Table Of Content1850
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
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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
