/SPi 543210 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 geometric and analytic work of Galileo, the Bernoullis, Cauchy and Steiner on the problems from antiquity. These problems were solved only in the nineteenth and early twentieth century by Cauchy, Schwarz and Dehn. Results without antecedents in antiquity include Euler’s polytope formula and Brunn’s inequality. Much of modern convexity came into being with Minkowski. Important later contributors are Blaschke, Hadwiger, Alexandrov, Pogorelov, and Klee, Groemer, Schneider, McMullen together with many further living mathematicians. Modern aspects of the subject include surface and curva- ture measures, the local theory of normed spaces, best and random approximation, affine-geometric features, valuations, combinatorial and algebraic polytope theory, algorithmic and complexity problems. Kepler was the first to consider problems of discrete geometry, in particular pack- ing of balls and tiling. His work was continued by Thue, but the systematic research began with Fejes Toth´ in the late 1940s. The Hungarian school deals mainly with packing and covering problems. Amongst numerous other contributors we mention Rogers, Penrose and Sloane. Tiling problems are a classical and also a modern topic. The ball packing problem with its connections to number theory, coding and the VI Preface theory of finite groups always was and still is of great importance. More recent is the research on arrangements, matroids and the relations to graph theory. The geometry of numbers, the elegant older sister of discrete geometry, was cre- ated by Lagrange, Gauss, Korkin, Zolotarev, Fedorov, the leading figures Minkowski and Vorono˘ı, by Blichfeldt, by Delone, Ryshkov and the Russian school, by Siegel, Hlawka, Schmidt, Davenport, Mahler, Rogers and others. Central problems of mod- ern research are the theory of positive definite quadratic forms, including reduction, algorithmic questions and the lattice ball packing problem. From tiny branches of geometry and number theory a hundred years ago, con- vexity, discrete geometry and geometry of numbers developed into well-established areas of mathematics. Now their doors are wide open to other parts of mathematics and a number of applied fields. These include algebraic geometry, number theory, in particular Diophantine approximation and algebraic number theory, theta series, error correcting codes, groups, functional analysis, in particular the local theory of normed spaces, the calculus of variations, eigenvalue theory in the context of partial differential equations, further areas of analysis such as geometric measure theory, potential theory, and also computational geometry, optimization and econometrics, crystallography, tomography and mathematical physics. We start with convexity in the context of real functions. Then convex bodies in Euclidean space are investigated, making use of analytic tools and, in some cases, of discrete and combinatorial ideas. Next, various aspects of convex polytopes are stud- ied. Finally, we consider geometry of numbers and discrete geometry, both from a rather geometric point of view. For more detailed descriptions of the contents of this book see the introductions of the individual chapters. Applications deal with measure theory, the calculus of variations, complex function theory, potential theory, numer- ical integration, Diophantine approximation, matrices, polynomials and systems of polynomials, isoperimetric problems of mathematical physics, crystallography, data transmission, optimization and other areas. When writing the book, I became aware of the following phenomena in convex and discrete geometry. (a) Convex functions and bodies which have certain prop- erties, have these properties often in a particularly strong form. (b) In complicated situations, the average object has often almost extremal properties. (c) In simple situations, extremal configurations are often regular or close to regular. The reader will find numerous examples confirming these statements. In general typical, rather than refined results and proofs are presented, even if more sophisticated versions are available in the literature. For some results more than one proof is given. This was done when each proof sheds different light on the problem. Tools from other areas are used freely. The reader will note that the proofs vary a lot. While in the geometry of numbers and in some more analytic branches of convex geometry most proofs are crystal clear and complete, in other cases details are left out in order to make the ideas of the proofs better visible. Sometimes we used more intuitive arguments which, of course, can be made precise by inserting addi- tional detail or by referring to known results. The reader should keep in mind that all this is typical of the various branches of convex and discrete geometry. Some proofs are longer than in the original literature. While most results are proved, there are Preface VII some important theorems, the proofs of which are regrettably omitted. Some of the proofs given are complicated and we suggest skipping these at a first reading. There are plenty of comments, some stating the author’s personal opinions. The emphasis is more on the systematic aspect of convexity theory. This means that many interest- ing results are not even mentioned. In several cases we study a notion in the context of convexity in one section (e.g. Jordan measure) and apply additional properties of it in another section (e.g. Fubini’s theorem). We have tried to make the attributions to the best of our knowledge, but the history of convexity would form a complicated book. In spite of this, historical remarks and quotations are dispersed throughout the book. The selection of the material shows the author’s view of convexity. Several sub-branches of convex and discrete geometry are not touched at all, for example axiomatic and abstract convexity, arrangements and matroids, and finite packings, others are barely mentioned. I started to work in the geometry of numbers as a student and became fascinated by convex and discrete geometry slightly later. My interest was greatly increased by the study of the seminal little books of Blaschke, Fejes Toth,´ Hadwiger and Rogers, and I have been working in these fields ever since. For her great help in the preparation of the manuscript I am obliged to Edith Rosta and also to Franziska Berger who produced most of the figures. Kenneth Stephenson provided the figures in the context of Thurston’s
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