1850 Lecture Notes in Mathematics Editors: J.--M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris V. D. Milman G. Schechtman Geometric Aspects of Functional Analysis 2002--2003 Israel Seminar 2002--2003 123 Editors VitaliD.Milman DepartmentofMathematics TelAvivUniversity RamatAviv 69978TelAviv Israel e-mail:[email protected] http://www.math.tau.ac.il/˜milman GideonSchechtman DepartmentofMathematics TheWeizmannInstituteofScience P.O.Box26 76100Rehovot Israel e-mail:[email protected] http://www.wisdom.weizmann.ac.il/˜gideon LibraryofCongressControlNumber:2004108636 MathematicsSubjectClassification(2000):46-06,46B07,52-0660-06 ISSN0075-8434 ISBN3-540-22360-6SpringerBerlinHeidelbergNewYork DOI:10.1007/b98686 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springeronline.com (cid:1)c Springer-VerlagBerlinHeidelberg2004 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:Camera-readyTEXoutputbytheauthor Printedonacid-freepaper 41/3142/du-543210 Preface Duringthe lasttwodecadesthe followingvolumescontainingthe proceedings of the Israel Seminar in Geometric Aspects of Functional Analysis appeared: 1983-1984 Published privately by Tel Aviv University 1985-1986 Springer Lecture Notes, Vol. 1267 1986-1987 Springer Lecture Notes, Vol. 1317 1987-1988 Springer Lecture Notes, Vol. 1376 1989-1990 Springer Lecture Notes, Vol. 1469 1992-1994 OperatorTheory:AdvancesandApplications,Vol.77,Birkhauser 1994-1996 MSRI Publications, Vol. 34, Cambridge University Press 1996-2000 Springer Lecture Notes, Vol. 1745 2001-2002 Springer Lecture Notes, Vol. 1807. Of these, the first six were edited by Lindenstrauss and Milman, the sev- enth by Ball and Milman and the last two by the two of us. As in the previous volumes, the current one reflects general trends of the Theory. Most of the papers deal with different aspects of Asymptotic Geo- metric Analysis, which includes classical “Local Theory of Normed Spaces” type questions, but now with the main emphasis on geometric problems. In addition, the volume contains papers on related aspects of Probability, clas- sical Convexityand also PartialDifferential Equations and BanachAlgebras. Last but not least, the volume contains two expository articles on two sub- jects which seem at first sight quite far removed from the main topic the Seminar deals with but whichprovedto be verymuch connected.One is Sta- tistical Learning Theory and the other is Models of Statistical Physics. We aregratefulto thetwoauthorsforagreeingtocontributethesesurveysto our volume,andweverymuchhope this inclusionwillsubstantiallycontributeto the developmentof the connectionof these fields with Asymptotic Geometric Analysis. All the papers here are original research papers (or invited expository papers) and were subject to the usual standards of refereeing. VI Preface As in previous proceedings of the GAFA Seminar, we also list here allthe talks given in the seminar as well as talks in related workshops and confer- ences. We believe this gives a sense of the main directions of research in our area. We are grateful to Ms. Diana Yellin for taking excellent care of the type- setting and some text editing aspects of this volume. Vitali Milman Gideon Schechtman The Start of GAFA Seminar Notes: Some Memories After 20 Years of Activity In 1983/1984Tel Aviv University published the first volume of what became the well-knownseriesofGAFA – GeometricAspects ofFunctional Analysis– IsraelSeminarNotes.SevenyearslaterthesameacronymGAFAwasadopted foranewjournal,GeometricAndFunctionalAnalysis,whichwasfoundedby M. Gromov andV. Milman. Now,20 yearson, I wouldlike to relate how and whytheGAFAseminarstarted,andhowthejournalGAFAcametothesame name. GillesPisieroncesaidtomewhileholdingacopyofthe1983/84volumeof GAFA, published privately in Tel Aviv: “It looks exactly like our ‘S´eminaire D’AnalyseFonctionnelle’;you‘stole’it!”.Yesandno.Wedidn’t“steal”itbe- causewepurposelycopiedit(whichisnotto saythatwelikeditsappearance ornot)!ThevolumesofSeminarNotesofEcolePolytechniquewhichwerefirst producedunderthetitle“S´eminaireMaurey-Schwartz”andthen,whenPisier took overfromMaurey,under the title “S´eminaireD’Analyse Fonctionnelle”, were privately published by the Centre de Math´ematiques, Ecole Polytech- nique. These volumes played an enormously important role in collecting and spreadingthemostup-to-dateknowledgeandinformationonGeometricFunc- tional Analysis,contributing to important developments in this theory in the 1970’s and early 1980’s. So, when G. Pisier declared that he was “tired” of producing the Seminar Notes andthat the 1980/81volume hadbeen his last, I was most upset and decided to act. Actually, since the 1980/1981academic year, Joram Lindenstrauss and I had been holding regular Israeli Seminars in Tel Aviv, the geographical center of Israel. The stream of foreign visitors brought important scientific news, and so the research center for Geometric Functional Analysis shifted to Israel. So, in 1983, Joram Lindenstrauss and I agreedto collect all the talks we had that year and to continue in the French traditionbycopyingthe appearanceof“S´eminaireD’Analyse Fonctionnelle”. I waslucky;an excellentstudent by the name ofHaim Wolfson,was finishing his Ph.D.that yearunder my supervision.He alreadyhada coupleofresults, knowntodaytoeveryexpertinthefield(unfortunatelyforus,hechangedhis VIII Start of GAFA direction to Computational Biology, and became a top expert in that field). So, Haim Wolfson was put in charge of the actual production (and later the distribution) of the volume. It was his idea to create the name “GAFA”. He feltthatthe name shouldsomehowbe connectedto TelAviv,andGAFA was the closest we could come to Jaffa, the oldest and most historical part of Tel Aviv.I acceptedthe idea immediately,butrealizedonlylaterhowstrongand appealing this short expression is. The production of the first issue was too costly for us. Having no way of funding the next issue, Joram Lindenstrauss and I agreed to turn to Springer Lecture Notes. Today, it seems strange that we were reluctant to do so, and were actually forced into it. As for the logo of GAFA, it was created by one of the most remarkable and distinguished Israeli painters, Jan Rauchwerger.Though a greatpainter, he was not a graphic artist, but being my closest friend, it was difficult for him to refuse my request for help. Years later, he also controlled the design of the cover of the journal GAFA, as well as any slides I used, or posters of conferences I organized. Today, his taste can be seen in many places in Mathematics, and some people think it is my taste. Now,continuingtothejournalGAFA,in1987,aBirkha¨userrepresentative approached me with the suggestion of producing a new journal. It was also around that time, in 1988/89, that Gromov and I decided that it would be worthwhile to create a forum to discuss the results related to what we called “Geometric Analysis”. I must emphasize here that our understanding of this combination of words,which we first made up in the mid 1980’s,was not the mechanical use of Analysis in Geometry or vice versa. It was the influence of analytical ideas in Geometry on a conceptual level and the asymptotic study of Geometry (asymptotics and estimates are the standard way of thinking in Analysis).Theonlymeaningforthisistheasymptoticsbydimension,namely, the study of geometric problemsin highdimensionalspaces,with special em- phasis on understanding the asymptotics by dimension. The Concentration Phenomenon and the immense progress in the asymptotic study of normed spaces during the mid 1980’s showed that the theory already existed and re- quired attention. To our dismay another group decided to launch a journal with the same name“GeometricAnalysis”just six monthsbefore ourjournal was scheduled to start. We tried negotiating but to no avail, and we had to urgently find a new name. I was in Zurich at that time for a 2-month visit andhalfthattimeIdevotedtotryingtoresolvethematter.Then“GAFA”,a nameIalreadylikedverymuch,sprungtomindandGromovinstantlyagreed. The dilemma was resolved and GAFA was born. Of course, this time instead of “Geometric Aspects of Functional Analysis”, GAFA became the acronym for “Geometric And Functional Analysis”. Vitali Milman Contents A Topological Obstruction to Existence of Quaternionic Plu¨cker Map S. Alesker....................................................... 1 Hard Lefschetz Theorem for Valuations and Related Questions of Integral Geometry S. Alesker....................................................... 9 SU(2)-Invariant Valuations S. Alesker....................................................... 21 The Change in the Diameter of a Convex Body under a Random Sign-Projection S. Artstein ...................................................... 31 An Elementary Introduction to Monotone Transportation K. Ball ......................................................... 41 A Continuous Version of the Brascamp–Lieb Inequalities F. Barthe ....................................................... 53 Inverse Brascamp–Lieb Inequalities along the Heat Equation F. Barthe, D. Cordero-Erausquin .................................. 65 Pinched Exponential Volume Growth Implies an Infinite Dimensional Isoperimetric Inequality I. Benjamini, O. Schramm ........................................ 73 On Localization for Lattice Schr¨odinger Operators Involving Bernoulli Variables J. Bourgain ..................................................... 77 X Contents Symmetrization and Isotropic Constants of Convex Bodies J. Bourgain, B. Klartag, V. Milman................................101 On the Multivariable Version of Ball’s Slicing Cube Theorem E. Gluskin ......................................................117 Geometric Probability and Random Cotype 2 E. Gluskin, V. Milman ...........................................123 Several Remarks Concerning the Local Theory of Lp Spaces W.B. Johnson, G. Schechtman ....................................139 On John-Type Ellipsoids B. Klartag ......................................................149 Isomorphic Random Subspaces and Quotients of Convex and Quasi-Convex Bodies A.E. Litvak, V.D. Milman, N. Tomczak-Jaegermann .................159 Almost Euclidean Subspaces of Real (cid:1)n with p an Even p Integer Yu.I. Lyubich....................................................179 Geometric Parameters in Learning Theory S. Mendelson....................................................193 Essential Uniqueness of an M-Ellipsoid of a Given Convex Body V.D. Milman, A. Pajor ...........................................237 On the Thermodynamic Limit for Disordered Spin Systems L. Pastur .......................................................243 On Read’s Proof that B((cid:1)1) Is Not Amenable G. Pisier .......................................................269 An Isomorphic Version of the Busemann–Petty Problem for Gaussian Measure A. Zvavitch .....................................................277 Seminar Talks (with Related Workshop and Conference Talks) ...............285 A Topological Obstruction to Existence of Quaternionic Plu¨cker Map S. Alesker School of Mathematical Sciences, Tel AvivUniversity,Tel Aviv 69978, Israel [email protected] Summary. It is shown that there is no continuous map from the quaternionic GrassmannianHGrk,n(k≥2, n≥k+2) to thequaternionic projective space HP∞ such that the pullback of the first Pontryagin class of the tautological bundle over HP∞ is equal to the first Pontryagin class of the tautological bundle overHGrk,n. In fact some more precise statement is proved. 1 Introduction This note is a bi-product of an attempt to understand linear algebra over the (noncommutative)fieldofquaternionsH.Forthe basicmaterialonlinear algebra over noncommutative fields we refer to [Art], [GGRW]. For quater- nionic linear algebra, see [As], [Al1], [GRW], and for further applications of it to quaternionic analysis, see [Al1], [Al2]. The main results of this note are Theorem 1 and its refinement Theorem 2 below. But first let us discuss what motivated them. Roughly speaking, it is shown that there is a topological obstruction to fillin the lastrowin the lastcolumnofthe followingtable (comparewiththe table in [Ar]). R C H w c p 1 1 1 rat. equiv. RP∞ =K(Z/2Z,1) CP∞ =K(Z,2) HP∞ −→ K(Z,4) (cid:1) (cid:1) (cid:2) (cid:3) w (V)=w ( topV) c (V)=c ( topV) p (V)=p ? 1 1 1 1 1 1 Inthistable,V denotesavectorbundle(respectivelyreal,complex,orquater- nionic) over some base, and K(π,n) denotes the Eilenberg–MacLane space. S.Aleskeretal.:LNM1850,V.D.MilmanandG.Schechtman (Eds.),pp.1–7,2004. (cid:1)c Springer-VerlagBerlinHeidelberg2004