Integral Geometry and Valuations PDF
Preview Integral Geometry and Valuations
Advanced Courses in Mathematics CRM Barcelona Semyon Alesker Joseph H.G. Fu Integral Geometry and Valuations Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Carles Casacuberta More information about this series at http://www.springer.com/series/5038 Semyon Alesker • Joseph H.G. Fu Integral Geometry and Valuations Editors for this volume: Eduardo Gallego, Universitat Autònoma de Barcelona Gil Solanes, Universitat Autònoma de Barcelona Semyon Alesker Joseph H.G. Fu Department of Mathematics Department of Mathematics Tel Aviv University University of Georgia Tel Aviv, Israel Athens, GA, USA I SSN 2297-0304 ISSN 2297-0312 (electronic) ISBN 978-3-0348-0873-6 ISBN 978-3-0348-0874-3 (eBook) DOI 10.1007/978-3-0348-0874-3 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2014952675 Mathematics Subject Classification (2010): Primary: 52B45, 53C65; Secondary: 52A39 © Springer Basel 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com) Foreword This book contains a revised and expanded version of the notes of the lectures givenbySemyonAleskerandJosephH.G.FuintheAdvancedCourseonIntegral Geometry and Valuation Theory that took place from September 6th to 10th, 2010 at the Centre de Recerca Matem`atica (CRM) in Bellaterra, Barcelona. This activitywasintendedasamodernintroductiontointegralgeometry,withaspecial emphasis on the new ideas coming from the theory of convex valuations. Valuations are finitely additive functionals on the space of convex bodies. Hadwiger’s famous theorem characterizes continuous, rigid-motion-invariant val- uations as linear combinations of the intrinsic volumes. This deep result allowed an axiomatic approach to integral geometry, although only in Euclidean space. After Hadwiger, the study of valuations has been an active subject, with essential contributionsbyP.McMullen,D.Klain,R.Schneiderandothers.Abreakthrough inthetheoryhappenedin2001,whenS.Aleskerprovedhisirreducibilitytheorem. Thisisastrongresultthatledtothediscoveryofseveralnaturalstructuresonthe spaceofvaluations.Amongthem,thealgebrastructureonthespaceofvaluations is probably the most useful. Indeed, this structure is in a sense dual to the kine- matic formulas that lie at the base of integral geometry. As shown by A. Bernig and J. Fu, the study of this algebraic structure of the valuation space provides a way to successfully determine the integral geometry of many spaces that were out of reach with the classical methods. The Advanced Course on Integral Geometry and Valuation Theory reported onthisrecentprogress,providingatthesametimeanintroductiontothesubject. There were two series of lectures, delivered by Semyon Alesker and Joseph Fu. These lectures were complemented with several invited and contributed talks by J.C. A´lvarez-Paiva, A. Bernig, L.M. Cruz-Orive, N. Dutertre, F. Fodor, D. Hug, T. Leinster, E. Vedel-Jensen, and S. Willerton. We would like to express our gratitude to the director and the staff of the CentredeRecercaMatema`ticaformakingpossiblethisactivity.Wealsothankthe Ministerio de Ciencia e Innovacio´n of the Spanish government (refs. MTM2009- 0876-E and MTM2009-06054-E) and the Consolider Ingenio Mathematica pro- gramme (ref. MIGS-C5-0328) for providing financial support. Special thanks are due to S. Alesker and J. Fu for the enthusiasm they showed towards this course and the careful preparation of these notes. Eduardo Gallego and Gil Solanes v Contents 1 New Structures on Valuations and Applications 1 Semyon Alesker Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Translation-invariant valuations on convex sets . . . . . . . . . . . 3 1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 McMullen’s theorem and mixed volumes . . . . . . . . . . . 4 1.1.3 Hadwiger’s theorem . . . . . . . . . . . . . . . . . . . . . . 5 1.1.4 Irreducibility theorem . . . . . . . . . . . . . . . . . . . . . 6 1.1.5 Klain–Schneider characterization of simple valuations . . . 7 1.1.6 Smooth translation-invariant valuations . . . . . . . . . . . 8 1.1.7 Product on smooth translation-invariant valuations and Poincar´e duality . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.8 Pull-back and push-forward of translation-invariant valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.9 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.10 Hard Lefschetz type theorems . . . . . . . . . . . . . . . . . 14 1.1.11 A Fourier-type transform on translation-invariant convex valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.12 General constructions of translation-invariant convex valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.1.13 Valuations invariant under a group . . . . . . . . . . . . . . 21 1.2 Valuations on manifolds . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.1 Definition of smooth valuations on manifolds and basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.2 Canonical filtration on smooth valuations . . . . . . . . . . 25 1.2.3 Integration functional . . . . . . . . . . . . . . . . . . . . . 25 1.2.4 Product operation on smooth valuations on manifolds and Poincar´e duality . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.5 Generalized valuations and constructible functions . . . . . 27 1.2.6 Euler–Verdier involution . . . . . . . . . . . . . . . . . . . . 29 1.2.7 Partial product operation on generalized valuations. . . . . 30 1.2.8 A heuristic remark . . . . . . . . . . . . . . . . . . . . . . . 31 1.2.9 A few examples of computation of the product in integral geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.2.10 Functorial properties of valuations . . . . . . . . . . . . . . 33 1.2.11 Radon transform on valuations on manifolds . . . . . . . . 36 1.2.12 Khovanskii–Pukhlikov-typeinversionformulafortheRadon transform on valuations on RPn . . . . . . . . . . . . . . . 38 vii viii Contents Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2 Algebraic Integral Geometry 47 Joseph H.G. Fu Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 About these notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . 49 2.1 Classical integral geometry . . . . . . . . . . . . . . . . . . . . . . 49 2.1.1 Intrinsic volumes and Federer curvature measures. . . . . . 50 2.1.2 Other incarnations of the normal cycle . . . . . . . . . . . . 53 2.1.3 Crofton formulas . . . . . . . . . . . . . . . . . . . . . . . . 53 2.1.4 The classical kinematic formulas . . . . . . . . . . . . . . . 54 2.1.5 The Weyl principle . . . . . . . . . . . . . . . . . . . . . . . 57 2.2 Curvature measures and the normal cycle . . . . . . . . . . . . . . 60 2.2.1 Properties of the normal cycle . . . . . . . . . . . . . . . . 60 2.2.2 General curvature measures . . . . . . . . . . . . . . . . . . 61 2.2.3 Kinematic formulas for invariant curvature measures . . . . 62 2.2.4 The transfer principle . . . . . . . . . . . . . . . . . . . . . 64 2.3 Integral geometry of Euclidean spaces via Alesker theory . . . . . . 67 2.3.1 Survey of valuations on finite-dimensional real vector spaces 67 2.3.2 Constant coefficient valuations . . . . . . . . . . . . . . . . 70 2.3.3 The FTAIG for isotropic structures on Euclidean spaces . . 73 2.3.4 The classical integral geometry of Rn . . . . . . . . . . . . . 76 2.4 Valuations and integral geometry on isotropic manifolds . . . . . . 79 2.4.1 Brief definition of valuations on manifolds . . . . . . . . . . 79 2.4.2 First variation, the Rumin operator, and the kernel theorem 81 2.4.3 The filtration and the transfer principle for valuations . . . 85 2.4.4 The FTAIG for compact isotropic spaces . . . . . . . . . . 86 2.4.5 Analytic continuation . . . . . . . . . . . . . . . . . . . . . 87 2.4.6 Integral geometry of real space forms . . . . . . . . . . . . . 91 2.5 Hermitian integral geometry . . . . . . . . . . . . . . . . . . . . . . 94 2.5.1 Algebra structure of ValU(n)(Cn) . . . . . . . . . . . . . . . 95 2.5.2 Hermitian intrinsic volumes and special cones . . . . . . . . 97 2.5.3 Tasaki valuations and a mysterious duality . . . . . . . . . 100 2.5.4 Determination of the kinematic operator of (Cn,U(n)) . . . 102 2.5.5 Integral geometry of complex space forms . . . . . . . . . . 103 2.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Chapter 1 New Structures on Valuations and Applications Semyon Alesker Introduction The theory of valuations on convex sets is a classical part of the topic of onvexity, with traditionally strong relations to integral geometry. During the roughly last 15yearsaconsiderableprogresswasmadeinvaluationtheoryanditsapplications to integral geometry. The progress is both conceptual and technical: several new structuresonvaluationshavebeendiscovered,newclassificationresultsofvarious specialclassesofvaluationshavebeenobtained,thetoolsusedinvaluationtheory anditsrelationswithotherpartsofmathematicshavebecomemuchmorediverse —besidesconvexityandintegralgeometry,onecanmentionrepresentationtheory, geometricmeasuretheory,elementsofcontactgeometry,andcomplexandquater- nionic analysis. This progress in valuation theory has led to new developments in integral geometry, particularly in Hermitian spaces. Some of the new structures turned out to encode in an elegant and useful way important integral geometric information: for example, the product operation on valuations encodes somehow the principal kinematic formulas in various spaces. Quiterecently,generalizationsoftheclassicaltheoryofvaluationsonconvex setstothecontextofmanifoldswereinitiated;thisdevelopmentextendstheappli- cability of valuation theory beyond affine spaces, and also covers a broader scope ofintegralgeometricproblems.Inparticular,thetheoryofvaluationsonmanifolds providesacommonpointofviewonthreeclassicalandpreviouslyunrelateddirec- tions of integral geometry: Crofton-style integral geometry, dealing with integral geometric and differential geometric invariants of sets and their intersections, and with projections to lower-dimensional subspaces; Gelfand-style integral geometry, © Springer Basel 2014 1 S. Alesker, J.H.G. Fu, Integral Geometry and Valuations, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-0348-0874-3_1
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