The Volume of Convex Bodies and Banach Space Geometry

94

THE VOLUME OF CONVEX BODIES AND BANACH SPACE

GEOMETRY

CAMBRIDGE TRACTS IN MATHEMATICS General Editors B. BOLLOBAS, F. KIRWAN, P. SARNAK, C. T. C. WALL

94

The volume of convex bodies and Banach space geometry

GILLES PISIER Universite Paris VI

The volume of convex bodies and Banach space geometry

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press The Edinburgh Building, Cambridge C132 8RU, UK

Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521364652

© Cambridge University Press 1989

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 First paperback edition 1999

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data Pisier, Gilles, 1950The volume of convex bodies and Banach space geometry (Cambridge tracts in mathematics; 94) Bibliography: p. Includes indexes. 1. Banach spaces. 2. Inequalities (Mathematics) 1. Title. II. Series. QA322.2.P57 1989

515.7'32

88-35356

ISBN 978-0-521-36465-2 hardback ISBN 978-0-521-66635-0 paperback

Transferred to digital printing 2007

Contents

Introduction

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Chapter 1. Notation and Preliminary Background . Notes and Remarks . . . . . . . . . . . . . Chapter 2. Gaussian Variables. K-Convexity Notes and Remarks . . . . . . . . . . Chapter 3. Ellipsoids Notes and Remarks

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27 40

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41

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57

Chapter 5. Entropy, Approximation Numbers, and Gaussian Processes . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . .

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61 84

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Chapter 4. Dvoretzky's Theorem Notes and Remarks . . . . .

Chapter 6. Volume Ratio Notes and Remarks .

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Chapter 7. Milman's Ellipsoids Notes and Remarks . . . .

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Chapter 8. Another Proof of the QS Theorem Notes and Remarks . . . . . . . . . . . Chapter 9. Volume Numbers Notes and Remarks . . .

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Chapter 10. Weak Cotype 2 Notes and Remarks . . . Chapter 11. Weak Type 2 Notes and Remarks . .

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Chapter 12. Weak Hilbert Spaces Notes and Remarks . . . . .

Chapter 13. Some Examples: The Tsirelson Spaces Notes and Remarks . . . . . . . . . . . . . v

89 97 99 123 127 137 139 148 151 168

169 187 189 204

205 215

Contents

vi

Chapter 14. Reflexivity of Weak Hilbert Spaces . Notes and Remarks . . . . . . . . . . . .

Chapter 15. Fredholm Determinants . Notes and Remarks . . . . . . . Final Remarks Bibliography . Index . . . .

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217 222 223 234 235 237 249

Introduction

The aim of these notes is to give a self-contained presentation of a number of recent results (mainly obtained during the period 1984-86) which relate the volume of convex bodies in R" and the geometry of the corresponding finite-dimensional normed spaces. The methods we employ combine traditional ideas in the Theory of Convex Sets (maximal volume ellipsoids, mixed volumes) together with Probability (Gaussian processes), Approximation Theory (Entropy numbers), and the Local Theory of Banach spaces (K-convexity, cotype and type). During the last decade, considerable progress was achieved in the Local Theory, i.e. the part of Banach Space Theory which uses finitedimensional (f.d. for short) tools to study infinite dimensional spaces. The paper [FLM] was one of the first to relate the analytic properties of a Banach space (such as type and cotype) with the geometry of its f.d. subspaces. More precisely, let B C R" be a ball (by this we mean a compact convex symmetric body admitting the origin as an interior point). In [FLM] the following question is studied: Fix e > 0, for which integer k does there exist a (1 + e)-Euclidean section of B? Equivalently, for which k does there exist a subspace F C R" with dimension k and an ellipsoid D C F such that (0.1)

DCBnFC(1+e)D?

By a celebrated result of Dvoretzky, this is always true for k = k(e, n), with k(e, n) oo when n -+ oo (with e > 0 fixed). More precisely ([M6]), this holds for k = [O(e) Log n], with 0(e) > 0 depending only on e. This estimate is the best possible in general; indeed, where

B is the unit cube B = [-1,1]' (i.e. the unit ball of f') 00 then this Log n bound cannot be improved. One of the main discoveries of [FLM] is that if B and all its sections are far from cubes, in a certain analytic sense, then the estimate can be improved to k(e, n) = [O(e)na]

for some a > 0. vii

Introduction

viii

Precisely, if the normed space E for which B is the unit ball has cotype q(2 < q < oo) with constant C then this holds with a = 2/q and O(e) depending only on e and C. The most striking case is the case q = 2 (hence a = 1) for which we find in answer to (0.1) an integer k proportional to n. For instance, this case covers the case of E = PP for 1 < p < 2. By duality, (0.1)

implies that B°, the polar of the ball B, admits a projection onto F which is (1 + e)-equivalent to an ellipsoid. Namely, if PF is the orthogonal projection from Rn onto F, we have (0.2)

(1 + e)-1D° C PF(B°) C D°.

Since B is arbitrary, we can replace B by B° in (0.2). Thus, Dvoretzky's Theorem says that any n-dimensional body B admits both kdimensional sections and k-dimensional projections which are almost ellipsoids with k -* oo when n - oo. However, in general k must remain small compared with n. One of the striking recent discoveries of Milman [M1] is that if one considers the class of all projections of sections of B (instead of either

projections or sections) then one can always find a projection of a section of B (say PF2 (Fl fl B) with F2 C F1 C Rn) which is (1 + e)equivalent to an ellipsoid and has dimension k = [V)(e)n] with 0(e) > 0 depending only on e > 0. Thus we find again k proportional to n, but this time without any assumption on B.

This surprising result gives the impression that in a number of questions an arbitrary ball in Rn should behave essentially like an ellipsoid. We will see several illustrations of this in the present volume. We now describe the contents.

This book has two distinct parts. The objective of the first part (Chapters 1 to 8) is to present selfcontained proofs of three fundamental results: (I) The quotient of subspace Theorem (Q. S. -Theorem for short) due to Milman [Ml]: For each 0 < 6 < 1 there is a constant C = C(6) such that every n-dimensional nonmed space E admits a quotient

of a subspace F = E1/E2 (with E2 C El C E) with dimension dim F > 6n which is C-isomorphic to a Euclidean space.

(II) The inverse Santal6 inequality due to Bourgain and Milman [BM]: There are positive constants a and Q (independent of n) such that for all balls B C Rn we have a/n < (vol(B) vol(B°))1/n < /3/n

Introduction

ix

(The upper bound goes back to a 1949 article by Santald [Salt.) (III) The inverse Brunn-Minkowski inequality due to Milman [M5J: Two balls B1, B2 in Rn can always be transformed (by a volume preserving linear isomorphism) into balls B1i B2 which satisfy vol(Bl + B2)'1' < C [vol(B1)1jn + vol(B2)1hm"]

where C is _a numerical constant independent of n. Moreover, the polars B1, B2 and all their multiples also satisfy a similar inequality.

We present two different approaches to these results which can be read essentially independently. In Chapter 7, we reverse the chronological order; we prove (III) first and then deduce (II) and (I) as easy consequences. In Chapter 8 we prove (I) by essentially the original method of [Ml]

and then give a very simple proof that (I) implies (II). In the second part of the book (Chapters 10 to 15) we give a detailed exposition focused on the recently introduced classes of Banach spaces of weak cotype 2 or of weak type 2 and the intersection of these classes, the class of weak Hilbert spaces. The previous chapters contain complete proofs of all the necessary ingredients for these results and various related estimates. Let us now

review the contents of this book, chapter by chapter. Chapter 1 introduces some terminology, notation, and preliminary background. In Chapter 2 one of our main tools-the majorization of the Gaussian K-convexity constant of an n-dimensional space-is discussed in detail. The connections between Gaussian measures and volume estimates are numerous. For instance, consider the following classical formula where B is any ball in Rn, ryn the canonical probability measure on

R" and D the Euclidean unit ball in R': (0.3)

vol(B + tD) - vol(tD) tlim00

to-1 vol(D)

- cn1/2 J sup < t, x > dryn(x), tEB

where an is a constant tending to 1 when n --+ oo (see Remark 1.5 and the beginning of Chapter 9 for more information). We will see more connections when we come to Chapter 5. In Chapter 3 we present several properties of the maximal volume ellipsoids. This goes back to Fritz John who proved in a 1948 paper

Introduction

x

[Joh] that any ball B C Rn contains a unique ellipsoid of maximal volume. As observed more recently by D. Lewis [L], one may impose on the class of ellipsoids D various different types of constraint (instead of D C B) and consider in each case the ellipsoid of maximal volume.

In particular, for a given ball B, we can consider the ellipsoid of maximal volume among all ellipsoids D such that the quantity (0.3) is < 1. This ellipsoid (called the e-ellipsoid in the sequel) plays a crucial role in the proofs of (I), (II), and (III) above.

In Chapter 4 we present the proof of Dvoretzky's Theorem and several related facts. We follow the usual concentration of measure approach (cf. [M6], [FLM]) but we use Gaussian measures instead of the Haar measure on the orthogonal group. This approach underlines the close connection between geometric characteristics of a space (here

the dimension of the almost Euclidean sections of the unit ball) and Gaussian random variables. In Chapter 5 we present results from the theory of Gaussian processes, mainly the Dudley-Sudakov Theorem (Theorems 5.6 and 5.5) which gives both a lower and an upper bound for integrals such as (0.4)

E sup Xt tEB

when (Xt) is a Gaussian process indexed by a set B. These estimates are given in terms of the metric d(s, t) = 11 Xt - X9112 on B and involve the smallest number of balls of d-radius e which are enough to cover B. Such bounds for (0.4) can be related to volume estimates, for instance via (0.3).

These inequalities can be reformulated in the language of entropy numbers of compact linear operators. We present a brief introduction

to the theory of these numbers in Chapter 5. In particular, several results due to B. Carl will be very useful in subsequent chapters. In Chapter 6 we present the notion of volume ratio. The volume ratio of an n-dimensional space E with unit ball B is defined as vol(B)

( vol(Dma,,) /

1/n '

where Dmax C B is the maximal volume ellipsoid. This was introduced by Szarek [S], [ST] following earlier work of Kasin [Kal] in Approximation Theory. Szarek observed that the 11-balls have a volume ratio

uniformly bounded (when n -+ oc) and that this implies the striking orthogonal decomposition of ein (due to Kasin) as E1+E2 with El, E2

Introduction

xi

both uniformly isomorphic to e2. This is now often called the Ka"sin decomposition of £1. We present Szarek's proof of this in Chapter 6 as well as several properties of spaces with bounded volume ratio. To give the flavor of what goes on in this book, we wish to record here several interesting inequalities about the volume ratio. Let us denote by II II B the gauge of B. Let B2 be the canonical Euclidean

ball with its normalized surface measure a on the boundary. Note that we have for any norm on R' (0.5)

c,,n-1/2

f IIxIIB da(x) =

f IIxIIB d'Y.(x)

with c,,, as above (cn 1 when n --* oo). Integrating in polar coordinates, we have (see Chapter 6)

(

(0.6)

i/n

vol(B) vol(B 2)

)

=

U

By a classical inequality of Urysohn (see Chapter 1), we have 1/n

(0.7)

Y

IIxIIB'

da(x))

5 dim E which is C-isomorphic to a Euclidean space. As mentioned above, it was proved in [FLM] that this holds if X has cotype 2. Moreover, an example of Johnson (cf. [FLM]) shows that conversely (*) does not imply cotype 2, but only cotype 2 + e for every e > 0. Independently of [FLM] and almost simultaneously, Kasin ([Kal]) proved that (*) holds for E = £ (uniformly over n) and-following

Szarek's proof [S]-this holds more generally if X has uniformly bounded volume ratio, i.e. if (0.9)

sup{vr(E) I E C X dim E < oo} < oo.

The question was then raised whether these two different approaches are related and specifically whether cotype 2 implies (0.9). This was proved by Bourgain and Milman in [BM]. Shortly after, this was continued (with somewhat different methods) by Milman and the author who showed in particular that (*) implies (0.9) (cf. [MP1]). In the same paper, a weakened version of the notion of cotype 2 (weak cotype 2) was introduced and was shown to be equivalent to (*) and (0.9). Chapter 10 is mainly devoted to these results of [MP1]. In Chapter 11, we present the notion of weak type 2 which is somewhat dual to the preceding. A space X is weak type 2 if its dual X* is K-convex and weak cotype 2, cf. [MP1]. We also include a characterization of weak type 2 in terms of volume ratio due to A. Pajor [Pal].

Introduction

xiv

In Chapter 12 we discuss the weak Hilbert spaces, i.e. the spaces which are both weak cotype 2 and weak type 2. This chapter is based on [P5]. For instance, we show that X is a weak Hilbert space if one of the following properties (a), (b) holds.

(a) There is a constant C such that for all f.d. subspaces E C X there are ellipsoids D1, D2 in E such that

D1CBXnECD2 and

vo1(D2)

1/n

< C.

vol(D1)

(b) There is a constant C such that for all n and all x1, ... , xn, xi, ... , xn in the unit balls respectively of X and X* we have Idet(<xa,xj>)I1/n

IA2I >, ... , then any nuclear operator satisfies supra nlAnl < 00In other words, instead of (An) in t, we find (An) in weak-L1 (the space t1,,) and this characterizes weak Hilbert spaces. Analogous statements hold for weak cotype 2 and weak type 2 (see [P5] for a broader notion of weak-P when P is a given property of Banach spaces). This explains in part the choice of the adjective weak common to all these notions.

We should warn the reader, however, that the space often called weak-L2 (often denoted L2,,,) is not a weak Hilbert space; it is well known that this space contains an isomorphic copy of In general, we do not give references in the text, but in the Notes

and Remarks following each chapter. There are a number of exceptions to this rule, but, in any case, we warn the reader always to consult the Notes and Remarks section to find out to whom a given result should be credited. We apologize in advance for possible errors or for references which may have been omitted for lack of accurate information. This book is based on a series of lectures given at the University of Paris VI (spring 1985) and on graduate courses given in Paris VI (spring 1986) and at Texas A&M University (fall 1986). I am grateful to all those who participated in these lectures and especially to Alvaro Arias, who also did a careful proofreading of the first draft. I am also grateful to N. W. Naugle for his expert advice on 1X and for his help in dealing with diagrams. Finally, special thanks are due to Jan Want for her excellent typing. This work was partly supported by the NSF.

Chapter 1

Notation and Preliminary Background

Let E, F be Banach spaces. If E, F are isomorphic, we define their Banach-Mazur distance as (1.1)

d(E, F) = inf{IITII IIT-111}

where the infimum runs all isomorphisms T : E , F. If they are not isomorphic, we set d(E, F) = +oo. If E, F are isomorphic and if d(E, F) < A, we will say briefly that E and F are A-isomorphic (or that E is A-isomorphic to F). For all Banach spaces, E, F, G we have obviously

d(E, G) < d(E, F)d(F, G).

We denote by BE the unit ball of E and by IE the identity operator on E. In most of these notes we work with finite dimensional Banach spaces E, F. We will abreviate finite dimensional by f.d. The above distance is then particularly useful since any two spaces with the same dimension are isomorphic. Let K(n) be the set of all normed (or Banach) spaces of dimension n. For E, F in K(n), a simple compactness argument shows that the infimum is attained in (1.1) so that E, F are isometric if d(E, F) = 1. The relation E is isometric to F is clearly

an equivalence relation on K(n). Let us denote by K(n) the set of all classes modulo this equivalence. Then it is not hard to check that K(n) equipped with the metric 6(E, F) = Log d(E, F) is a compact metric space, sometimes called the Banach-Mazur compactum.

We will denote by $p the space Rn equipped with the norm IIxii = (Ei Ixtilp)1/p. In particular, .f2 is the n-dimensional Euclidean

space. The latter space plays a central role among the elements of K(n) (or of K(n)). In particular, we show in Chapter 3 a classical result of F. John [Joh]

d(E, ez) < n112 for all E in K(n). 1

Chapter 1

2

There has been in recent years a great deal of progress concerning the local theory of Banach spaces. This is the part of Banach space theory which uses mainly finite-dimensional tools and methods. In this theory an infinite-dimensional space is studied through the collection of all its finite-dimensional subspaces. For instance, by a fundamental theorem of Dvoretzky (cf. Chapter 4), every-infinite dimen-

sional space X contains for each n and e > 0 a subspace E such that d(E, BZ) < 1 + E. Recently these methods have been successfully applied to prove several inequalities on the volume of convex symmetric bodies in Rn. We will call these simply balls. More precisely, throughout the sequel, a ball will be a compact convex symmetric subset B C Rn with non-empty interior. Let II B be the gauge of B. For any ball B C Rn, the space Rn equipped with IIB is a Banach space admitting B as its unit ball. Conversely, any n-dimensional normed space E can be identified (in more than II

II

one way) with Rn and hence we may associate to E its unit ball BE C Rn. Note that two balls B1, B2 in Rn correspond to two isometric Banach spaces if there is a linear isomorphism u : Rn Rn such that u(Bl) = B2. If the normed space associated to a ball B C Rn is a Hilbert space, then B is an ellipsoid, i.e. there is an isomorphism u : Rn -* Rn which maps B onto the canonical Euclidean ball BI n. It will be useful to recognize geometrically the balls of subspaces and quotient spaces of an n-dimensional normed space E with unit ball BE C R. For subspaces this is immediate: let F be a subspace of E; clearly the section F fl BE can be viewed as the unit ball of the normed subspace F. We may also consider the quotient space E/F. Geometrically, this space corresponds not to sections of BE, but to linear projections of BE. Indeed, let P : E E be any lin-

ear projection such that kerP = F. Let G be the range of P. We equip G with the norm which admits P(BE) as its unit ball. Then G is isometric to E/F. In particular, we may wish to use the orthogonal projection PG into G, orthogonal with respect to a fixed scalar product on Rn (for instance the usual one). Then PG(BE) can be naturally identified with the unit ball of the normed space E/G -. In the sequel, we always denote by E* the dual space of E equipped

with the dual norm. Recall that for K C Rn, the polar set is defined as

K°={2ERnI <x,y> 1 and assume the lemma proved for n - 1. Consider f, g from R" into [0, oo]. Let y E R be fixed, define 0y : Rn-1 [0, oo] by cy(t) = 0(t, y) and similarly for fy and gy. Clearly, we have, if y = .\y1+(1-A)yo (yo, y1 E R),

¢y(Ar+(1-X )s) > fy1(r)\9yo (s)1-'\

for any r, s in Rn-1. Therefore by the induction hypothesis, fRn-1 qy >

(

1-a

a \fRn-1

fyl) (fRn-1 gy')

Finally, applying (1.2) one more time (here for n = 1), we obtain

f .0 dm = f

(fR1(dy)dy>

U f dm)A( f gdm)1-A.

This completes the proof of Lemma 1.2.

Proof of Theorem 1.1: The inequality (1.1) follows immediately from (1.2) with 0 = lAA+(1-A)B,

f = 1A,

9 = 1B.

Then the Brunn-Minkowski inequality (BM) follows by setting A = (vol(A))1/"(vol(A)1/n +

vol(B)1/n)-1

Notation and Preliminary Background

5

Note that if A' = vol(A)-1/"A and B' = vol(B)-1/nB, then (1.1) implies

vol(AA' + (1 - A)B') > 1,

(1.3)

and, since

AA' + (1 - A)B' =

n B (vol(A)1

+

1(B)1/n)'

(1.3) immediately implies (BM) by homogeneity.

Remark: For a different proof of (BM) and more information, we refer the interested reader to [Be], [BZ], [Eg] [BF], [Hal] and [Ha2].

The reader who wishes to learn about recent developments in the Classical Theory of Convex Sets should consult for instance [Sa2] and the collection of surveys [GW]. As a corollary, we can prove the classical isoperimetric inequality

in R. We recall that the area of the boundary of a convex compact subset of Rn can be defined by an approximating procedure from the simpler case of polytopes (cf. e.g. [Be] or [Eg].)

Corollary 1.3.

(Isoperimetric inequality) Let C be a compact convex subset of R' with non-empty interior. Let us denote by a(C) the area of its boundary. Let B2 be the canonical euclidean ball in Rn. Then

(a(C) \ 1/n-1

1/m

C vol(C)

vol(B2))

-

a(B2) ft

Proof: Indeed, the area can be derived from the volume in a simple way, since we have (1.4)

a(C) = liymt-1(vol(C + tB2) - vol(C)).

By (1.4), we have a(B2) = n vol(B2) and by (BM) vol(C + tB2) > (vol(C)l/n + t vol(B2)1/n)n > vol(C) + nt vol(B2)l/n vol(C) nn 1 + o(t)' hence

a(C) >-

Vol(C) vol( c)

n1 n

a(B2).

We can also deduce from (BM) a classical inequality of Urysohn [U] which we do not really use in the sequel, but which clarifies the relationship between our methods and classical inequalities such as (BM). We denote by S the Euclidean unit sphere of Rn and by o its normalized area measure.

Chapter 1

6

Corollary 1.4.

(Urysohn's Inequality) Let K be a compact subset

of R. Let IIXIIKO = sup{< x, y >, y E K}

for all x E Rn. Then

(vo l(B ))1/n 0 we have mi vol(Ai)1/n
, v) then we have (1.5)

f vol(At)1dv(t) < (vol

(fAt dv(t)) )

1/n

Now let (1, v) be the orthogonal group 0(n) equipped with its normalized Haar measure. Let At = t(K) for all t in 0(n). Then vol(At) = vol(K) for all t. On the other hand, f t(K) dv(t) is clearly (by symmetry) a multiple of the Euclidean ball B2. Hence

ft(K)dzi(t)

(1.6)

= AB2

for some number A > 0. Now let 1= be a fixed element of S. (For instance, let e = e1 the first basis vector of Rn.) Writing that the images under l; of both sides of (1.6) coincide, we find

f

e(t(K)) dv(t) = A[-1, 1].

Clearly, l;(t(K)) = [at, bt] with at = inf e(t(x)) and bt = sup (t(x)). xEK

This implies

f

xEK

bt dv(t) = A,

so that (1.5) implies vol(K)1/n
1 we have (0 < p < oc): (1.18)

c1(p)n-1/P < (vol (Bep))

1/n

1 of independent identically distributed (i.i.d. for short) Gaussian random variables (r.v. for

short) with the standard normal distribution. This means that {g-} are independent and for each n we have VtER

P(9n > t) =

(21r)-1/2

ft

e-X2/2 dx.

More generally, for any Borel subset B C Rn we have P({(91,

9n.) E B}) =

(2ir)-n/2

n

expxa) dxl IB 1

... dxn1

The property which will be of crucial importance in the sequel is the rotational invariance of the Gaussian distribution. Namely, if (aZ.j) is an orthogonal n x n matrix, then the sequence n

9j =

i = 1, 2, ... , n

ai.j gj

j=1

has the same distribution on Rn as the original sequence (gl, ... , 9n) This means that (2.1)

-

P((91, ... , 9n) E B) = P (91, ... , 9n) E B)

for any Borel subset B of Rn. This basic fact (2.1) is an immediate consequence of the invariance under O(n) of the standard Gaussian probability measure on Rn, which we denote by 'yn. (2.2)

1

1'n = exp - 2

n

z=1 13

xi dxl ... dxn.

Chapter 2

14

Let X be a random variable with values in a separable Banach space E. We will denote by dist(X) the distribution of X, i.e. the probability measure on E defined, for all Borel subset B C E, by dist(X)(B) = P(X E B). Recall that if X and Y have the same distribution, then we have Ecp(X) = EW(Y) for all Borel measurable functions cp : E -o R such that W(X) is integrable. The identity (2.1) has many consequences. For instance, for any

(al, ... , an) in R', we have (2.3)

dist(a19i + ... + an9n) = dist (9i ' (El

aiI2)1/2)

1

Indeed, we may, by homogeneity, assume E ai = 1 and then we just

use the fact that there is an A in 0(n) which maps a = (al, ... , an) into el (the first vector of the canonical basis of Rn). This formula (2.3) can also be deduced from the classical Fourier transform formula (2.4)

ei£X-x2/z

f

de E R

dx(27r)-1/2 = e-C2/2,

for all j > 1. By (2.3) (and the remark preceding it) we have n

n

i/z

P=1191ip1Elailz for all 0 < p < oo.

Let -y(p) = II9i II,,. Clearly II9i Ih < oo for all p < oo. Moreover, we have (2.6)

7(p) < K pl/z

if 1 < p < 00,

for some numerical constant K. By (2.5), the closed span of {gn} in Lp(f2, A, P) is isometric to Bz. Indeed, if (en) is the canonical basis of £2i we can define an isometric embedding J :£z - LP which maps en into gn for all n. By (2.5), the

range of J (i.e. J(P2)) is independent of 0 < p < oo. Let G be the closed span of {gn} in L2. Then, by (2.5), G = J(ez), G is closed in all Lp spaces for all 0 < p < oc, and we have (2.7)

`d x E G

IIxIIp = 7(p)IIxII2

Gaussian Variables. K-Convexity

15

It will be of interest to note that the closed subspace G C LP is

complemented in LP for 1 < p < oc. Let Q1 : L2 -+ G be the orthogonal projection. Then Q1 is bounded also from LP onto G for 1 < p < oo. Indeed, if 2 < p < oo, for any x in L2, we have by (2.7) 'Y(p)-1IIQ1x!Ip = IIQlXII2 O

is a positive contraction on L2.

For Qk we can take the orthogonal projection onto the span of the Hermite polynomials of degree exactly k on RN.. To check Lemma

Chapter 2

16

2.1, we need to recall first some properties of Hermite polynomials. We refer e.g. to [N] for more details. The Hermite polynomials in one real variable x will be denoted by (hn(x))n>o. They are determined by the generating formula /

(2.8)

dAER exp(Ax-

A2A2

An

\

I=

/

n>0

1t.

hn(x).

Note that ho(x) - 1 and hi(x) = x.

The sequence (hn)n>o is an orthogonal basis of L2(R,-yi). (-y1 is defined in (2.2) above.) More generally, the collection {hai (xi) - ha2 (x2) ...

han (xn)I (al, ... , an) E Nn}

forms an orthogonal basis of L2(Rn,-yn). We need more notation to extend this to RN.. We will denote simply by A the set of all a = (al, a2.... ), with an E N such that only finitely many an's are non zero.

Let lal = Ei>1 Jail and a! = al!a2!...

.

As usual, we denote by

R(N.) the space of all sequences (An)n>1 with An E R such that only finitely many An's are non zero. Then, for x in RN we write 00

< A,x > = E Anxn, n=1

and, for a in A, .. Aa = Aa1 1 Aa2 2 Now, we can define, for all a in A,

b' x e RN H,(x) = hai (x1)ha2(x2) ... and we have a multinomial generating formula generalizing (2.8): (2.9)

`d x E RN. VA E RN'

exp < A, x > - 2

An

= 1: ai H. (x). aEA

One then checks easily that {Ha I a E Al is an orthogonal basis of L2(RN, y ) (i.e. of L2(1,1P) with (S2, P) standard).

Let Hk be the closed span in L2 of the functions {Hal la! = k}. This is called the Wiener chaos of degree k. Let Qk : L2 -* Hk be

Gaussian Variables. K-Convexity

17

the orthogonal projection. Clearly, Qo(() = f (dP for all ( in L2. Moreover, when k = 1 (recall h1(xn,) = xn), H1 is the span in L2 of the sequence {xnIn > 1} and QI is (as above) the projection onto the span of the Gaussian i.i.d. sequence {xnIn > 1} on (R'-, -yam). It will be convenient to use the fact that the collection of functions

l

1

W,\(x) = exp(< A,x > -2 n

Anl

AER(N.) is total in L2, so that if two linear operators coincide on {W,\ IA E

then they coincide on L2. (The density of the span of {W,\} follows from the Stone-Weierstrass Theorem.)

Proof of Lemma 2.1: Let S be the linear span of {Hales E A}. Equivalently, S is the space of all polynomials in the variables x1, x2i .... We will write simply LP for Lp(1l, P). Note that S is dense

in LP for all0 0 for all f > 0. It remains to check that T(c) =

£kQk k>o

For that purpose, consider cpa as above.

Chapter 2

18

It is very easy to check that 7 eA

On the other hand, by (2.9), we have

(kQk

E1al

SPA =

k>0

aEA

a1

H.;

hence by (2.9) again

This shows that T(e) coincides with >ekQk on {V,\}; hence it coincides with it on L2. We now come to the notion of K-convexity which was introduced in

[MaP] and further studied in [P3]. In order to introduce this notion, we need to clarify a notation. Let (52, m) be any measure space, let L2 = L2(Sl, m), let T be an operator on L2 and let X be a Banach

space. We denote by Ix the identity operator on X. Then clearly T ® IX is a linear operator well defined on L2 ® X as follows:

`dcp1EL2 Vx1EX T®Ix(Evi®x1) 1

/

1

For oD in L2 ® X, we define, as usual, the L2(X)-norm by 1/ 2

Y

dm(w))

and we define L2 (X) as the completion of L2 ® X with respect to this

norm. However, in general, the operator T 0 IX is not bounded on L2 ® X equipped with this norm and does not extend to L2(X). Actually, by a theorem of Kwapien [K2], the only spaces X for which T ® IX is bounded on L2 (X ), for all T bounded on L2, are those which are isomorphic to a Hilbert space. More generally [K2], the only spaces X for which T ® Ix is bounded on Lp(X), for all T bounded on LP, are those which are isomorphic to a subspace of a quotient of LP (here 1 < p < oo is fixed). We will use the following elementary and well-known fact:

Gaussian Variables. K-Convexity

19

Lemma 2.2.

Consider T : L2 -> L2 and a Banach space X as If T is positive (in the lattice sense) or if X is isomorphic to a Hilbert space, then T ® Ix extends to a bounded operator on above.

L2 (X ). We denote its norm by IIT ® Ix II . If T is positive we have

IIT ® Ix II = IITII

If X is isomorphic to a Hilbert space, we have IIT (9 Ix II < d(X, H)IITII.

(2.11)

Proof: (Sketch) Assume first that T is positive. Then for all sequences (coi) in L2 we have (2.12)

sup IT(wi)I < T(sup Iwi1). i i

Let -0 be an element of L2 ® X, and let co(w) = Clearly, (2.12) implies (using w(w) = sup{I < x*, 4)(w) > I x* E X* 11x* 11 0 and every n > 1 there is a subspace Xn C X such that d(Xn, in) < 1 + e. When (ii) holds we say that X contains el's uniformly. If a space X is isomorphic to a Hilbert space H, we have clearly

(cf. Lemma 2.2) K(X) < d(X, H). Actually, it turns out that this estimate can be improved considerably.

Theorem 2.5.

Assume X isomorphic to a Hilbert space H. Then K(X) < K(1 + Log d(X, H)), where K is a numerical constant.

This result is one of the crucial tools for the volume inequalities that we will prove in subsequent chapters. To prove Theorem 2.5, we will use

Lemma 2.6.

Let {xk I k > 0} be a sequence in a Banach space B, with only finitely many non-zero terms. Assume 00

max

-1<en

< 2n(1 + C2-n)

Hence, choosing n = [Log C] + 1, we obtain Lemma 2.6. (Note that if Log C is the logarithm in base 2, then we can take K = 4.)

Proof of (2.13):

By translation, it

is sufficient to prove

IQ'(0)l co(k)a,=iI:cjQ(27r4n)

so that

IQ'(o)I < E Icjl IIQII.= nIIQII.. Remark: Actually the inequality IIQ'Iloo < 2nIIQII". is enough for our purposes and is easier to prove. Indeed, let Fn be the Fejer kernel

Fn(t) = E (1 - nI+

1)eijt

li l

IIIxIII=sup

L2(X*)

Proof: Note that Gn*(X) and Gn(X*) can be naturally identified respectively with Xn and X*n, and the duality relation is the obvious

one < E gixi, x >= > < xi, xi >. Let us check that the norms fit. Let 4D be an arbitrary An-measurable element of L2(X). Define

xi = E(gi-D) for i = 1, 2,... , n. Then zb = D - En gixi is clearly in Nn(X) and E(< En gixi, 4D >) i

< xi,xi >.

Since we have obviously

E1

=sup{ L2(X*)

we obtain immediately

Y

*

gixi

= sup

{

-11 EL2(An;X) 1I'D 112

K(X)-1

t

2

2

Proof: Since Q1 ®Ix (V)) = 0 for all 0 in N, (X), we have < K(X )

E 9ixi 2

2

hence

< K(X)IllxIII . The first inequality then follows from Proposition 2.7. The second one is trivial.

Notes and Remarks Lemma 2.1 is a well-known classical property of the so-called Hermite semi-group (sometimes called also the Ornstein-Uhlenbeck semigroup). The kernel of the operator defined in (2.10) is often called the Mehler kernel. Lemma 2.2 is well known and elementary. The definition of the notion of K-convexity goes back to the last remarks in the paper [MaP]. The original definition used the Rademacher functions (i.e. an i.i.d. sequence of symmetric ±1-valued random variables) instead of Gaussian variables. Nevertheless, it was realized

very early that the two definitions are equivalent and the proofs of most results can be done in either setting with minor changes. Let Kr(X) be the K-convexity constant of a Banach space X, but using the Rademacher functions intead of (g,a) in Definition 2.3. Then it follows from known simple facts that K,.(X) < (7r/2)K(X) (see [FT] for details). Conversely, it was observed in [TJ1] that K(X) < Kr(X). Theorem 2.4 appeared in [P3]. Theorem 2.5 first appeared in [P7]. The simpler proof which we present via Lemma 2.6 is taken from [BM]. Proposition 2.7 is a simple observation going back to the introduction of K-convexity [MaP]. Similarly for Corollary 2.8.

Chapter 3

Ellipsoids

Historically, the ideas in this chapter originated in the 1948 work of Fritz John ([Joh]). John considered an ellipsoid of maximal volume included in the unit ball BE of an n-dimensional normed space E. By compactness the existence of such an ellipsoid is rather obvious but John also proved its unicity. Let us denote this unique ellipsoid (often

called the John ellipsoid of E) by D". F. John also proved that BE C ,rLDEax V

which implies immediately (since Dmax C BE by definition)

d(E, ) < Vfn_. By duality, this clearly implies the existence of a unique ellipsoid of minimum volume containing BE. We will denote it by DE"'. We have necessarily by polarity from the preceding inclusion n-1/2Dmin C BE C Dmin E E

More recently, D. Lewis formulated a generalization of F. John's result which played an important role in the recent development which we want to present below. To describe Lewis' idea, we first make precise what we mean by an

ellipsoid in an n-dimensional space E. We call ellipsoid in E every subset D C E which is the image of the canonical Euclidean ball Bt by a linear isomorphism. Hence, D = u(Be2)for some invertible

u: R' -+E. Note that if we have an other representation D = v(Bp2) for some isomorphism v : R" -+ E then necessarily v-1u and u-1v both preserve Bj2, which means that v-1u is an orthogonal transformation of R'n. Thus u is unique rnodulo an orthogonal transformation. In analytical terms, the ellipsoid of maximal volume Dm. included

into BE corresponds to an operator u : e2 -+ E with lull < 1 such that vol(u(B1 )) is maximal. 27

Chapter 3

28

It is well known that there is only one notion of volume on E, up to a multiplicative constant. Moreover, if E is equipped with a fixed linear basis, we may consider the determinant, denoted by det(u), of

any linear map u : R' -+ E. Then we have necessarily for some constant c > 0 vol(u(Bt )) = cl det(u)I. The idea of Lewis is to study the operators u which maximize I det(u)I when u runs over all operators such that IIJulII < 1, where I is now an arbitrary norm on B(PZ , E) (and not only the operator norm as in John's Theorem). Let E, F be two vector spaces. We will denote by £(E, F) the space of all linear operators from E I

I

II

I

into F. Let a be a norm on the space £(R' , E) of all linear operators from Rn into E. There is a well-known duality between L(Rn, E) and £(E, Rn) defined by `du E Q R n,E)

Vv E L(E,Rn)

< v, u >= tr(vu). Let e 1, ... , en be the canonical basis of Rn. To any u in £(Rn, E), we can associate x1, ... , xn on E defined by xi = u(ei). Similarly, to any v in £(E, Rn) we can associate xi, ... , xn in E* by letting xi = u*(ei). It is then easy to check that n

< v,u >= tr vu =

(3.1)

xi(xi).

We can introduce on £(E, Rn) a dual norm a* as follows: (3.2)

Vv : E -+ Rn a*(v) = sup{tr(vT)IT : Rn -+ E a(T) < 1}.

We now state Lewis' Theorem:

Theorem 3.1.

Let E be a vector space of dimension n and let a be a norm on £(R', E). Then there is an isomorphism u : Rn -+ E such that

a(u) = 1 and a*(u-1) = n. Remark 3.2: Using the correspondence u --> (u(el),... ,u(en)) between £(R' , E) and En, we can reformulate Theorem 3.1 as follows.

Ellipsoids

29

Let a be a norm on E'. Then there is a linear basis (x1, ... , x,,,) of E such that the biorthogonal functionals (xi, ... , 4) satisfy

a((xl,... , xn))a*((xi, ... , xl)) = n. Proof: Let u , det(u) be a determinant function (associated to a fixed linear basis of E). Let K = {u E C(Rn,E)ja(u) < 1}. Since K is compact, the determinant attains its supremum on K; hence there exists u in K such that I det(u) I = sup{I det(v) I I v E K}.

This implies for all T in £(Rn, E) det

u+T

a(u+T)))

I < I det(u)1;

hence, by homogeneity, (3.3)

1 det(u + T)l < det(u)I(a(u + T) )n.

Clearly, det(u) # 0, so that u is invertible, and dividing (3.3) by I det(u)I we find I

det(1 + u-1T)j < a(u + T)n;

hence by the triangle inequality < (1 + a(T) )n. Since T is arbitrary, we have for all e > 0 (3.4)

1 det(1 + eu-1T)I < (1 + ea(T))n.

But when e -* 0 det(1 + eu-1T) = 1 + etr u-1T + o(e); hence (3.4) implies

tr u-1T < na(T) for all T : E - Rn; equivalently, we have by (3.2)

a*(u-1) < n. On the other hand, we have trivially n = tru-lu < a(u)a*(u-1), hence a(u) = 1 and a*(u-1) = n. As an illustration, we derive a classical result of Auerbach.

Chapter 3

30

Corollary 3.3.

Let E be a normed space of dimension n. There is a basis x1, ... , xn of E such that n

(3.5)

V(c) E Rn

sup kaiI 0 such that

II EAieiIIE = 1 and

II

.i lei IIE = n.

Equivalently, we have n

V(ai) E Rn n-1

jail
0 small enough A + Eµ E Rn and O(A + ,Fu) < 11 E(A + Eµi)ei jjnO(A). Dividing by O(A), we get

1 + e E Ai 1µi + o(e) < 1 + nell E µiei ll + o(E), hence

µiei11 for all µ in R+.

Ai 1µi < n1l

By the unconditionality condition (3.6), this is equivalent to A2 7'e2!

< n-

On the other hand, n > II E Aiej (I E Ai lea II so that we obtain the announced result. We leave as an exercise to the reader to recognize that Corollary 3.4 is a special case of the following more abstract statement.

Corollary 3.5.

Let a be a norm on £(Rn, Rn). Let G be a compact subgroup of the group GL(n) of all invertible linear maps g : Rn -+Rn. Assume that for every g in G we have V u : Rn --+ Rn

a(u) = a(gug-1).

Then there is an isomorphism u : Rn -+ Rn which commutes with G such that a(u) = 1 and a*(u-1) = n.

Proof: (Sketch.) Consider T in £(Rn, RI). Let T = f gTg-1 dg where dg is the normalized Haar measure on G. Then T is in £(Rn, Rn)

and T commutes with G. We have (3.7) a(T) < a(T). Let then KG = {u : Rn -+ Rn ea(u) < 1, gu = ugVg E G}, and consider u in KG such that Idet(u)I = sup{jdet(w)jIw E KG}. We find tr u-1T < n for all T in KG. Using (3.7) and the identity tr u-1T = tru-1T for all T : Rn -a RI, we obtain Corollary 3.5.

Chapter 3

32

To recover Corollary 3.4 from Corollary 3.5, take for G the group of all diagonal matrices with diagonal coefficients ±1 and take

a(u) = sup{II E aju(ej)IIEI sup Iai4 < 1}. We now discuss the unicity of the ellipsoid which appears in Theorem 3.1.

Proposition 3.6. In the situation of Theorem 3.1, assume that a is invariant under the orthogonal group 0(n), more precisely that (3.8)

a(u) = a(uT) `d T E 0(n) V u : R' - E.

Then the operator u : R' -f E such that a(u) = 1 a*(u-1) = n is unique modulo 0(n). More precisely, if an isomophism v : R' E satisfies a(v) = 1 a*(v-1) = n, then necesssarily u-1v is an orthogonal transformation. Proof: Let v be as above and let u be as in the proof of Theorem 3.1. By the polar decomposition of u-1v, we have

u-1v = TA with T E 0(n) and A hermitian positive. Let A 1, ... , A,,, be the positive eigenvalues of A. We have v = uTA. The definition of u implies I det(v)I < det(u) 1; hence

det(u)

I det(uTA)I

and since I det(T)I = 1 we find

H Ai t2 is arbitrary, then (3.12)

f(uT) < t(u)IITII

Indeed, t(uT) is maximal on an extreme point of

IT: ti

--+ tz , IlTII L2(E)

hence by Corollary 2.8 we have t(v*) =

E

*

< K(E*)t*(v),

9ixi

L2(E*)

so that we can state (recalling K(E) = K(E*)):

Lemma 3.10. For any f.d. Banach space E and any v : E --+ t2 we have t(v*) < K(E)t*(v).

Remark. It is easy to extend the preceding result to the case of an infinite-dimensional Banach space E. Indeed, in that case let us denote by.F the class of all f.d. subspaces of E. For F in.F, we denote by iF : F --+ E the inclusion mapping. Obviously, for all

in E*, we have

IIeii = FEP IIfjFII = FEP IIZFeII

From this, it follows that for any v : E -+ t2

t(v*) = sup t((viF)*). FEY

Now by Lemma 3.10, we clearly have

t((viF)*)

K(F)t*(viF);

hence a fortiori

< K(E)t*(v). Therefore we obtain t(v*) < K(E)t*(v) in the infinite-dimensional case also.

The final statement is a recapitulation; it is a crucial tool in the sequel.

Ellipsoids

37

Theorem 3.11.

Let E be an n-dimensional Banach space. Then there is an isomorphism u : in -p E such that

£(u) = n1/2 and

£((u-1)*) < n1/2K(E).

Equivalently, there is a basis x1i... , x, in E with biorthogonal functionals xi, ... , x* such that

= n1/2 and

E giXi L2(E)

gixi IL(E-)

n1/2K(E).

Moreover, we have K(E) < K(1 + Log (d(E, i2 n))) for some numerical constant K.

Proof: By Theorem 3.1, there is an isomorphism u : e2 that £(u) = e*(u-1) = n1/2. By Lemma 3.9, we have £((u-1)*)
P2 such that (w) = tr vw for all w P2_-> X. Since e xtends v must extend u-1 and we have we conclude £*(v) _ $*(u-1). Since t*(v) = £(B2 ,E). Then

Chapter 3

38

We now define the B-norm in the infinite dimensional case. Consider

a (possibly infinite dimensional) Hilbert space H. Let u : H -> X be an operator with values in a Banach space X. For any f.d. subspace S C H with an orthonormal basis (el,... , en) we may identify S with BZ so that B(ins) is unambiguously defined. Note that by (3.12) it does not depend on the choice of the basis e1, ... , e,,, of S with which we identify S and B2. Then we can define the (possibly infinite) number (3.13)

B(u) = sup{B(uj,) I S C H, dim S < oo}.

(If H is finite dimensional, this coincides by (3.12) with the preceding definition (3.11).) The operators u for which f(u) < oo will be called B-operators and

the set of all such operators will be denoted by B(H, X). This set becomes a Banach space when equipped with the norm B. Note that (3.13) may be equivalently rewritten (using (3.12)) as B(u) = sup{B(uT)I n > 1,T : 0 -4 H, IITII H we have B(SuT) < IISIIB(u)IITII.

To give some examples of B-operators, it is easy to check using (1.15)

that every 2-summing operator u : H - X is an B-operator and that we have (3.14)

B(u) < 7r2(u).

(Indeed, consider S C H with orthonormal basis el,... , e,d and let = > giei. By (1.15) we have B(ins) = II Eg1u(ei)II L2(x) < However, in general, B-operators are more general than 2-summing operators. (To see this, it is enough to show that the norms 7r2 and t are not equivalent. For that purpose, consider the inclusion mapping j,,, : 0 B', then it is easy to check that 7r2(jn) > n1/2 while B(jn) is O((Log n) 1/2), so that 7r2 and f are not equivalent norms on the space Nevertheless, if we restrict of finite rank operators from B2 into ourselves to Hilbert spaces, we recover Hilbert Schmidt operators.

Ellipsoids

39

Proposition 3.13. Let H1, H2 be Hilbert spaces and let X be a Banach space isomorphic to H2. Then every £-operator u : H1 -+ X is 2-summing and we have ir2(u) < d(X, H2)t(u).

(3.15)

Moreover,

(3.16)

(u)2)1/2

X an

< d(X, H2)t(u).

Proof: Let x1,... , xn in H1 be such that (3.17)

r 2 1/2 I;EBH1} 0, there is a number rl(e) > 0 with the following property. Let E be an f, d. Banach space of dimension N. Then E contains a subspace F C E of dimension n = [ri(e) Log N] such that d(F,f) < 1 + E.

Remarks: (i) In geometric terms, the above theorem says that if n = [71(e) Log NJ

then any ball B C RN admits a section which is (1+e)-equivalent to an ellipsoid, i.e. there is a subspace F C RN with dim F = n

and an ellipsoid D C F such that

DCFfBC(1+e)D. 41

42

Chapter 4

Clearly, by duality, a similar statement holds for projections of B: There is an n-dimensional projection of B which is (1 + e)equivalent to an ellipsoid. (ii) In other words, every normed space E of dimension N admits, for n = [ij(e) Log N], an n-dimensional quotient space which is (1 + e)-isomorphic to f2'. This trivial dualization applies also to Theorem 4.1 and Corollary 4.2: every infinite dimensional

Banach space E admits, for each e > 0, a sequence {E,,,} of quotient spaces such that d(E., t 2n) 0 there is a number 7j1(e) > 0 for which the following statement holds. As before, let (gk) be an i.i.d. sequence of Gaussian normal random variables on some probablity space (Sl,A,P). Let E be a Banach space and let (Zk) be a sequence of ele-

ments of E. We assume that the series 00

X=Egkzk k=1

converges in L, (Q, A, P; E). Let

0' (X) = sup { (EI((zk)I2)1/2 (e'E* I

II(II 0, E contains a subspace F of dimension n= [,11(e)d(X)] which is (1 + e)-isomorphic to 42 .

Dvoretzky's Theorem

43

Note: We have EI((X)I = EIg1I (E I((zk)I2)1/2 by (2.5), and EIg1I = (2/7x)1/2, hence o-(X) < (ir/2)1/2EIIXII Remark 4.5: The preceding theorem is proved in detail below, but we wish to give first here a general idea of the proof: Let X1i X2, ... , X" be i.i.d. copies of the variable X on some probability space (1, A, P). Let 1l(n, e) be the set of all w in Sl such that

d (ai) E Rn

(1

+e)-1/2 (E

(ail2\1/2 J

< (IEaiXi(w)(EIIXII)-1I) x-1 1/2

(,+6)1/2 (1: IaiI2/

We will show below that P(Sl(n, e)) > 0, provided that n is not too large and precisely provided n < [77, (E) d(X)]. This clearly yields Theorem 4.4.

Remarks: (i) We call d(X) briefly the dimension of X (perhaps the term concentration dimension would be more appropriate). Note, however, that this notion of dimension depends not only on X but also on the norm of the Banach space E into which X takes its values. Also note that d(X) is a real number and not necessarily an integer. (ii) In [P1], we have defined d(X) as the ratio EIIXII2/o(X)2. Note that by Corollary 4.9 below this is equivalent to the above definition.

(iii) If E is f.d. with dim E = N, we know (cf. Chapter 3) that 7r2(IE) < N1/2. This implies, using (1.15) that EIIXII

(EIIXII2)1/2 < N1/2 SUP {(EI((X)I2)1/2I( E BE. }

.

Therefore we have d(X) < N = dim E. (iv) The above Theorem 4.4 reduces the task of proving Dvoretzky's Theorem for E to that of exhibiting E-valued Gaussian variables X with large dimension d(X). More precisely, let us denote by ne(X) the largest integer n such that there is an n-dimensional subspace F C E satisfying d(F, 4) < 1 + e. Also, let S(E) = sup d(X) where the supremum runs over all Evalued Gaussian variables X as in (4.1). Then by Theorem 4.4 we have for some function 7)2(E) > 0 (4.2)

ne(E) ? 72(E)6(E)

44

Chapter 4

There is also a converse estimate which shows that this measure theoretic approach to Dvoretzky's Theorem is somewhat sharp. This is much easier to check as follows.

Proposition 4.6. 8(E) \ 21

(1 + e)2 > ne(E).

Proof: Let F C E be such that d(F, f) < 1+e. Let T: e2 -- F be such that IITII IIT-' II < 1 + E. We define

X=

giT(ei) i=1

Then v(X) = IITII and

II

T-'E ll X II

II

> _ E IIT-'(X)II = E

Igi12

1/2

(i)

>

=n 1/2 (2/7r) 1/2.

1

This implies d(X) > (2/ir)n(IITII yields Proposition 4.6.

IIT-'II)-2 > (2/7r)n(1+e)-2, which

For the proof of Theorem 4.4, the following estimation of the deviation of 11X II from its mean will be crucial. In Milman's terminology, this is a concentration of measure phenomenon.

Theorem 4.7. (4.3)

Let X be as in Theorem 4.4. We have

`d t > 0 P {I IIXII - EIIXIII > t} < 2exp(-Kt2/cr(X)2),

where K is a numerical constant (K = 27r-2 in the proof below).

Proof: We may clearly assume that E is f.d. and that the series (4.1) defining X is actually a finite sum,

X = > gkzk 1

(zk E E).

Dvoretzky's Theorem

45

We may also assume non-degeneracy so that the distribution of X

is equivalent to the Lebesgue measure on E. Then let u : t2 - E be the linear operator defined by

VxERmm u(x)

xkzk.

Let -y.. be the canonical Gaussian probability measure on R. Then (4.3) reduces to the following inequality (4.4)

l m Sx E R-jjjju(x)II - f jIu(x)IId'YmI > ty 1 (EIIXIIP)11P

EIIXII 0. There is a subset A C S which is a 6-net of S with respect to the norm III III with cardinality

)fl. card(A) < (1 + 2 Proof: Let (yi)i 6 for all i j. Clearly, by maximality, (yi)i 4(K62)-1), otherwise there is nothing to prove. Hence we can assume the right side of (4.9) < 1. We then obtain that with positive probability we have

`/aEA IM-1IIEa$Xi(w)II-1I 0. Hence if F,,, is the span of (Xi(w), ... , Xn(w)) and (recalling (4.10)) if n = [4-1K 6(e)3d(X )], we have proved that with positive probability we have d(F,,, t2) < 1 + C.

We now turn to the proof of Theorem 4.3. For this we will use the following variant of a classical lemma of Dvoretzky-Rogers:

Lemma 4.13.

Let E be an N-dimensional space. Let N = [N/2]. Then there are N elements (xk)k 1/N for all N > 1. Hence

P suNgk 0 such that for all N > 1 we have (4.19)

a(e) Log N < ne(PN) 00 < ,Q(e) Log N.

(ii) For each 2 < q < oo, there are functions a(q, e) > 0 and ,3(q, E) > 0 such that for all N > 1 a(q, e)NZ/q 1 cb(e)N < ne(P9) < N.

Dvoretzky's Theorem

55

Proof: Fix q such that 1 < q < oo. Let (ek) be the canonical basis of IN and let X = EN gkek. If we consider X as a random variable with values in IN , then we have, by Lemma 4.14, Log N, for q = oo;

d(X)

N2/q, N,

for 2 < q < oo;

for1 0. Then there is a finite set A C B, with (4.21),

card (A)
1.

On the other hand, (4.23) implies, for all 0 < e < 1, E-n

< N(BE, EBE),

and therefore

2-(k-1)/n < ek(IE)

(5.8)"

for all k > 1. We refer to [Pil], [Pi3], or [Ko] for more details on the numbers

(sk(n)) We will need the following elementary fact:

Proposition 5.1.

Let X, X1, Y, Y1 be Banach spaces. Assume that

X is isometric to a quotient of X1, and that Y is isometric to a subspace of Y1. We denote by q : X1 -+ X the quotient map and by j : Y --+ Y1 the isometric embedding. Then for any operator u : X -+ Y we have ek(u) = ek(uq)

and

1 ek(u) < ek(ju) < ek(u) for all k > 1.

Moreover, if X1 = £1(I) for some set I we have dk(U) = ak(uq), while if Y1 = PE(I) we have ak(ju) = ck(u).

Proof: The first part is immediate since q(BX3) = BX. Also, ek(ju) < Ilillek(u) = ek(u). Note that if ju(BX) is covered by 2k-1 balls of radius e in Y1, then u(BX) is the union of 2k-1 subsets of diameter

Chapter 5

64

2e, hence ek(u) < 2 ek(ju). Finally, the last assertions are immediate consequences of the lifting property of f, (I) and the extension property

of t, (I). The following result of Carl will be very useful in the sequel. It shows that the numbers ck, dk, or ak dominate in a certain sense the entropy numbers.

Theorem 5.2.

Let sk denote either Ck, dk, or ak. For each a > 0 there is a constant pa such that for all operators u : X -+ Y between

Banach spaces we have (5.9)

sup kaek(u) < pa sup kask(u) for all n > 1. km 0 to be specified later. We deduce from (5.7) since Km C IIDmIIBY,

that (5.11) de>O

2rk(°m) N(Km,eIIAmllB')
0 be arbitrary. We make the choice Em = This gives N

E emIIAmII + 2-Na < 2-Na(C1t + 1) 0

and

N

2

m-0

Em

r 2m+1

r


0, we denote simply

Nz(e) = N(Kz,eBL2) Then we can state:

Theorem 5.6.

There are absolute constants Ci > 0 and C2 such that for any Gaussian process Z = {Z=Ii E I}, indexed by a finite or countable set I, we have (5.13)

Clsupe(Log Nz(E))1"2 < EsupZi < CZ iEI

e>0

J0 00

(Log Nz(e))112de.

Remark: More generally, the preceding result still holds if we merely

assume Z separable, i.e. such that there is a countable subset J C I

for which k = {Zili E J} is dense in Kz. Indeed,we have then N(K, eBL2) = Nz(E). If we define supiEl Zi as the supremum in the Banach lattice L1 then clearly supiEl Zi = SUPiEJ Zi a.s., and (5.13) still holds. For the lower bound in (5.13), the following lemma is crucial. It originates essentially in the work of Slepian.

Lemma 5.7.

Let {Xi,1 < i < n} and {Yi,1 < i < n} be two

Gaussian processes such that (5.14)

V iJ IIYi-Y,II2 c1}U...U{Yn >cn})cl}U...U{xn>c,l}). Remark: The above result is known without the factor 2 but we do not use this improvement in the sequel. We refer the reader to [BC] and [Fell or [Kahl] and [Kah2] for more information. Remark: It is worthwhile to observe by symmetry that

Esup IXi - Xj = Esup(Xi - Xj) = 2E supXi.

(5.17)

ij

i,j

Proof of Lemma 5.7: We assume (5.14) and (5.15). We will first prove (5.16). We may clearly assume without loss of generality that

X = (Xi)i 0

P{Zk < s} < (als)k

`dt>2 P{Zk>t}<exp-()31kt2).

The proof of (5.31) and (5.32) are outlined below. It is easy to derive Lemma 5.11 from these estimates. Indeed, for each fixed non-zero y

in f2 we have for all0<s K we have

I co(x) o (dx) = 1G,k m(dF) 1SF

oF(dx).

This is immediate since a is the unique probability on S invariant under rotations.

Step 3. Let x E SF, F C R, dim F = k. Then for all 0 < 6 < f we have

QF({yESF,Ix-yJ (ir) Indeed, this measure is equal to F with F(s) = fo (sin t)k-2 dt and A with a determined by the equality 2 = sin(2 ), 0 < a < Z . The reader can check this easily by referring to the picture.

Chapter 6

92

x

0 Then we note that F(ir) < it and (assuming k > 1)

F(a) 1 j(sin t)k-2 cost dt = (k -

1)-i(sina)k-1;

hence

F(a) > (k - 1)-1 (2 sin 2 cos

i/J2J k-i

62 8(1_

=(k-1)-i

> (k - 1)1

21k-i

(o)k_i

4/

J

/

so that we have

F(a) > it-1(k - 1)-1 1 F(7r)

(,)k

72)k-i

>

(,)k ( r k-i - 7r

)k

2

kk-

1

This establishes Step 3. Let us now complete the proof. oF(dx) < (2A)"}. Let 1 o = {Fj fF By Steps 1 and 2, and Markov's inequality, we have IIXII-n

m(Slo) > 1 - 2-n.

Volume Ratio

93

Consider F in SZo. Again, by Markov's inequality, (6.3)

o'F ({y E SF, IIyII :5r}) < (2Ar)".

`d r > 0

Now we choose r so that

(r)k

(2Ar)"=

(6.4)

2:

Let b = 2. Then the set L,. = {y E SFI IIyII : r} cannot contain (by (6.3) and Step 3) any ball of radius S induced on SF by the Euclidean metric.

In other words, V x E SF 3 y E SF - Lr such that (x - yI < b. A fortiori, we have fix - yII < b =

2,

hence

IIxii ? IIyII - IIx - y1I

>r--r =2.r 2

Thus we have shown (by homogeneity)

Vx E SF

IIxII >- 2Ixi.

Finally, we analyze the choice of r made in (6.4). (r )k; hence r"-k = (2- )" so that (2A(2x)". Therefore, we conclude . > (47r)= z

(2Ar)"

Corollary 6.2. Let E be an n-dimensional normed space. Then for any k = 1, 2, ... , n -1, E contains a subspace F with dim F = k such that

d(F,EZ) < (4irvr(E))' . This is an immediate consequence of Theorem 6.1.

Remark: We can also reformulate Theorem 6.1 for operators u : E -' PZ . Note that Cvol(uBE)11/" < 2e"(u). vol(Bjn) J Hence we obtain from Theorem 6.1 the following inequality: ck+1(u) < (41r

\

vol(uBE)1 ^`vol(Bz-) )

< (8ire"(u))

.

In the particular case when u is a diagonal operator from £ into itself, it is easy to see that this cannot be significantly improved.

Chapter 6

94

Corollary 6.3. Assume n = 2k in the situation of Theorem 6.1. Then there is a decomposition R" = El ® E2 with El, E2 orthogo-

nal with respect to the inner product structure associated to D, dim El = dim E2 = k and El, E2 satisfy

VxE ElUE2 C'IxI which imply m({F, F E SZo and Fl E Slo}) > 0. Letting El = F and E2 = Fl with F and Fl in SZo, we obtain the announced result. In particular, recalling (6.1) we obtain the following:

Corollary 6.4.

Let us denote by LP the space R" equipped with the

norm 1/P

IlxllP =

n >1 IxilP

.

Then if n = 2k there is an orthogonal decomposition LZ = El ® E2 with dim El = dim E2 = k such that

Vx E El U E2

Ilxlll 0 such that for each k there is a 2k x 2k orthogonal matrix A satisfying A2 = I and such that Vx E L2'`

C3IIxII2 < 2(11x111+IIAxIII) max(IIPixIII, IIP2xIIi) > (C")-I max(IIPixII2, IIP2xII2)

> 2-IIIxII2. Remark: There is no constructive proof of a matrix A satisfying the above. Note that the proof shows actually that we can obtain (for some constant C3) a set of matrices A with large measure in 0(2k). This result has a rather striking infinite-dimensional application obtained by Krivine [Kr] and independently by Kasin [Ka2]. We follow

Kain's argument.

Corollary 6.6. Let LI, = LP([0,1]). There is an orthogonal decomposition L2 = El ® E2 such that the L2 and L, norms are equivalent both on E, and on E2. Proof: We use the Haar orthonormal basis of L2, which we denote by

{h,,, n > 1}. Here hl = 1, and for all m > 0,1 < j < 21 the function hem+, is supported by [(j -1)2-m, j2-m] and takes the value +2 on the left half of this interval and -2'l on the right half. Let E0 be the span of h, and let Em+I be the span of {hn, 2'" < n < 2m+I}(m > 0). Then Em can be identified with L2m-' and Em equipped with the L, Lim-1 norm can be identified naturally with This implies that there is for all m an orthogonal decomposition Em = E,m ® E,2m such that (6.5)

`dx E Em U Em

IIxii,

1, any ball B C R" satisfies 1/n

C-1 < vol(B)vol(B°)

(

vol(Bl2 )2

)

< C.

Proof: By Theorem 7.1 we have clearly

C-1s(D) < s(B) < Cs(D) and by (7.3) this implies Corollary 7.2. The classical Brunn-Minkowski inequality states that if A1i A2 ar.

two compact (or suitably measurable) subsets of R" we have (se, Theorem 1.1) (vol(A1))1/" + (vol(A2)) n < (vol(A1 + A2))1/".

Milman's Ellipsoids

101

It is easy to see that we cannot expect any inequality in the converse

direction with a fixed constant and this even if we restrict ourselves to balls. However, Milman discovered that if Al, A2 are balls, there is always a relative position of Al and A2 for which a converse inequality holds. _ We will say that a ball b is equivalent to a ball B if there is a linear

isomorphism u : Rn -> R' such that I det(u) I = 1 and b = u(B). Note that since u preserves volumes, we have vol(B) = vol(B). Clearly, if D is a Milman ellipsoid for B, then u(D) is a Milman ellipsoid for u(B). Hence, for any ball B there is always an equivalent ball B admitting for its Milman's ellipsoid amultiple of the canonical ellipsoid Bt2 . In that case we will say that B is in a regular position. With this terminology, any ball is equivalent to a ball in a regular position, so that the next statement is very general.

Corollary 7.3.

Let B1, ... , Bk be balls in a regular position in R'.

Then for all tl > 0,...,tk > 0

(vol(t1B1 + ... +( tkBk))l/n

< C(3C)k [tl(vol(B1))1/n + ... + tk(vol(Bk))1/nI

(vol(t1Bi +... + tkBk))l/n < C(3C)k [tl(vol(Bi ))l/n + ... + tk(vol(B*))1/nI where C is as in Theorem 7.1.

Remark 7.4: Note that if Bl,... , Bk are in a regular position then the same is true for t1B1i ... tkBk and t1Bi, ... , tkBk. Therefore, it is enough to prove (7.4) and this only for tj = ... = tk = 1. To prove this corollary (and also in the sequel) we will need several elementary facts relating the covering (or entropy) numbers and volumes. Let Al, A2 be subsets of R. Recall that we denote by

N(A1i A2) the smallest number N such that Al can be covered by N translates of A2. Clearly, we have (7.5)

vol(A1) < N(A1, A2) vol(A2).

Chapter 7

102

More generally, we record here the following obvious facts.

For any subset A C R" (we also assume that all the sets appearing below are measurable) we have vol(Ai + A) < N(Ai, A2) vol(A2 + A).

(7.6)

For arbitrary balls B1, B2 in R" we have (7.7)

N(B1, 2(B1 n B2)) < N(B1, B2)

Indeed, if B1 C Ui 1 be a fixed integer. Let Cn be the smallest constant C for which the statement of Theorem 7.1 is

true, that is to say

C. =

sup

BCR° B ball

inf

DCR°

M(B, D).

D ellipsoid

We will show by an a priori estimate that Cn has to remain bounded

when n -p oo. (Of course, we know that Cn is finite since for any B there is an ellipsoid D-the John ellipsoid-such that D C B C /iD, which implies Cn < 2n1/2(n1/2 + 1).) Let D be an ellipsoid such that M(B, D) < 4Cn.

(7.14)

Clearly, we have M(B, D) = M1M2 where M1 __ and

1/n (vol(B + D) vol(B°) vol(B) vol(B° n D°)

Chapter 7

106

M2

__

(vol(Bc + D°) vol(B°)

vol(B) \ 1/n vol(B fl D) /)

Note that M2 is obtained from M1 by replacing B, D by the polars

B° D°. By (7.14) we have either M1 < 2(Cn)1/2 or M2 < 2(C.) 1/2. We will

show that M1 < 2(Cn)1/2 implies that there is an ellipsoid D1 C Rn satisfying M(B, D1) < K2(Cn)1/2(1 + Log C)2 for some numerical constant K2. Since M(B, D1) = M(B°, Di), the other case M2 < 2(Cn)1/2 leads to the same conclusion by simply exchanging the roles of (B, D) and (B°, D°). Hence we assume M1 < 2(Cn)1/2 Let

a

+ D) )1/n

__

J

vol(B) (vo1(B

so that M1 = a,3
1

dk(w) < A(n/k)a and ck(w-1) < AO(9)(n/k)a,

and if we replace w by w = 0(9)1/2w, we find (7.39)

V k > 1 dk(w) < AO(0)1/2 (n/k)' and

ck(w-1)
1 we have max {N(B, tD), N(D, tB), N(B°, tD°), N(D°, tB°)} < exp(KKnt-P).

Proof: This follows immediately from Corollary 7.15 (at least for

t > Cl) applied to the normed space E associated to B, letting D = u(Bt) and p = 1/a. Using Lemma 4.16 we see that by suitably adjusting the constant KK we can obtain a similar estimate for all t > 1. Remark: Let us briefly review here the improvements that Theorem 7.13 brings to corollaries 7.3 and 7.10. Fix a number a > 1/2. Let B be a ball in R' with associated normed space E. We will say that B is a-regular if we can choose the isomorphism u in Theorem 7.13 equal to a multiple of the identity on R'. Equivalently, the ellipsoid associated to u is a multiple of the canonical Euclidean ball. With this terminology, Theorem 7.13 says that every ball is equivalent to a ball b which is a-regular. (To verify this, we simply apply Theorem

7.13 to E and let b = u-1(B).) Again, we emphasize that if B is a-regular, then its polar B° and all the multiples of B and B° are also a-regular. We may refine corollary 7.3 as follows: Let B1, ... , B,,,, be a finite set of a- regular balls in R. Then we have (7.46)

vol (B1 + ... + Bm)1/'L
1

(7.47)

max {N(Bi, t Di), N(Di, t Bi)} < exp(Knt-P).

Hence

N (Bl + ... + B.,,,,, t(D1 + ... + Dm)) < II N(Bi, t Di) < exp(Knm t-P) so that

vol (B1 +... + Bm)1/n < t exp(Kmt-P) vol (Dl +... + Dm)1/" Now since we assume B1,... , Bm a-regular, the ellipsoids Dl,... , Dm are all multiples of a fixed ellipsoid so that (trivially)

vol (Dl +... + Dm) 1/n = vol (D1 )1/n +... + vol (Dm)1/n Moreover, by (7.47) with t = 1 we have vol (D1)1/" < exp(K) vol (Bi)1/n,

hence we conclude that for all t > 1

vol (Bl +... + Bm)1/n



8(a)m-1-ainf {vol(B1)1/n,... ,vol(Bm)1/n},

for some positive constant 6(a) depending only on a > 1/2. Indeed, this is easy to deduce from (7.46) and Corollary 7.2.

Remarks: (i) In the proof of Theorem 7.13, we can use Theorem 5.14 instead of Theorem 5.8; we then obtain the same result but only for all

Chapter 7

122

a > 1. Of course, this suffices to obtain (7.40) and hence to complete the second proof of Theorem 7.1. (ii) By a simple modification of the proof of Theorem 7.13 one can give a direct proof of Corollary 7.15 using the entropy numbers

ek(u) instead of dk(u) and dk(u-1). The ingredients for this alternate route are reduced to the fact that sup k112 max {ek(u), ek(u*)} < K 1(u) k>1 (for some absolute constant K) and the fact that the entropy

numbers satisfy both inequalities (7.27) and (7.28).

Remark: It is worthwhile to mention that Theorem 7.13 is invalid for a < 1/2. Indeed, by Theorem 5.5 and Theorem 5.2 we have for all

u:P2-+ E `d a < 1/2 1(u) < ii(a)n1/2sup(k/n)atck(u*), k>1

for some constant Vi(a) depending only on a. Therefore, if a < 1/2 then (7.36) implies

1(u) < Vi(a)Cn1/2 and

.t(u-1*)


0 such that for all n and all isomorphisms u : In -, Pn we have (7.48)

8(u)2(u-1*) > En(Log

n)1/2.

This follows from the following facts: For some constant K1 we have (i)

sup(Log k)1/2ak(u) 1

For some constants K2 and K3 we have, for all v :1. -> in21 sup k1/2ak(v) < K21(v*) k

1r2(v) C K3IIvII.

Milman's Ellipsoids

123

The first point (i) follows easily from Lemma 1.8, (4.14) and (4.18). The second point (ii) follows from the fact that e1 is of cotype 2 (see the subsequent definition 10.1), while the third point (iii) follows from a weak form of Grothendieck's theorem (cf. e.g. Theorem 4.3 in [P2], p. 54).

Then we deduce from (iii) that for all subspaces F C £ with dim F = d we have d1/2

= 72(1F)
2 -2" vol(PE2 (E1 n B)) vol(E2 n E1 n B) vol(PEi (B)).

Another Proof of the QS-Theorem: The Iteration Procedure

135

Proceeding similarly for B°, we find

vol(B°) > 2-2" vol(E2 n PE, (B°)) vol(PE2 PE, (B°)) vol(Ej n B°). Let

Al=PEl(B) and A2=E2 nElnB, and dl = dim Ei , d2 = dim El n EZ . Note that k = d, + d2. Taking the product of the last two inequalities, we obtain s(B)n 2-4n(s(PE2(El n B)))n-ks(Al)dls(A2)d2; > hence, by (8.11) and the definition of SN, s(B)n > 2-4n (K-1)n-k(Vn-k)(SN)d1Vd, (sN)d2Vd2; therefore by (8.12), since k = d1 + d2,

2-4(K-1)1 n (sN)* and since SN < Si = 1 and k < n/2, we have (SN)k/n >- (SN) 1/2, So that we have proved SN = inf{s(B)(Vn)-1/"} > 2-4K-1(SN)1/2.

Dividing by (sN)1/2 and squaring we finally obtain

sN > 2-8K-2

(Note that we need to know that sN > 0 for all N, but this is clear since by John's Theorem we have a priori SN > N-1/2.) This shows the lower bound for SN; we obtain the upper bound by an entirely similar reasoning left as an exercise for the reader. We note the following immediate consequence.

Corollary 8.9.

Let B1, B2 be balls in Rn. Then (with a > o and > o as in Theorem 8.7) we have 1

(vol(Bl) vol(Bl )11/n vol(B2)vol(B2O)

< Qa

1

'

hence rvol(B2))1/n

f vol(B )l1/n

`vol(B2))

(vol(B°) 1/n

- a-1vol(Bi))

To conclude this chapter, we also indicate how a result of KonigMilman [KM] can be deduced from Theorem 8.7.

Chapter 8

136

Theorem 8.10.

There is a numerical constant c such that for all

balls B1, B2 in Rn we have

n N(Bz, BO) 1, we define the n-th volume number of K as (9.1)

(vol(P(K))

vn(K)=sup

1/n

vol(P(BH) ))

'

where the supremum runs over all orthogonal projections P : H --+ H

with rank equal to n. If dimH < n, we set vn(K) = 0. Let T : X -> H be a compact operator defined on a Banach space X. We define the volume numbers of T as

vn(T) = vn(T(Bx))

In (9.1) above, vol denotes the volume in P(H) (the range of P) equipped with the Euclidean structure induced by H. In particular, vol(P(BH)) is equal to the volume of the Euclidean ball of R' which we have denoted earlier by Vn. We have clearly vl(T) = 1IT11. These numbers behave very much like the entropy numbers, as we shall see. We first note that {vn(K)} is a non-increasing sequence. Indeed, this follows from the famous inequalities of Fenchel and Alexandrov on the mixed volumes (cf. [Al]). Let B be a compact subset of Rn. We define Wk(B)

U

vol(P(B)) dP)1/k,

tik = J where the integral is with respect to the uniform probability on the

set of all the orthogonal projections P : Rn -+ Rn of rank k. Clearly,

(vol(B)) 1/n

wn(B)

Vn

and

wi(B) =

J

supt(x) do(x), tEB 139

Chapter 9

140

where o, is the normalized area measure on the Euclidean sphere (i.e. the unit sphere of 2 ). For a convex body B, w,,,_1(B) can be identified with the ratio Ca(B) a(Bt2

nl l

\

of the areas of the surfaces respectively of B and of the Euclidean unit ball. The inequalities of Alexandrov simply say that the sequence {wk(B)} is a non-increasing sequence. We refer to [Al], [BF], [BZ], or [Eg] for a proof.

In particular, it is worthwhile to observe that this implies wn(B) < w1(B)

(Urysohn's inequality)

and

wn(B) < w,i_1(B)

(the isoperimetric inequality).

See Chapter 1 for a different approach to these two inequalities. A fortiori, the inequalities of Alexandrov imply that for any k < n we have

Ivol(B) l 1/n

< sup

n

Ivol(P(B)) ))1/k rk(P) =

k}

JJ'

which immediately implies, for all K C H as above,

vn(K) < Vk(K) if k < n. Actually, the isoperimetric inequality (cf. Corollary 1.3) suffices to prove this since it gives us wn(B) < wn_1(B) for any B C Rn, hence vn(K) < vn_1(K) for any K C H. The numbers wk(B) are especially useful via the following classical Steiner-Minkowski formula (cf. again [Al], [BF], [BZ], or [Eg]): t1 t > 0

vol(B + tBt2) = Vn

ll n_k wk (B) E \k/ t O 1 let fk(u) = inf{1(u - uP)},

where the infimum runs over all orthogonal projections P on H

with rank P < k. There are numerical constant C' > 0 and C" > 0 such that for all Hilbert spaces H and all u : H -+ X we have for each k > 1 (9.3)

4(u) < C" E vn(u*)n-1/2(1 + Log n).

n>Cx In particular, if En>1 vn(u*)n-1/2(1 + Log n) < oo then 1(u) < oo and u belongs to the closure of the finite-rank operators in the space t(H, X) (iii) Finally, if X is K-convex, the preceding statements hold without the factor (1 + Log n).

The crucial lemma in the proof may be formulated like this.

Lemma 9.2. There is a numerical constant C1 such that the following holds. Let n be an arbitrary integer and let B be any ball in Rn. Let k < m < n. There is a subspace F C Rn with dim F = m - k + 1 such that

diam(F f1 B) < C1vk(B) f (- 1 where f (6) = (1 - 6)-1[1 + Log (1 - 6)-1].

,

Chapter 9

142

Proof: We first consider the particular case where B is an ellipsoid. Then we may assume (without loss of generality) that there are A, _! A2 ... >An > 0 such that n

B= x E R n E(Xi/Ai)2 < 1

.

Then clearly < vk(B)k, so that Ak < Vk(B). Now, if we let F be the span of ek, ek+i, ... , en we find dim F =

n - k + 1 and diam(B f1 F) < vk(B). We now turn to the general case. We will use the QS-Theorem proved in the preceding chapter. Let k < m < n. By the Remark after Theorem 8.4., we know that there is an m-dimensional projection of a section of B which is Cl f (m/n)-equivalent to an ellipsoid D, where Cl is a numerical constant. Hence there are E2 C El C Rn and D C E2 such that (9.4)

DCPEZ(E,fB)CClf(m)D. n

Now we will compare vk(D) and vk(B). Clearly, PE2 (El fl B) C PE2 (B); hence

Vk(PE2(E1 fl B)) < vk(PE2(B)) < vk(B) where the second inequality follows from definition (9.1).

By (9.4), this implies

vk(D) < vk(B).

(9.5)

By the first part of the proof, there is a subspace F C E2 with dim F =

m - k + 1 such that diam(F fl D) < vk(D); hence by (9.4) and (9.5)

diam(F fl B) < Cl f (n) vk (D) < Cl f

n

vk(B).

As a typical application, we can state (take k = [ ] and m = [ 34 ] ). a

Volume Numbers

143

Corollary 9.3. There is a numerical constant C2 > 0 such that, for all n, for any ball B C R' there is a subspace F C Rn of dimension [n] such that 2 diam(F f1 B) < C2v[ ] (B). 4

We will also need the following lemmas which are related to the bounds for the K-convexity constant presented in Chapter 2.

Lemma 9.4.

Let S be a closed subspace of a Banach space E. Let Q : E -+ E/S be the quotient map. Assume that E is K-convex. Let xn in E/S be such that En =1 gnxn converges in L2(E/S). Then for

each e > 0 there are xn in E such that a(xn) = xn and the series

Eono= 1

gnxn converges in L2(E) to a limit satisfying co

00

II E9nxn1IL2(E) < K(X)(1 +e)II Egnxn 1

(L2(E/S).

Proof. This clearly reduces to the case of a finite sequence (x1, ... , xn). Then, let _ En gixi. We consider 0 as an element of L2(E/S), i or equivalently L2(E)/L2(S). Clearly, there is in L2(E) such that

Q¢ =0 and (9.6)

II4IIL2(E) :5 (1 + e)II4'IIL2(E/S)

Moreover, we may clearly assume that q5 depends only on g1, ... , gn.

With the notation of Chapter 2, let V, _ (Q1 ®IE)(?).

Then

admits an a priori expansion n

_

gixi with xi E E.

Obviously, o o = ¢, hence we must have o-xi = xi for all i; and on the other hand, by definition, IIbIIL2(E) S K(E)IIcbIIL2(E);

and by (9.6),

< K(E)(1 + e)II'IIL2(E/S) N

Chapter 9

144

Remark 9.5: Recall that for an arbitrary n-dimensional normed space E we have proved in Chapter 2 that K(E) < K(1 + Log n). Thus, Lemma 9.4 can be useful also in that case, as we shall see in the proof of Theorem 9.1. Note also that Lemma 9.4 may be reformulated

as follows: For each e > 0, every operator v : tz -+ E/S admits a lifting v : tz -+ E, satisfying ov = v and t(v) < K(E)(1 + e)t(v). We need one more lemma.

Lemma 9.6.

Let n be an integer. Let u : ten --* X be an operator into a Banach space X. Then there is an orthogonal projection P on Pen with rank 2n such that f(uP) < C3vn(u*)n1/2(1 + Log n), where C3 > 0 is a numerical constant.

Proof: We may clearly assume dim X < 4n. By Corollary 9.3 we know that there is a subspace F C R4n with dimension 2n such that I1urU.-1(F)II X/Fl be the quotient map. Then llvull = IIu*F1 l) < C2vn(u*). We have, trivially,

t(vu) < (4n)1/2IIaull < (4n)1/2C2vn(u*).

By Lemma 9.4 and Remark 9.5, there is an operator iii

:

n --p X such

that vii = ou and t(u) < 2K(1 + Log 4n)t(ou). Hence

t(u) < C3vn(u*)n1/2(1 + Log n) for some numerical constant C3 > 0.

Finally, we note that o(u - u) = 0 implies that the image of u - u lies in u(F), hence rk(il - u) < dim F _< 2n, which implies dimKer(u - u) > 2n. Let P be the orthogonal projection onto Ker(u - u). We have (ii - u)P = 0, hence t(uP) = t(iiP) < C3vn(u*)n1/2(1 + Log n).

Volume Numbers

145

Proof of Theorem 9.1: Consider an operator u : H -* X. We will first prove (9.2). Let A(2n) = sup{t(uQ)JQ E Pen}. Note that t(u) = supn>0) (2n). By Lemma 9.6, for any projection Q in Pen there is a projection P in Pen-, with P < Q such that (Qu*)(2n-2)1/2(1

f(UP) < C3v2n-2

+ Log

2n-2)

< C3v2n-2(u*)2nJ2(1 + Log 2n).

Clearly, e(uQ) < e(uP) + 2(u(Q - P)); hence, since Q - P E Pen-1, 2(uQ) < C3v2n-2(u*)2n/2(1 + Log 2n)

+A(2n-1).

Thus we have proved A(2n) < C3v2n-2 (u*)2n/2(1 + Log 2n) +

A(2n-1),

which implies

A(2n) < C3 E v2n-2(u*)2n/2(1 + Log 2n) +.A(2). n>2

Clearly, )(2) < 211u11 = 2v1(u*), so that we can write £(u) = sup)(2n) < C4 E v2k(u*)2k/2(1 + Log 2k) k>O

< C5 E vn(u*)n-1/2(1 + Log n) n>1

for some numerical constants C4, C5. This proves (9.2). Thus we have proved that (9.7)

E vn(u*)n-1/2(1 + Log n) < 00 n>1

implies t(u) < oo. By the results of Chapter 5 (cf. Theorem 5.5), it follows that u is compact. Hence, for each e > 0, there is a projection Q in P such that Ilu - uQ11 < E. Obviously this implies vn((u - uQ)*) < e. On the other hand, clearly

vn((u - uQ)*) < vn(u*), hence

vn((u - uQ)*) < min{e, vn(u*)}.

Chapter 9

146

By (9.2), this shows that AU - uQ) < C/3(e)

(9.8)

where

/3(e) = E min{e, vn(u*)}n-1/2(1 + Log n). n>1

Obviously (9.7) implies /3(e) -+ 0 when a -p 0. For simplicity, we may (and do) assume without loss of generality

that H is infinite dimensional and that rk(Q) = 2K where K is an integer depending on e. Then we can view uQ essentially as an operator from elk into X to which we apply Lemma 9.6. This shows that there exists P1 in P2K-1 such that P1 < Q and

t(uP1)
C'k

Hence, by Theorem 5.8,

ck((1 - P)u*) < C7 E vn(u*)n-1/2(1 + Log n). n>C'k

Clearly, C2k(U*) < 4((1 - P)u*) since rk(P) < k, therefore we can easily conclude that the above claim is valid for suitable constants

6>0andC6. From this claim, we deduce that if vn(u*) is 0(na) with a < -1/2, then cn(u*) is 0(naLog n), hence cn(u*) is 0(na+6) for all 6 > 0, and by Theorem 5.2 this implies a fortiori that en(u*) is 0(n' ).

Notes and Remarks This chapter is mainly based on [MP2], but our exposition is different. We follow the approach suggested in the note added in proof to [MP2]. I am indebted to A. Pajor for several conversations which considerably clarified the presentation of this chapter. Except for Lemma 9.4, which is a basic property of K-convex spaces (see [P6] for a more refined result), all the results come from [MP2]. The motivation for this chapter came from the 1967 paper of Dudley [Dul] where he connected the geometric study of compact subsets of Hilbert space with the continuity of the paths of Gaussian processes. We can describe briefly Dudley's viewpoint as follows. Let H be a (separable) Hilbert space and let (Xt)tEH be a Gaussian process indexed by H such that (9.11)

`d t, s E H

JIXt - X8112 = lit - sJJH.

A subset K C H is called a GC-set (resp. GB-set) if the process (Xt)tEK has a version which is continuous on K (resp. bounded on K). _ A process (Xt) is called a version of Xt if we have each Xta=a'Xt for t. Since the marginal distributions of Gaussian processes are determined by their covariance, it is easy to see that these definitions do not depend on the particular choice of the process (Xt) as long as it

Volume Numbers

149

satisfies (9.11). It is well known that there exists a Gaussian process satisfying (9.11) and such that t -+ Xt is a linear map from H into L2 (Q, A, P). To see this, just take H = £2 and let Xt = E tngn>

(9.12)

with (g,,,) i.i.d. normal Gaussian variables as usual. Actually, the same argument (consider the marginal distributions of Xt restricted to K) shows that if there is a (not necessarily linear) isometry from a set K1 onto a set K2 then K1 is a GC-set (resp. GBset) if K2 is also one. More generally, it was later proved by Sudakov [Su] that if there is a (non-linear) contraction from K1 onto K2 and if K1 is a GC-set (resp. GB-set) then K2 is also one. In his fundamental 1967 paper, Dudley proved the sufficiency of the metric entropy condition as proved in Chapter 5. He also proved that the condition sup,, n1/2vn(K) < oo is necessary for K to be a GB-set and in the converse direction he conjectured that sup,, n1/2+Evn(K)
0 is sufficient for K to be a GC-set. This was first proved in [MP2]. (It clearly follows from Corollary 9.7, which was also conjectured by Dudley.) The necessity of sup,, n1/2vn(K) can be proved (essentially as Dudley) using the Urysohn inequality (cf. Corollary 1.4). Indeed, we have

vn(K) < f sup < t, x > dor(x). S tEK

But, with (Xt) as in (9.12),

f

sup I < t' X >

tEK

I2

do-(x) = n-'E sup IXt12; tEK

hence for all K C H we have IXtl2)1/2 vn(K) < n-1/2 sup (E sup PEP tEPK < n-1/2 (E sup IXt I2)1/2. tEK

Finally, we may write for all u : £2 -+ X

vn(u*) < n-1/2t(u).

(Recall also that (vn(u*) < 2en(u*), as noted in the beginning of Chapter 9, so that this can be viewed also as a consequence of the lower bound in Theorem 5.5.)

150

Chapter 9

We refer to [PT2] and [PT4] for more results using volume numbers. We also refer to [Pa2] and [Pa3] for more information on the mixed volumes when the Euclidean unit ball is replaced by the unit ball of $""'

or of ?. In particular, Pajor proved a version of Urysohn's inequality (cf. Corollary 1.4) for random choices of signs. Namely, he proved that for all compact subets K C R' we have yol(K)

(vol(Bt,))

1/"


1

Now we introduce weak cotype 2. 151

Chapter 10

152

Definition 10.1.

Let X be a Banach space. We say that X is a

weak cotype 2 space if there is a constant C such that, for all n and all operators u : t -p X, we have (10.1)

supk112ak(u) < C 1(u). k>1

The smallest constant C for which this holds will be denoted by wC2(X). Clearly, the above preliminary remarks show that cotype 2 implies weak cotype 2 and wC2(X) < C2(X). Of course, (10.1) immediately extends to all operators u : f2 - X for which 1(u) < oe. The notion of weak cotype 2 is clearly inherited by the subspaces of a Banach space (but not by the quotient spaces in general; see below). Actually, it is stable by finite representability. Recall that a Banach space Y is said to be finitely representable into X (in short, Y f.r. X) if for every e > 0 and every f.d. subspace E C Y there is a subspace

F C X such that d(E, F) < 1 + e. It is easy to see that Y f.r. X and X of weak cotype 2 implies Y weak cotype 2 also. (Thus weak cotype 2 is a super-property in the sense of James [J].) Clearly, the direct sum of two (or a finite number of) weak cotype 2 spaces is again a weak cotype 2 space. We will show below that weak cotype 2 characterizes the class of Banach spaces for which the dimension of the spherical sections of

the unit ball (as in Dvoretzky's Theorem) is always essentially the largest possible. More precisely, to measure these dimensions and their asymptotic behavior, we introduce the following (possibly infinite) number for 0 < 6 < 1:

dx(6) = sup inf{dFIF C E dimF > 6 dimE}. ECX

E f.d.

In other words, dX (b) is the smallest constant C such that every f.d.

subspace E C X contains a subspace F C E with dim F < 6 dim E such that dF < C. Clearly, if X is a Hilbert space, then dX (b) = 1 for all 0 < 6 < 1. In general, of course, dX (b) may be infinite. We first connect the notion of weak cotype 2 with the existence of

Euclidean subspaces of proportional dimension in every E C X, as follows.

Weak Cotype 2

Theorem 10.2.

153

Let X be a Banach space. The following properties

are equivalent.

(i) X is a weak cotype 2 space. (ii) There is 0 < 60 < 1 such that dx(6o) is finite. (iii) For each 0 < 6 < 1,dx(6) is finite. (iv) There is a constant C such that for all 0 < 6 < 1 we have dX (6) < C(1 - 6)-1 [1 + Log 1

C 6] .

Note: In this chapter the properties numbered from (i) to (viii) will all be equivalent to weak cotype 2.

Remark: The proof below of (ii)

(iv) gives an estimate of

C < C'dx(6o)60 1 for some numerical constant C'. For the proof of Theorem 10.2, we will need the following:

Lemma 10.3.

Let u : H -+ X be an operator on a Hilbert space H with values in a Banach space X.

Let 0 < a < 1, 3 > 0 and A > 0 such that a dim H >_ 1. ,

As-

sume that for any n and any n-dimensional subspace S C H we have (denoting [ ] the integral part) (10.2)

a[.n](uIs)
1;

.f[an]-13

then we have (10.3)

supk13ak(u) < AC

with C = 313 (1 - a213)-1/2.

k>1

Proof. We may clearly assume A = 1 by homogeneity. Let Cn be the best constant C for which the preceding statement holds for all u : H X with rank < n. Clearly Cn < oo. We will show by an a priori reasoning that Cn remains bounded when n --+ oo. More precisely, we claim that Cn < 30(1 -

a20)-1/2.

To prove this claim, we consider u : H -> X with rank < n. Let m = [k/a]. By definition of amn+1(u) there is a subspace S C H such

that (10.4)

ilulsIl = am.+1(u) < Cn(m +

1)-13

Chapter 10

154

We may identify Sl with 4 and apply the assumption of Lemma 10.3 to uls-L. Thus we find [am]-0

a[.,,,](u1sl)
_ 2). Since k > am, we have ak(u) < a[am] (u) and, moreover, m + 1 _> k/a and [am]> k - a - 1 so that (10.6) implies, for all k > 2, (10 7)

k2Oak(u)2 < C'a20 + (k(k - a - 1)-1)20 < Cina20 +3 20

On the other hand, there is obviously an integer no = no(a) such that [ano] = 1, hence (10.2) implies Dull < 1. This observation and (10.7) imply sup k20ak(u)2 < a20Cn +3 20. k>1

Therefore, we have (going back to the definition of Cn) Cn < a20Cn + 320;

hence (since a20 < 1) Cn < 30(1 - a20)-1/2, as claimed earlier. It is then easy to deduce that any operator u satisfying (10.2) must be compact and must satisfy (10.3).

Proof of Theorem 10.2: We will show that (iv) (iii) (ii) (i) (ii) are trivial. Let us show . (iv). The implications (iv) (iii) that (ii) ' (i). Assume (ii). Let H be a Hilbert space and consider u : H --+ X with $(u) = sup{e(uls), S C H} < oo. We will prove that (10.1) holds using Lemma 10.3. Let a = 1 - 80/2. Consider a subspace S C H with n = dim S. We claim that, if [an] > 1, we have (10.8)

a[an](uls)
1. On the other hand, the case [] < 1 is trivial. Indeed, if n < 2/8o then obviously sup k1/2ak(u) < n1/2IIuII < (2/6o)1/zt(u). k E satisfying t(u) = n1/2 and t(u-1*) < K(1 + Log dE)n1/2. By Theorem 5.8, there is a subspace F1 C E with codim F1 < k such that (10.9)

IIujF II < K(1 + Log

Consider Sl = u-1(Fl) and ups,

:

dE)(n/k)1/z.

Sl -+ E. By (10.1) there is a

subspace S2 C S1 with dim Si/S2 < k such that IIuIs2 II n - 2k + 1, F C F1, and, by (10.9) and (10.10), dF < IIUISz II . IIujF II

< wC2(X)(n/k)K(1 1/2 + Log

dE)(n/k)1/z.

Chapter 10

156

By an obvious adjustment of the constant K, we can replace 2k by k in the preceding estimate and we obtain for every k = 1,... , n a subspace

F C E with dimF = n-k such that dF < KwC2(X)(n/k)(1+LogdE) where K is a numerical constant. By the iteration argument of Chapter 8 (cf. Lemma 8.6), for every k = 1, ... n, there is a subspace F C E with dim F = n - k such that, for every k = 1, ... , n, there is a subspace F C E with dim F = n - k such that

dF < K'wC2(X) (n) (1 +Log(K'wC2(X)n)), where K' is a numerical constant. This concludes the proof of that (i) = (iv) and the proof Theorem 10.2 is complete.

Remark: In case X is of cotype 2, it is possible to prove a better estimate then (iv) above, namely

dx(6) < K(1 - b)-1/2(1 + log(1 - 6)-1) for some constant K. This was first proved in [PT1] (cf. [MP1] for a slightly more direct proof). We do not know whether such an estimate is valid in a weak cotype 2 space. We now come to the relationship with the notion of volume ratio introduced in Chapter 6.

Theorem 10.4.

Let X be a Banach space. The following are equiv-

alent.

(i) X is a weak cotype 2 space. (v) There is a constant C such that for all n, for all n-dimensional subspaces E C X and for all v : E --* t2 we have en(v) < C

7r2(v)n-1/2

(vi) There is a constant C such that for all f, d. subspaces E C X we have vr(E) < C.

Remark: The constants involved in the preceding equivalent properties can all be estimated in the usual manner. For instance, in the proof of (i) (vi) below, we will actually establish the following more precise statement: There is a function A f (A) such that any Banach

space X with wC2(X) < A must satisfy (vi) with C < f(A). For the proof we will use the following simple result.

Weak Cotype 2

157

Lemma 10.5. Let E be an n-dimensional space. Assume that there is a constant D and a > 0 such that for all k < n there is a subspace F C E with dimF > n - k such that dF < D(n/k)«. Let v : E -+ e2 be an operator. Then we have for all k ck(v) < D(n/k)7r2(v)k-1/2.

Proof: Let j be an integer with 1 < j < n. Let F C E satisfy dF < C(n/j)« and dimF > n - j. We may consider vIF : F -+ ez as an operator between Hilbert spaces and using (1.16) we find n

1/2

(E Ci(//VIF)2)

dF7r2(v),

i=1

hence

Ci(vIF)
2 and we have Cq(X)

Q(q)wC2(X ),

where,3(q) is a constant depending only on q.

Proof: We claim that there is a constant ,0 such that for all m and

all u: t2 - E we have (10.12)

7r2 (U) < Q(1 + Log n) sup k112ak(u).

This clearly implies (10.11) since we have, denoting xi = u(ei), M

1/z

< 0(1 + Log n)wC2(E)t(u).

Ilxill2)

To prove (10.12), we assume sup k1/2ak(u) = 1. Let vk : t2m -+ E

be operators satisfying rk(vk) < 2k and IIu - vkll S a2k(u). Let

Do = vo and Ak = Vk - vk-1 so that rk(Ak) < 2k + 2k+1 and IIokll s Ilvk - III + IIu - vk-111 < 2-k/2 +2 -(k-l)/2 < 4.2-k/2.

Note that Vk = u if 2k > n since dim E = n, so that u =

EO ... > lanl. We note that the symmetry of the independent variables (gi) implies that the sequence (gixi) is 1-unconditional in L2(X). Therefore we can write, for any k, n

n II

'1 ' aigixi

L2(X)

_ Elailgixi

IL2(X)

1

k

>

lailgixi

L2(X)

k

l ak l

9ixi

L2 (X)

hence by (10.15)

Iakl(2C)-1k12.

Chapter 10

162

Taking the supremum over k, we obtain (10.16). Let A >_ 1. Recall that a finite or infinite sequence (xi) in a Banach space is called A-unconditional if, for all finitely supported sequences of scalars (ai) and for all choices of signs Si = f1, we have

IIESiaixil < al Eaixi I.

(10.17)

The smallest A for which this holds is called the unconditionality constant of {xi}.

Corollary 10.9.

Let x1, ... , xn be a normalized A-unconditional sequence in a weak cotype 2 space X. Then we have, for all a in Rn, rn

(10.18)

llall2oo
s letixi(n E N,xi E X). Equivalently, Rad(X) is the space of all series

En enxn with coefficients in X which converge in =1

L2(X) = L2(1, P; X).

Similarly, we consider an i.i.d. sequence of Gaussian variables (gn)

on (1, A, P) and we denote by G(X) the closure in L2(X) of all the functions Enj=1 gixi. Again, G(X) coincides with the set of all the series EO_1 gnxn which converge in L2(X). Note that with the notation introduced at the end of Chapter 2, G(X) is the closure in L2(X) of unGn(X). In the next statement we denote simply L2 and L2(X) instead of L2 (1, P) and L2 (1l, P; X).

Proposition 10.11.

The following properties of a Banach space X

are equivalent. (a) X is a cotype 2 space.

(b) 22 (X) is a weak cotype 2 space. (c) L2(X) is a weak cotype 2 space. (d) G(X) is a weak cotype 2 space. (e) Rad(X) is a weak cotype 2 space.

Proof: It is easy to show that X is a cotype 2 space if the same (b) is easy. The implicais true for L2(X) or £2(X). Hence (a) tion (b)

(c) follows from the elementary fact that L2(X) is finitely representable into £2 (X) (see the comments following Definition 10.1). (d) is trivial. The implication (d) (e) follows The implication (c) from the fact that if (d) holds, then Rad(X) and G(X) are isomorphic.

Indeed (d) implies X weak cotype 2, hence (cf. Proposition 10.7) of cotype q for q > 2, and this implies by (10.19) that G(X) and Rad(X) are isomophic in a natural way.

Chapter 10

164

Thus it remains to show (e) = (a). Assume (e). We will use Proposition 10.8. Consider x1, ... , xn in X with Il xi jj = 1. Then, for any (ai) in Rn we have clearly sup l ai I
abn we conclude that X satisfies (vii) and this completes the proof.

Remark: The reader will easily check that we can replace L1(µ) in the preceding argument by any weak cotype 2 space L such that every operator B : L -+ £2 is 2-summing. We can thus relax the gB2-property by replacing L1(µ) by L, and the preceding statement remains valid.

Remark: Finally, let us briefly return to the case of cotype q spaces. Let X be a Banach space of cotype q (2 < q < oo). Then, as seen at the beginning of this chapter, there is a constant C such that for all n and all u : 22 -+ X we have (10.26)

sup kl/gak(u) < Ct(u).

Now let E C X be a subspace of dimension n. Let u : B2 -+ E be such that t(u) = B*(u-1 = n1/2) (cf. Theorem 3.1). Let P be any orthogonal projection on i2; we have, clearly,

rk(P) = tr P < f(uP)C*(u-1), hence

£(uP) > rk(P)n-1/2 for some numerical constant a > 0.

In other words, we have found an E-valued Gaussian variable

Z with d(Z) > a'n2/q for some constant a' > 0.

Equivalently,

6(E) > a'n2/q (with the notation of Chapter 4). Therefore, by Theorem 4.4, we can state

Chapter 10

168

Theorem 10.14.

Let X be a space of cotype q (or merely satisfying (10.26) above). Then there is a function 71(e) >0 such that every E C X

of dimension n contains a subspace F C E with dim F = [q(e)n2/9] which is (1 + e)-isomorphic to a Euclidean space. The preceding result shows that if a space X is roughly far from L,,, then the Log N estimate in Dvoretzky's Theorem (cf. Chapter 4) can be improved to a power of N. We recall that for example Lq is of cotype q if q > 2 (and of cotype 2, otherwise) so that Theorem 10.14 applies in that case. This refines the estimate given above in Proposition 4.15. Remark: It seems natural to call the spaces satisfying (10.26) weak cotype q spaces. But actually, the class of spaces which are really of interest to us from the point of view of these notes are the spaces which satisfy the conclusion of Theorem 10.14, and for these we know of no neat characterization so far.

Notes and Remarks In their paper on the inverse Santalo inequality (cf. Chapter 7), Bourgain and Milman proved that any space of cotype 2 has a uniformly bounded volume ratio. In other words, they proved that every space of cotype 2 satisfies the property (vi) in the preceding chapter. This result was the main motivation for the study of weak cotype 2 spaces which was developed in [MP1]. Thus all the statements from Definition 10.1 to Proposition 10.11 come from [MP1], with minor refinement occasionally. Note, however,

that we have incorporated the estimate of [PT1] to state the best known estimate for dx (b) in Theorem 10.2. Theorem 10.13 is due to Figiel and Johnson; it is implicit in [FJ1] and was (explicitly) communicated to us by W. B. Johnson. Finally, Theorem 10.14 is due to Figiel, Lindenstrauss, and Milman [FLM].

Concerning the notion of weak cotype q, it was observed by U. Matter and V. Mascioni that this is equivalent to the cotype q property restricted to vectors of equal norm (precisely: there is a constant C such that for any x1, ... , x,, in the unit sphere of X we have nllq < CII EgkxkIIL2(x)) It is known that the latter property is strictly weaker than cotype q if q > 2. This follows from an examplt of Tzafriri; see [CS] for more details.

Chapter 11

Weak Type 2

This chapter is based on [MP1] and [Pal]. We proceed as in the previous chapter and start by recalling the notion of type p. Let 1 < p < 2. A Banach space X is called of type p if there is a constant C such that for all n and all x1i... , xn in X we have 1 there is a subspace E C X such that d(E, t i) < 1 + e.

Weak Type 2

171

(In that case, we say that X contains ti 's uniformly.) Moreover, X is K-convex ifX is of type p for some p > 1. Also, X is K-convex ifX is locally 7r-Euclidean, which means the following: There is a constant

C and for each e > 0 and n > 1 an integer N(n, e) such that every subspace E C X with dim E> N(n, e) contains a subspace F C E with dim F = n such that dF < 1 + e and F is C-complemented in X (which means there is a projection Q : X -* F with J1Q11 < C).

Proposition 11.4. Let X be a Banach space.

Then X is

weak type 2 if X* is K-convex and weak cotype 2. Moreover, wC2(X*) < wT2(X) < K(X)wC2(X*). On the other hand, X* is weak type 2 if X is K-convex and weak cotype 2. (We recall that K-convexity is a self-dual property and K(X) = K(X*).) Proof: We have seen in Chapter 3 that if X is K-convex, then for all v : X -* t2 we have (11.3)

t(v*) < K(X)t*(v).

Thus, if X* is K-convex and weak cotype 2, we have supk112ak(v*) < wC2(X*)t(v*)

< wC2(X*)K(X)t*(v). This shows that X is weak type 2 and wT2(X) < wC2(X*)K(X). Conversely, assume X weak type 2. Then, by Proposition 11.2, X is of type p > 1, hence by Theorem 11.3 X is K-convex. Now using the trivial inequality t*(v) < t(v*) for all v : X --' t2, we deduce from (11.2) that X* is weak cotype 2 and wC2(X*) wT2(X). This proves the first part of the statement. It is easy to see that we can interchange the roles of X and X* in the above proof, whence the second part.

Remark 11.5: We have just proved that X is weak type 2 if it is K-convex and dx. (6) < oo for all 0 < 6 < 1. We should emphasize that if X is K-convex, we may drop the logarithmic term in the upper estimate of dx. (6) in Theorem 10.2 above, so that we have dx. (6) < K(1 - 6)-1 for some constant K. This is clear from the proof of Theorem 10.2 (recall that the log factor in Theorem 10.2 comes from the estimate of finite-dimensional K-convexity constants which are unnecessary if X and X* are K-convex.)

Chapter 11

172

While weak cotype 2 is related (via Theorem 10.2) to the existence of large Euclidean sections, the notion of weak type 2 is linked to the existence of projections of large ranks onto Euclidean subspaces. This direction was motivated by a result of Maurey [Mall, who proved that if X is type 2 then every operator u : S ` £2 defined on a subspace S of X admits an extension u : X -+ £2. Equivalently, there is a constant

C such that for all S C X and all u : S -+ i2 there is an extension u : X -+ in such that llufll < CJJuJJ. The converse is still an open question. We will see, however, that a weak form of this extension property characterizes weak type 2. (In this chapter the properties numbered from (i) to (ix) are all equivalent to weak type 2.)

Theorem 11.6.

The following properties of a Banach space X are

equivalent.

(i) X is weak type 2. (ii) There is a constant C satisfying for any subspace S C X and any u : S -+ $2 there is an extension l : X -* i2 such that for any k = 1, 2, ... , n there is a projection P £ -+ $2 of rank > n - k such that JlPuII < C(n/k)1/211 u11.

(iii) There is a 0 < 6 < 1 and a constant C such that for any S C X and any u : S --+ I2 there is a projection P :.f2 -+ in of rank

> on and an operator v : X -+ P2 with vIs = Pu such that (IvII < CIIuII

(iv) There is a constant C and 0 < 6 < 1 such that every f.d. quotient space Z of X satisfies the following: for every n, every n-dimensional subspace E C Z with dE < 2 contains a subspace F C E with dim F > On onto which there is a projection

Q:Z-*Fwith l1Q11 0 and all n, every operator v : i2 -+ XIS admits a lifting v : i2 -+ X such that Qv = v and

f(v) 8'n such that dE1/E2 < 2. Then E1*

Chapter 11

174

can be identified with Z = X/El. Also, (E1/E2)* can be identified with the subspace G of Z formed by the elements which vanish on E2. By (iv) there is a subspace G1 C G with dim G1 > b dim G which

is C-complemented in Z by a projection Q : Z -+ G1. This clearly implies that Z admits a quotient space, namely Z1 = Z/ Ker Q, which is C-isomorphic to G1 and hence satisfies dZ, < 2C. Dualizing again, this means that Z* = E1 admits a subspace F with dim F > 6 dim G (hence dun F > bb'n) such that dF < 2C. We have thus shown that X * satisfies (ii) in Theorem 10.2, hence X* is weak cotype 2. By Proposition 11.4, this completes the proof that (iv) (i).

Remark: Note that in (iv)

(i) we only use (iv) for E C Z with

dim E proportional to dim Z. Remark 11.8: In analogy with Proposition 10.11, we note the equivalence of the following properties:

(a) X is type 2. (b) £2(X) is weak type 2. (c) L2(X) is weak type 2. (d) G(X) is weak type 2. (e) Rad(X) is weak type 2. Indeed, when X is K-convex, we may identify G(X)* (resp. Rad(X)*) with G(X*) (resp. Rad(X*)), so that this follows immediately from Proposition 11.4 and Proposition 10.11. We will now dualize Corollary 10.9. For that we need to introduce a predual of 2200, namely the space 221. Let a = (an) be a sequence of numbers tending to zero at infinity. Let (a;,) be the non-increasing rearrangement of (l an l )n. We define IlaII21 =

00

an*n-1/2

n=1

We denote by 221 the space of all sequences (an) such that IIall21 < oo.

It becomes a Banach space when equipped with this norm. It is well known that 221 = 2200 with equivalent norms.

Proposition 11.9. Let x1, ... , xn be a normalized A-unconditional sequence in a weak type 2 space X. Then we have for all a in Rn IIE a'xzll r}.

Proof: Let E be the span of X1, ... , xn. Let xi,... , xn be the functionals in E* which are biorthogonal to x1i ... , xn. We have, by Proposition 11.4, wC2(E*) < wT2(E), hence by Corollary 10.9

daERn

IIaII2o, m - k such that owP = wP and IIuPII < C'(mlk)1/2IIwII

(11.6)

for some constant C. Then vi P : P2 -+ 4 satisfies (recall (1.13) and Proposition 1.6)

X ak(vwP)2)1/2 = IIvwPII xs = 7r2(vwP) < 7r2(v)II@PII; hence ak(vwP) < obtain

7r2(v)IIwPIIk-1/2.

ak(vwP)
f2 and B* : X * --p f2 as follows:

V X E X Ax = (x*(x))j>1 V x* E X* B*x* = (x*(xi))i>j.

Clearly B* is the adjoint of an operator B :£2

X. It is easy to

check that 1/2

"2(A)

(Ell x; II2)

lr(B*)

(E 11x;112)1/2.

and

Therefore, for any operator U : £2 -> £2i with norm 1, the composition T = BUA belongs to 172* (X, X) and (12.18)

'y (T)

llXll1z(x)IIx*Ilf2(X

)-

By (ix) we have sup nl An (BUA) I < Cry2 *(T), but we observe that the non-zero eigenvalues of BUA are the same (with multiplicity) as those

of UAB (cf. e.g. [Pill). Therefore (12.18) implies (12.19)

supnIAn(UAB)I
n/2 and dEl IE2 < K, when K is a numerical constant. By Step 1, the space El satisfies (xii). Moreover, El D E2 = (El/E2)* and dEJ < K. Note that dimE2 > n/2. Let 9 = (2CK)-1.

Chapter 12

202

By Step 2, there is a subspace F C E2 with dim F > On/2 and a projection P from El onto F with IIPII S 0-1. Clearly, dF n9(2CK)-1 and IIPII an for some a > 0. Then there is a subspace F C E and a projection P : X --+ F such that -y2 (P) < 2/a and dimF = [an/2]. Proof: This is a routine argument. By (12.13) and the comments after it, rye (i) > an if there is an operator S : X -+ E such that -y2(S) < 1 and tr Si > an. We can find a decomposition S = S1S2 with S2 : X -+ in and Sl : 4' -+ E such that IISIII < 1 and IIS2II < 1. Let R = S2,ES1. Then R : 4' -+ 4' satisfies

tr R = tr S1S2IE = tr Si > an.

Let R = UIRI be the polar decomposition of R.

We have trIRI > Itr RI > an and IIRII _< 1. Replacing S2 by U*S2i it is no loss of generality to assume that R= IRI. We may as well assume (for simplicity) IRI diagonal with coefficients 1 >_ Al > ... > An > 0.

Let k = [an/2]. Since F pi > an, we have µk > a/2. Let G = [el,... , e,] C 4' and let PG be the orthogonal projection onto G. Consider the operator w = (RIG)-'PG. Note that IIwII < 2/a since

a, > a/2 for i < k. It is then easy to check that P = S1wS2 is a projection from X onto a subspace F = S1(G) C E with dimF = k and y2(P) < IISIII IIwII IIS2II < 2/a. This completes the proof.

Corollary 12.8. For any weak Hilbert space X, there is a constant K such that, for all n > 1, every n-dimensional subspace E C X satisfies dE < K Log n. Actually, there is a projection P : X -+ E with -y2(P) < K Log n.

Weak Hilbert Spaces

203

Proof: Let E be as in the statement. Let i : E -4 X be the inclusion map. We claim that for some constant K we have for all T : E -> E Itr TI < K Log n(ry2(iT)).

Indeed, assume rye (iT) < 1; then there is an operator T : X --+ X extending iT and such that rye (T) < 1. By Theorem 12.6, for some constant C we have IAk(T)I E satisfying -y2(P) < 4C (resp. -y2(P) < 4ab). Indeed, by (13.12) and (13.13) (with Y instead of E) we have for all 3 : f2 --+ Y, 1r2(/3) < 4C7r2(,3*). This implies that if i : E -* Y is the inclusion mapping with

dimE = m, then for all u : E -> E we have N(u) < 4Cry2 (iu), hence tr u < 4Gy2 (iu). By the Hahn-Banach theorem, it is easy to see that this implies the existence of a projection P : Y -> E with -y2 (P) < 4C. If Y is as in (ii) we simply use C < ab. We can now prove

Lemma 13.5.

The space Xb is a weak Hilbert space. Actually, it is a weak cotype 2 space of type 2.

Proof: We first show that Xb is a weak cotype 2 space. Let Y,,, be the span of {e,,,,+,, em+2, ... } in Xb. Clearly, by Proposition 13.2, Y,,,, satisfies the assumption of Lemma 13.3 with a = 6-1 and b = 1. Therefore, every m-dimensional subspace E C Y,,,, satisfies dE < 46-1.

Now, consider a subspace El C X6 with dimE, = 2m + 1, and let E2 = E1 n Y,,,,. Clearly, dimE2 > m. Therefore E2 (and hence

El) contains an m-dimensional subspace E which must satisfy dE < 46-1. By Theorem 10.2, this proves that X6 is a weak cotype 2 space. Therefore, X6 is of cotype q for all q > 2 (by Proposition 10.7) and since it is 2-convex, it must actually be of type 2 by a result of Maurey (cf. [LT2], 1.f.3 and 1.f.9). We note in passing that if we use Remark 13.4 and the fact that is complemented in X6 by a norm one projection we obtain that the 46-1-complemensubspace E C E2 in the preceding argument is also ted in X6. This gives an alternate proof (based on Theorem 12.1) that X6 is a weak Hilbert space. To prove that X6 is not isomorphic to a Hilbert space, we will need several more specific lemmas concerning the norms 11

11,,,.

Some Examples: The Tsirelson Spaces

Lemma 13.6.

211

For all x in R(N) and all n > 0 we have

(13.15)

IIxI12 0 such that V x E R(N)

0 (I

Ixk12)1i2

IxkI2.

Choosing n large enough so that Stn < 92/2, we obtain b x E R(N)

(02)

1: IxkI2 < IIxiI,,

Some Examples: The Tsirelson Spaces

213

but this and (13.7) implies that X5,,, is isomorphic to £2i which contradicts Lemma 13.8. This contradiction concludes the proof.

Remarks: (i) The preceding argument shows that no subsequence of (ek) is equivalent to £2. Actually, it is known (cf. [Jo2]) that X6 contains no subspace isomorphic to £2. (The relation between X6 and Johnson's space is discussed in detail below.) (ii) In fact, it is proved in [CJT] that any sequence of normalized disjoint consecutive blocks on the basis (ek) is equivalent to a subsequence of (ek) in X6 (and hence cannot span an isomorph of l'2).

(iii) Morover, it is known (cf. [C]) that the spaces X6 are all mutually non-isomorphic, and in fact are totally incomparable, i.e. if b # 6', then X6 and X61 have no isomorphic infinite-dimensional subspaces.

Remark 13.9: The spaces X6 described above are part of a class of examples known as the "Tsirelson spaces". They are named after the Soviet mathematician Boris Tsirelson (sometimes spelt also Cirelson) who constructed the first example of a Banach space which contains no isomorphic copy of co or £P(1 < p < oo), cf. [T]. (See also [FJ2] or [LT1], 2.e.1). Many variants of his construction have been investigated and a lot of information is available concerning these spaces (cf. [CS]). In particular, several equivalent definitions of the norm are known. Let us say that a subset (yl, ... , y,,,.) is allowable if there are disjoint

subsets El,... , E,,,, of N such that yi coincides with the indicator function of Ei and such that ElUE2U...UEm. C [m+1,oo[. We will prove below the following Claim: If we repeat the preceding construction with allowable subsets instead of acceptable ones, we obtain the same norms II III and II II and the same space X6.

Indeed, let us denote by [ ] the sequence of norms constructed as above but with allowable subsets of R(N). We set [x] = as before. Clearly, [ ] still satisfies the 2-convexity property (13.8). Hence, if we define (13.17)

V x E R(N)

1114 1 = [E IxkI1/2ek] 2

,

Chapter 13

214

we obtain a 1-unconditional norm on R(N). By the defining property of [], we have (13.18)

V x E R (N)

62 > lilyi

X111 < ilixil1

i<m

for all y,, ... , y,n allowable. Now if we fix m and an integer N, we may consider the convex hull of all the allowable m-tuples (yl,... , y,,,,) with supp(yi) C [m + 1, m + N]. It is easy to check that this convex hull coincides with the set C of all the m-tuples (zl,... , z,,,) in R(N) such that supp(zi) C [m + 1, m + N], zi > 0 and 11 Ei<m, zi IioO < 1. (Indeed, the allowable

sets are the extreme points of C.) Therefore, by convexity we deduce from (13.18) that for all z1,... z,,,, in C 62

(13.19)

E iiizi xlii

Ilixill.

i<m

Let us denote for all x in R+Ni, xl/2 = E xk/2ek. Going back to [x], (13.19) implies (13.20)

6

(1: [zi /2

x,2

1/2

< [x].

.

But clearly, when (zl,... , zm) runs over C, then (zi/2, ... , over all acceptable subsets with support in [m + 1, m + N]. Therefore, since N is arbitrary, (13.20) implies

runs

1/2

SI

E[yi.x]2)

< [x]

for all (yi, . . . , ym) acceptable. This clearly implies jjxii < [x], but the converse implication is trivial (since allowable acceptable) so that we conclude as announced that [x] = 1lxii. Similarly, we have [x],,, = Ilxlln for each n. This proves the above claim.

We can restrict the notion of allowability further and still obtain the same space. More precisely, let us say that (yl,... , y,,,,) is an admissible subset of R(N) if it is allowable and if the sets El, ... , Em which are the supports of yl, ... , ym satisfy

max{k E Ei} < min{k E Ei+i} for all i.

Some Examples: The Tsirelson Spaces

215

Equivalently, this means that we restrict y,.... , y,,,, to be the indicator functions of consecutive disjoint subintervals of [m + 1, oo[. We can then repeat the original construction of X6 with admissible sets instead of acceptable or allowable ones, and we thus obtain a new norm on R(N). Casazza and Odell proved ([CO]) the surprising fact that this new norm is actually equivalent to the norm of X6 as defined above with allowable (or acceptable) sets.

Notes and Remarks In the abundant literature on Tsirelson's spaces (see [CS]) the completion of R(N) for the norm defined in (13.17) with S = 2-1/2 is traditionally denoted by T and is called the Tsirelson space. Tsirelson's original example is the dual space T*. The space X6 with 6 = 2-1/2 is denoted by TT and is called the 2-convexified version of T (or rather of the modified version of T) in the book [CS]. W. B. Johnson ([Jo2]) proved that this space TT has the properties required in Theorem 13.1. Many variations are possible. In particular, the consideration of subsequences of the basis yields examples with significant strengthenings of the weak cotype 2 property. Namely, the proportional dimension on of the Euclidean subspaces of an ndimensional subspace can be replaced by n - f (n) for any function f tending to infinity with n. See [FLM] for more details on this. The reader will also find in [Jo2] a proof that X6 does not contain any isomorphic copy of B2. (Actually, no quotient of a subspace of X6 is isomorphic to £2i cf. [Jo2], [Jo3].) If one uses the admissible version of the space X6i a proof that X6 does not contain 12 was given already in [FJ2] but it was realized only later in [CO] that this modification leads to the same Banach spaces. Many results on block basic sequences in these spaces can be found in [CJT] and [BCLT]. In particular, it is proved in [CJT] that T* is minimal, i.e. it embeds in every infinite-dimensional subspace of itself. In [BCLT], it is proved that, T has a unique unconditional basis (up to permutation). The subsequences of the basis of X6 which are equivalent to the original basis are characterized in [B2]; see also [B1]. The Tsirelson space is also used in Tzafriri's construction of a Banach space with equal norm cotype q which is not of cotype q (q > 2); see [Tz].

It is proved in [Jo3] that there is a subspace Y of X6 such that all of its subspaces have a basis. More precisely, Y is spanned by a subsequence (e,,,,k) of the basis of X6 with (nk) growing sufficiently

216

Chapter 13

fast so that every quotient of a subspace of Y has a basis (but is not isomorphic to 22). See [Jo3] for the details. (Actually, by the results of [CO] this subspace Y is isomorphic to X6; see [CS] for more precision.)

Theorem 13.1 and Proposition 13.2 are due to Johnson [Jo2]. We have modified the presentation to make it as brief and as transparent as possible, but the ideas (in particular, Lemmas 13.5, 13.6, 13.7 and their proofs) are all explicitly or implicitly either in [Jo2] or in [FJ2]. Finally, the proof of Lemma 13.3 is a combination of well-known results

of Kwapien [K2] and an inequality on 2-summing operators of rank m due to Tomczak-Jaegermann [TJ3], but the result first appeared (essentially) in [KRT].

Chapter 14

Reflexivity of Weak Hilbert Spaces

In this chapter we discuss the proof and some ideas related to the following result of W. B. Johnson (we incorporate an observation of Bourgain which lifted an unnecessary restriction).

Theorem 14.1.

([Jol]) Weak Hilbert spaces are reflexive.

Since being a weak Hilbert space is clearly a superproperty in the sense of James [J1], this implies that weak Hilbert spaces are superreflexive, hence by [E] have an equivalent uniformly convex and uniformly smooth norm. We know of no satisfactory estimates for the corresponding moduli of convexity or smoothness (see [P4] for related information).

The proof of Johnson's result leads naturally to at least two (a priori different) weakenings of the notion of Hilbert space. We first recall that (for A > 1) a finite or infinite sequence (xi) in a Banach space is called a A-unconditional basic sequence if for all finitely supported sequences of scalars (ai) and for all choices of signs ei = ±1 we have (14.1)

I E eiaixi I 1 this implies (14.1)'

sup jai I < A 11E aixil

We will say that a Banach space X possesses the property (H) if for each A > 1 there is a constant K(A) such that for any n and any Aunconditional basic sequence (xl,... , xn) with Ilxi lI = 1(i = 1,... , n) we have (14.2)


0 such that II

(14.3)

`d x = (xo,x21... ,) E E(N)

a sup IIXnII

xn

bE IlxnII

The completion of E(N) for such a norm will be called here an infinite sum of copies of E. We denote it by Z. Moreover, if we have for all integers nl, and all x in E(N) ,

IIxII = II(xo,... ,xn1,0,0,... ,0,xn1+1,xnj+2,...)II (with an arbitrary finite number of zeros), then Z is called a subsymmetric sum of copies of E. Also, if for all x in E(N) we have d (en) E {-1,1}N II(EOxo,... , Enxn, ...

, )II