Geometric Aspects of Functional Analysis: Israel Seminar 2001-2002

437 Pages·2003·3.057 MB·English

by  F. Barthe, M. Csörnyei, A. Naor (auth.), Vitali D. Milman, Gideon Schechtman (eds.)

1807
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
3
Berlin
Heidelberg
NewYork
HongKong
London
Milan
Paris
Tokyo
V. D. Milman G. Schechtman (Eds.)
Geometric Aspects
of Functional Analysis
2001--2002
Israel Seminar
2001--2002
1 3
Editors
VitaliD.Milman
DepartmentofMathematics
TelAvivUniversity
RamatAviv
69978TelAviv
Israel
e-mail:[email protected]
GideonSchechtman
DepartmentofMathematics
TheWeizmannInstituteofScience
P.O.Box26
76100Rehovot
Israel
e-mail:[email protected]
http://www.wisdom.weizmann.ac.il/˜gideon
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MathematicsSubjectClassification(2000):46-06,46B07,52-0660-06
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Preface
Duringthelasttwodecadesthefollowingvolumescontainingtheproceedings
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.
Of these, the first six were edited by Lindenstrauss and Milman, the sev-
enth by Ball and Milman and the last by the two of us.
As in the previous volumes, the current one reflects general trends of
the Theory. The connection between Probability and Convexity continues to
broadenanddeepenandanumberofpapersofthiscollectionreflectthisfact.
Thereisarenewedinterest(andhopeforsolution)intheoldandfascinating
slicing problem (also known as the hyperplane conjecture). Several papers in
this volume revolve around this conjecture as well as around some related
topics as the distribution of functionals, regarded as random variables on
a convex set equipped with its normalized Lebesgue measure. Some other
papersdealwithmoretraditionalaspectsoftheTheoryliketheconcentration
phenomenon. Finally, the volume contains a long paper on approximating
convex sets by randomly chosen polytopes which also contains a deep study
of floating bodies, an important subject in Classical Convexity Theory.
All the papers here are original research papers and were subject to the
usual standards of refereeing.
As in previous proceedings of the GAFA Seminar, we also list here all
the talks given in the seminar as well as talks in related workshops and
conferences. 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 aspects of this volume.
Vitali Milman
Gideon Schechtman

Table of Contents
A Note on Simultaneous Polar and Cartesian
Decomposition
F. Barthe, M. Cso¨rnyei and A. Naor................................. 1
Approximating a Norm by a Polynomial
A. Barvinok......................................................... 20
Concentration of Distributions of the Weighted
Sums with Bernoullian Coefficients
S.G. Bobkov......................................................... 27
Spectral Gap and Concentration for Some
Spherically Symmetric Probability Measures
S.G. Bobkov......................................................... 37
On the Central Limit Property of Convex Bodies
S.G. Bobkov and A. Koldobsky....................................... 44
On Convex Bodies and Log-Concave Probability
Measures with Unconditional Basis
S.G. Bobkov and F.L. Nazarov....................................... 53
Random Lattice Schr¨odinger Operators with Decaying
Potential: Some Higher Dimensional Phenomena
J. Bourgain.......................................................... 70
On Long-Time Behaviour of Solutions of Linear
Schr¨odinger Equations with Smooth Time-Dependent
Potential
J. Bourgain.......................................................... 99
On the Isotropy-Constant Problem for “PSI-2”-Bodies
J. Bourgain......................................................... 114
On the Sum of Intervals
E.D. Gluskin........................................................ 122
Note on the Geometric-Arithmetic Mean Inequality
E. Gluskin and V. Milman........................................... 131
Supremum of a Process in Terms of Trees
O. Gu´edon and A. Zvavitch.......................................... 136
Point Preimages under Ball Non-Collapsing Mappings
O. Maleva........................................................... 148
Some Remarks on a Lemma of Ran Raz
V. Milman and R. Wagner.......................................... 158
On the Maximal Perimeter of a Convex Set in Rn
with Respect to a Gaussian Measure
F. Nazarov.......................................................... 169
On p-Pseudostable Random Variables, Rosenthal
Spaces and ln Ball Slicing
p
K. Oleszkiewicz..................................................... 188
Ψ2-Estimates for Linear Functionals on Zonoids
G. Paouris.......................................................... 211
Maximal (cid:2)n-Structures in Spaces with Extremal
p
Parameters
G. Schechtman, N. Tomczak-Jaegermann and R. Vershynin.......... 223
Polytopes with Vertices Chosen Randomly from
the Boundary of a Convex Body
C. Schu¨tt and E. Werner............................................ 241
Seminar Talks
(with Related Workshop and Conference Talks)............... 423
A Note on Simultaneous Polar and Cartesian
Decomposition
F. Barthe1(cid:1), M. Cso¨rnyei2(cid:1)(cid:1) and A. Naor3(cid:1)(cid:1)(cid:1)
1 CNRS-Universit´e de Marne-la-Vall´ee, Equipe d’analyse et de Math´ematiques
appliqu´ees, Cit´e Descartes, Champs-sur-Marne, 77454, Marne-la-Vall´ee,
Cedex 2, France [email protected]
2 Department of Mathematics, University College London, Gower Street, London,
WC1E 6BT, United Kingdom [email protected]
3 Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
[email protected]
Summary. We study measures on Rn which are product measures for the usual
Cartesian product structure of Rn as well as for the polar decomposition of Rn
inducedbyaconvexbody.Forfiniteatomicmeasuresandforabsolutelycontinuous
measures with density dµ/dx = e−V(x), where V is locally integrable, a complete
characterization is presented.
1 Introduction
AsubsetK ⊂Rn iscalledastar-shapedbodyifitisstar-shapedwithrespect
to the origin, compact, has non-empty interior, and for every x (cid:2)= 0 there is
a unique r > 0 such that x/r ∈ ∂K. We denote this r by (cid:4)x(cid:4)K ((cid:4)·(cid:4)K is
theMinkowskifunctionalofK).Notethat(cid:4)x(cid:4)K isautomaticallycontinuous
(if xn tends to x (cid:2)= 0, then for every subsequence xn such that (cid:4)xn (cid:4)K
k k
convergestor,thecompactnessensuresthatx/r ∈∂K,sothatr =(cid:4)x(cid:4)K by
the uniqueness assumption). Any star-shaped body K ⊂Rn induces a polar
product structure on Rn\{0} through the identification
(cid:1) (cid:2)
x
x(cid:5)→ (cid:4)x(cid:4)K,(cid:4)x(cid:4)K .
InthisnotewestudythemeasuresonRn,n≥2whichareproductmeasures
withrespecttotheCartesiancoordinates,andtheabovepolardecomposition.
In measure theoretic formulation, we will be interested in the measures
µ on Rn which are product measures with respect to the product structures
(cid:1) Partially supported by EPSRC grant 64 GR/R37210.
(cid:1)(cid:1) Supported by the Hungarian National Foundation for Scientific Research, grant
# F029768.
(cid:1)(cid:1)(cid:1) Supported in part by the Binational Science Foundation Israel-USA, the Clore
Foundation and the EU grant HPMT-CT-2000-00037. This work is part of a
Ph.D. thesis being prepared under the supervision of Professor Joram Linden-
strauss.
V.D.MilmanandG.Schechtman(Eds.):LNM1807,pp.1–19,2003.
(cid:1)c Springer-VerlagBerlinHeidelberg2003
2 F. Barthe et al.
Rn = R×···×R = R+ ·∂K. Here × is the usual Cartesian product and
for R ⊂ R+, Ω ∈ ∂K, the polar product is by definition R·Ω = {rω; r ∈
R and ω ∈Ω}.Weadoptsimilarnotationforproductmeasures:⊗willbe
usedforCartesian-productmeasuresand(cid:9)forpolar-productmeasures.With
this notation, we say that µ has a simultaneous product decomposition with
respecttoK iftherearemeasuresµ1,...µn onRsuchthatµ=µ1⊗···⊗µn,
and there is a measure τ on R+ and a measure ν on ∂K such th(cid:3)at µ=τ(cid:9)ν
(in what follows, all measures are Borel). Notation like Ak or iAi always
refers to the Cartesian product.
For probability measures one can formulate the notion of simultaneous
product decomposition as follows. A measure µ on Rn has a simultane-
ous product decomposition with respect to K if and only if there are in-
dependent real valued random variables X1,...,Xn such that if we denote
X = (X1,...,Xn) then µ(A) = P(X ∈ A) and X/(cid:4)X(cid:4)K is independent of
(cid:4)X(cid:4)K.
The standard Gaussian measure on Rn is obviously a Cartesian product.
A consequence of its rotation invariance is that it is also a polar-product
measure for the usual polar structure induced by the Euclidean ball. Many
characterizationsoftheGaussiandistributionhavebeenobtainedsofar.The
motivations for such characterizations arise from several directions. Maxwell
proved that the Gaussian measure is the only rotation invariant product
probability measure on R3, and deduced that this is the distribution of the
velocities of gas particles. The classical Cramer and Bernstein characteriza-
tions of the Gaussian measure, as well as the numerous related results that
appearedintheliteraturearosefromvariousprobabilisticandstatisticalmo-
tivations. We refer to the book [Br] and the references therein for a detailed
account.ThemoremoderncharacterizationduetoCarlen[C]arosefromthe
need to characterize the equality case in a certain functional inequality.
To explain the motivation for the present paper, we begin by noting that
the Gaussian density is in fact one member of a wider family of measures
with simultaneous product decomposition, involving bodies other than the
Euclidean ball. They will be easily introduced after setting notation. The
cone measure on the boundary of K, denoted by µK is defined as:
(cid:4) (cid:5)
µK(A)=vol [0,1]·A .
This measure is natural when studying the polar decomposition of the
Lebesgue measure with respect to K, i.e. for every integrable f : Rn → R,
one has (cid:6) (cid:6) (cid:6)
+∞
f(x)dx= nrn−1 f(rω)dµK(ω)dr.
Rn 0 ∂K
(cid:7)For the particular case K = Bpn = {x ∈ Rn; (cid:4)x(cid:4)p ≤ 1}, where (cid:4)x(cid:4)p =
( ni=1|xi|p)1/p,afundamentalresultofSchechtmanandZinn[SZ1](seealso
Rachev and Ru¨schendorff [RR]), gives a concrete representation of µK: