Geometric Inequalities
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Editors M. Artin S. S. ehern 1. M. Fröhlich E. Heinz H. Hironaka F. Hirzebruch L. Hörmander S. MacLane C. C. Moore 1. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai 1. Tits M. Waldschmidt S. Watanabe Managing Editors M. Berger B. Eckmann S. R. S. Varadhan
Yu. D. Burago V. A. Zalgaller
Geometrie Inequalities Translated from the Russian by A. B. Sossinsky With 43 Figures
Springer-Verlag Berlin Heidelberg GmbH
Yuri'f Dmitrievich Burago Viktor Abramovich Zalgaller Leningrad Branch of the Steklov Mathematical Institute Fontanka 27, 191011 Leningrad, USSR
Title ofthe Russian original edition: Geometricheskie neravenstva Publisher Nauka, Leningrad 1980 This volume is part of the Springer Series in Soviet Mathematics Advisers: L. D. Faddeev (Leningrad), R. V. Gamkrelidze (Moscow)
Mathematics Subject Classification (1980): 52-02, 52A40, 53-02, 53A05, 53 A07, 53C20, 53C40, 53C42, 53C45, 53C65, 58A25
ISBN 978-3-642-05724-3
Library of Congress Cataloging in Publication Data Burago, IU. D. (IUriI Dmitrievich) Geometrie inequalities. (Grundlehren der mathematischen Wissenschaften = Aseries of comprehensive studies in mathematics ; 285) Translation of: Geometricheskie neravenstva. I. Geometry, Differential. 2. Isoperimetrie inequalities. I. Zalgaller, v. A. (Viktor A.) 11. Title. 111. Series: Grundlehren der mathematischen Wissenschaften; 285. QA641.B7813 1988 516.3'6 86-10214 ISBN 978-3-642-05724-3 ISBN 978-3-662-07441-1 (eBook) DOI 10.1007/978-3-662-07441-1 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions ofthe German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1988 Originally published by Springer-Verlag Berlin Heidelberg New York in 1988 Softcover reprint of tbe hardcover I st edition 1988 Typesetting: Asco Trade Typesetting Ltd., Hong Kong 214113140-543210
Foreword
Geometrie inequalities have a wide range of applieations-within geometry itself as weIl as beyond its limits. The theory of funetions of a eomplex variable, the ealculus of variations in the large, embedding theorems of funetion spaees, a priori estimates for solutions of differential equations yield many sueh examples. We have attempted to piek out the most general inequalities and, in model eases, we exhibit effeetive geometrie eonstruetions and the means of proving sueh inequalities. A substantial part of this book deals with isoperimetrie inequalities and their generalizations, but, for all their variety, they do not exhaust the eontents ofthe book. The objeets under eonsideration, as a rule, are quite general. They are eurves, surfaees and other ma
