Asymptotic Theory of Finite Dimensional Normed Spaces: Isoperimetric Inequalities in Riemannian Manifolds (Lecture Notes in Mathematics)

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1200

Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

Vitali D. Milman Gideon Schechtman

Asymptotic Theory of Finite Dimensional Normed Spaces With an Appendix by M. Gromov "Isoperimetric Inequalities in Riemannian Manifolds"

Springer

Authors Vitali D. Milman Department of Mathematics Tel Aviv University Ramat Aviv, Israel Gideon Schechtman Department of Theoretical Mathematics The Weizmann Institute of Science Rehovot, Israel

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Milman, Vitali D.: Asymptotic theory of finite dimensional normed spaces / Vitali D. Milman; Gideon Schechtman. With an appendix Isoperimetric inequalities in Riemannian manifolds / by M. Gromov. - Corr. 2. printing. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2001 (Lecture notes in mathematics ; 1200) ISBN 3-540-16769-2

Corrected Second Printing 2001 Mathematics Subject Classification (1980): 46B20, 52A20, 60FlO ISSN 0075-8434 ISBN 3-540-16769-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfi Ims or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Typesetting: Camera-ready TEX output by the authors SPIN: 10797471 41/3142-543210 - Printed on acid-free paper

INTRODUCTION This book deals with the geometrical structure of finite dimensional normed spaces, as the dimension grows to infinity.

This is a part of what came to be known as the Local

Theory of Banach Spaces (this name was derived from the fact that in its first stages, this theory dealt mainly with relating the structure of infinite dimensional Banach spaces to the structure of their lattice of finite dimensional subspaces). Our purpose in this book is to introduce the reader to some of the results, problems, and mainly methods developed in the Local Theory, in the last few years. This by no means is a complete survey of this wide area. Some of the main topics we do not discuss here are mentioned in the Notes and Remarks section. Several books appeared recently or are going to appear shortly, which cover much of the material not covered in this book. Among these are Pisier's [Pis6] where factorization theorems related to Grothendieck's theorem are extensively discussed, and Tomczak-Jaegermann's [T-Jl] where operator ideals and distances between finite dimensional normed spaces are studied in detail. Another related book is Pietch's [Pie]. The first major result of the Local Theory is Dvoretzky's Theorem [Dv] of 1960. Dvoretzky proved that every real normed space of finite dimension, say n, contains a (1 + c)-isomorphic copy of the k-dimensional euclidean space l~, for k = k(c,n) which increases to 00 with n (see Chapter 5 for the precise statement). Dvoretzky's original proof was very complicated and understood only by a few people. In 1970 Milman [MIl gave a different proof which exploited a certain property of the Haar measure on high dimensional homogeneous spaces, a property which is now called the concentration phenomenon: Let (X, p, J.L) be a compact metric space

(X, p) with a Borel probability measure J.L. The concentration function a(X, c), c > 0, is defined by

a(X,c) = 1- in!{J.L(A.)j J.L(A) 2:

~,

A

~X

Borel}

where

A.={XEXj

p(x,A)~c}.

It turns out that for some very natural families of spaces, a( X, c) is extremely small. For

example, it follows from Levy's isoperimetric inequality that for the Euclidean n-sphere sn, with the geodesic distance p and the normalized rotational invariant measure J.L,

a(sn,c)

~

jiexp(-c 2 n/2) .

It follows from this inequality (see Chapter 2) that any nice real function on sn must be very

close to being a constant on all but a very small set (the exceptional set being of measure

VI

of order smaller than exp( -c 2 n/2)). This last property is what is called the concentration phenomenon. It has proved to be extremely useful in the study of finite dimensional normed spaces. Going back to the concentration function, we define a family (Xn , Pn, J-Ln) of metric probability spaces to be a Levy family if a(Xn , Cn diam X n ) n~ 0 (diam X n is the diameter of X n ). Chapter 6 below contains a lot of examples of such natural families. Many of these examples have deep applications in the Local Theory. It is usually quite a difficult task to establish that a certain family is a Levy family, the methods are different from one example to the other and come from diverse areas (including methods from differential geometry, estimation of eigenvalues of the Laplacian, large deviation inequalities for martingales, isoperimetric inequalities). Levy families, the concentration phenomenon and their applications to the asymptotic theory of normed spaces are the main topics of the first part of this book. We have already mentioned one application, namely Dvoretzky's Theorem. In the same direction we deal with estimation of the dimension of euclidean subspaces of various large families of normed spaces, that is, with the evaluation of the function k(c, n) mentioned above when restricted to some (wide) families of normed spaces. (This study originated in [MIl and [F.L.M.].) Here is an example: There exists a function c(c)

>

=

>

0, such that if X n

= (m. n , II . II)

is a family of normed

0 and n, X n contains a (1 + c)-isomorphic [c(c)n"], or for any integer k and any c > 0, there is an n such that

spaces, then either for some a copy of l~ with k

0, c

>

0 and any c

>

X n contains a (1 + c)-isomorphic copy of l~. (The proof of this result uses, besides the concentration phenomenon, also the notions of type and cotype introduced below.) We also deal, in the first part, with packing high dimensionall;, 1 l~

... ,ek))ie; E IR , (e;, ej)

with the metric p(e, f)

= (L:7=1 d(e;, /;)2) ~,

metric. Note that Wn,n

=

On, Wn,l

=

= o;j,l:S i,j:S k}

d being either the euclidean or the geodesic

sn-l and Wn,n-l

=

SOn

=

general Wn,k may be identified with On/On-k via the map rp:On

{T E Oni det T -+

=

I}. In

Wn,k, rp(el, ... ,e n ) =

(el ... ek).

c. The Grassman manifolds Gn,k, 1 :S k :S n, consists of all k dimensional subspaces of IR n with the metric being the Hausdorff distance between the unit balls of the two subspaces

p(c,~)=

sup :l:Esn-ln~

p(x,sn-1nc).

4

The equivalence Gn,k = On/(Ok x On-k) is again easily verified. d.

If G is any group with invariant metric p and Go is a subgroup we may define a

metric d on M

= G/G o

by

d(t,s)

= inf{p(g,h)jtp(g) = t,tp(h) = s}

where tp: G ---; M is the quotient map. In this way M becomes an homogeneous space of G. Note that in all previous examples the metric given on the homogeneous space of On is equivalent, up to a universal constant (not depending on n), to the metric given here. 1.6. The uniqueness of the normalized Haar measure allows us to deduce several interesting consequences. a. The first remark is that for any A

~

sn-l and

Xo

E sn-l.

JL{T EOn; Tx o E A} = JL(A). b. Next we give two identities. Fix I the (k - I)-dimensional sphere of E. Then

~

k ~ n, for E E Gn,k we denote S(E) = sn-l

nE

for all f E c(sn-l) where JLe is the normalized Haar measure on S( E) (by our convention JL on the left is the normalized Haar measure on sn-l and on the right on Gn,k' We identify m2n with (J: n (by introducing a complex structure in one of the possible ways). For each k we denote the collection of complex k-dimensional subspaces of (J: n by

Gn, k and the unit sphere of any E E (J: Gn,k by (J: S( E) (which can be identified with S2k-l). (J: Gn,k is again an homogeneous space and we get an identity similar to the previous (J:

one

Jr

S2n-l

fdJL =

J J (J:

Gn.k

(J:

sW

f(t)dJLe(t)dJL(E).

Gn, k, than the one in the first identity which, adjusted to the dimensions here, would be G 2n ,2k' Note that here one integrates on a much smaller space,

(J:

2. THE ISOPERIMETRIC INEQUALITY ON sn-l AND SOME CONSEQUENCES 2.1.

We begin with the statement of the classical isoperimetric inequality on the sphere. A

proof is given in Appendix 1. A simple proof of a version of Corollary 2.2. (which is sufficient for the applications here) is given in Appendix V. For a set A in a metric space (M,p) and e> 0 we denote A£ = {tiP(t,A) ~ e}. In what follows we use the geodesic metric on sn-l. THEOREM: For each 0 < a < 1 and e > 0, min{J.L(A£); A ~ sn-l, J.L(A)

= a}

exists

and is attained on A o - a cap of suitable measure (i.e., A o = B(xo,r) for any Xo E sn-l and r such that J.L(B(r)) = a, where B(r) = B(xo,r) = {XiP(X,X o) ~ r}).

Using this for the case a = 1/2 (in which case A o is half a sphere) we get:

2.2. COROLLARY: if A ~ sn+l with J.L(A) 2: 1/2 then J.L(A£) 2: 1- V7r/8e-£'n/2 PROOF:

By theorem 2.1 it is enough to evaluate J.L(B(7r /2

+ e)).

Note that cosnO is

proportional to the n-volume of Se - the set of points on sn+l which are of distance 0 from

B(1/2) (this is an n-dimensional sphere of radius cos 0). It follows easily (draw a picture) that

h(e, n)

cos n OdO /17 0 and an integer n let k(e,f),n) = [e 2 n/(2log 4/f))J. Let IE c(sn+l) then, for all e,f) > 0, there exists a subspace E ~ IR n + 2 with

=k

dim E

(i)

~ k(e, f),

= sn+l n E

n) and a f) - net N in SeE)

I/(x) - Mfl ~ Wf(e)

for all

such that

x EN

and x E En sn+l.

for all

(ii) follows from (i). The proof of (I) consists of the following two lemmas. As in 2.3. A

= {xjf(x) = M f }·

2.5.

LEMMA: For any N ~ sn+l with

INI < V1rfi ee

2

n/2 there exists aTE On+2 such

that TN ~ A e . Consequently, for all x E TN, I/(x) - Mfl ~ Wf(e).

PROOF: The lemma follows immediately from 1.6.a: for each x E sn+l J.L{T E On+2; Tx E A e }

= J.L(A e ) ~ 1 - V1rfi e-e

2

n/2. Therefore,

7 JL{T E On+2i Tx E A£ for every x E N} 2: 1-INlv1l"/2 e-£'n/2

>

°. o

2.6. For later use we state the next lemma in a more general framework. LEMMA:

S(X) =

For every normed space X with dim X = k there exists a 0 - net {x E Xi Ilxll = 1} with INI ~ (1 + 2/0)k ~ e k log3/8.

{Xdi::l

PROOF: Let

i

i= j.

be a maximal set in S(X) with the property that

Ilxi -

N

In

xiII 2: 0 for

Then {xi}f=l is an 0 - net. The open balls B(Xi,O/2) are pairwise disjoint and are all

contained in B(O, 1 +0/2). Comparing the volume of B(O, 1 + 0/2) with that of Uf=lB(Xi, 0/2) we get

or

o 2.7.

REMARK: An inspection of the proof shows that, taking a bit smaller k in Theorem

2.4. one can get the conclusion for most subspaces of dimension k : For k ~ [e: 2 n/(410g 4/0)] the measure of the set Ek ... xn)1 =

yn

(l~

is the n-dimensional canonical Hilbert space, i.e., (L~=l xW/ 2).

mn

with

We only sketch the proof briefly. Since D ~ B(X), Ilxll ::; Ixl. We want to show that B(X) ~ y'n D. Applying a linear transformation, we may assume that D = {(XI""'X n ) E m.nj L~=lx; ::; I}. If

B(X) ~ ..;n D then there exists apE B(X) with Ipl > ..;n. Since B(X) is convex, also K = conv D U (±p) ~ B(X). We want to show that K contains an ellipsoid of volume larger than the volume of D, in contradiction to the assumption. Without loss of generality we may assume that p = (d, 0, ... ,0), d > ..;n. The ellipsoid n

{x E

m. n jxUa 2 + LxUb2::; I} i=2

has volume a· bn -

Vol (D) and as long as a 2/d 2 + (1-1/d 2)b2 ::; 1 it is contained in K (check). The pair a = d/..;n, b = l/n / 1/ d 2 satisfies this last requirement and a' bn - l > 1. l

.

VI -

VI -

o 3.4. THEOREM (Dvoretzky-Rogers): Let X = (R n , 11·11) be an n-dimensional normed space

and let D be the ellipsoid of maximal volume inscribed in B(X). Then there exists a basis (Xi)~=l orthononnal with respect to D such that 1 ~

IIxili ~ 2- n /(n- i+l), i = 1, ... , n -

PROOF: We choose an orthonormal system inductively in the following way:

x;).L n D with maximal possible

Ixl::; 1, IIxii

is any

IlxIiI = 1). Given XI, ... ,Xi we choose Xi+! in II ·11 -norm. Then for any X E span(xi,"" x n ) with

vector in D with maximalll·ll-norm (clearly (Xl' ••• '

Xl

1.

::; Ilxill·

Consider the ellipsoid

~i-l 2 ~n 2 } wI aj +~ 0 be such that

1

0

1 - 20

0

1-0

2

1-0

2'

- - < 1 + - and - - > 1 - let

0:'

> 0 be such that 1 + e;' 1 8 nd 1 - 29 - e;' 1_ 0 1-9 < + a 1-9 > ,

*

and let

c(o)

0:

=

£I.~r. By Theorem 2.4., there exists an

and a O-net N in En

sn+1

E

~

n

R + 2 with dim E;:::

[~ lo~2:18]

=

such that

for all x EN. It follows from Lemma 4.1. that, for all x E E,

o REMARKS: a.

It is easy to see from the proof that the constant c(o) satisfies c(o) ;:::

c0 2/10g% for some absolute constant c. Using Remark 2.7. one may show that the collection of all subspaces E

b.

having the property d(E, l~) ~ (1 4.3.

+ 0)/(1 -

~

G n + 2 ,k

0) has measure;::: 1 - exp( -c(o) . n . M; /b 2 ).

We turn now to the problem of finding large euclidean sections in a finite dimensional

normed space. Theorem 4.2. reduces this problem to the problem of finding euclidean norms for which Mr/b is as large as possible. For a finite dimensional normed space X and for we denote by k(X,o:) the dimension of the largest Hilbert space which is (1

0:

> 0,

+ o:)-isormorphic

to a subspace of X. THEOREM: Let X = (m n , II·

III

and assume d(X,l~) ~ d n , then

k(X, 0:) where c(o:) ;::: c· 0: 2/1 0g

Ilx'l

c(o:)· n/d~

c an absolute constant.

There exists an euclidean norm on IR n such that a . b

PROOF:

a-llxl

i,

;:::

blxl

for x E IR n . Obviously M r ;::: d;;l. Now apply Theorem 4.2. ~

~

min{llxll; Ixl = I}

;:::

a-l

~

d n and

so that Mr/b;:::

o

14 If for a family of spaces {Xn}::"=l we have sUPn d(Xn , (2) = M < 00 then we get from the Theorem above that k(Xn,e)/n is bounded from below by a constant depending on e and M alone. For this reason it is enough (for the purpose of getting asymptotic estimates) to deal only with k(Xn , 1) (say). We shall denote k(X, 1) by k(X). Before continuing with the investigation of the quantity Mr/b for some special families of spaces, we would like to gather some facts about the relation between k(X) and k(X·). 4.4. X

Ilxll',

Applying Theorem 4.2. to the dual norm r'(x) =

= (m. n +2 , II . II)

we get from 3.1.2. that if

and xE

then there exists a subspace F ~

m. n +2 with

(1 - e5) . M .. ·Ix\

~

Ilxll'

k'

= dim

m. n + 2

F ~ c(e5) . n· (M.. /a)2 and

(1 + e5) . M ..

~

'Ixl

for x E F.

Moreover, if k ~ c(e5) . n' min{(Mr /b)2, (M.. /a)2} then Remark b in 4.2. implies that one can find a subspace E ~ m. n with dim E = k such that simultaneously

and

(1- e5) . M ..

·Ixl

~

tlxll'

~ (1

+ e5) . M .. 'Ixl

for all x E E.

We summarize these two remarks (specifying a e5) in the following theorem. THEOREM: There exists an absolute constant c such that for every normed space X =

(m n , II . II)

and euclidean norm

I· I on mn

satisfying

we have

(i)

~ c. n2 . (Mr. M .. ) 2

k(X) . k(X')

a·b

and

(ii) There exists a subspace E

~

mn

with dim E ~ c· n' min{(Mr /b)2, (M.. /a)2} such

that 2

3" . M r 'Ixl

~

II xii

~

4

2

3" . M r 'Ixl and 3"' Mr. 'Ixl

4.5. Note that always Mr' M .. an x E sn-l with

Ilxll

~ M r and

~

~

Ilxll

•

~

4

3" . M .. ·Ixl, x

E

E.

1. Indeed, by the definition of M r and M .. , there exists

Ilxll'

~ M ... Then,

Mr' M .. ~

Ilxll' 'lIxll

~ (x,x) = 1.

15 Next, one may choose 4.4.

1·1 in such a manner that a·b =

d(X,q), therefore we get from Theorem

COROLLARY:

In particular,

k(X) . k(X') 2: c· n . The last statement follows from Theorem 3.2. 4.6. The next proposition evaluates the norm of a projection onto the euclidean section one

obtained in Theorem 4.4. (ii). PROPOSITION: Let E ~

mn

be a subspace sa'tisfying (ii) of Theorem 4.4. (all is needed

is Ilxll :::; ~ . M r 'Ixl, Ilxll* :::; ~ . M" 'Ixl x E E). Let P 1·1), projection on E. Then 16 IIPII :::; 9 . M r . M r *

(IIPII

be the orthogonal (with respect to

is the norm of the projection as an operator from X to X).

PROOF: For any x E IR n ,

IPxl

2

= (Px, Px) = (Px, x)

:::;

4

IIPxll" . Ilxll :::; 3' . M" . IPxl . IIxll.

Thus,

IPxl :::;

4

3' . M" . Ilxll

and 4

IIPxl1 :::; 3' . M r

·IPxl :::;

16

9' Mr' M" ·lIxll· o

4.7. Putting together Theorem 4.4. and Proposition 4.6. we get the inequality 4.7.1.

k(X) . k(X*) 2: c· (n

·IIPII/(a· b))2

where P is a projection onto a subspace satisfying (ii) of Theorem 4.4. Moreover, choosing the ellipsoid that gives the distance of X to l2, we get

4.7.2.

k(X)· k(X") 2: c· (n '1IPII/d(X,l2))2 .

= (IR n , 11·11) be a normed space and let 1·1 be an euclidean norm on IRn satisfying 3.1.1., a-llxl :::; Ilxll :::; blxl. We related above the search for euclidean subspaces of X with the median M r of r(x) = IIxll on the sphere sn-l = {x; Ixl = I}. The next theorem 4.8. Let as before, X

relates this search of subspaces of X, closer to being euclidean then X itself, to the median of r*(x) = IIxl'". In the next chapter we shall use this information to obtain good euclidean subspaces and quotient spaces of X.

16

THEOREM: Let X satisfy 3.1.1. Then for any 0 < ¢J < 1 there exists a subspace E

~

X

with k

= dim

E 2: (1 - ¢J)n

and

for all x E E where c > 0 is an absolute constant.

For the proof we need two lemmas the first of which is a different version of 2.2.

i

4.9. LEMMA: a. Let A ~ sn-1 with JL(A) 2:

and let 0
1

intersect any 8 - cap in Sk, that is, A t -£

n Sk

is a 8 - net

Take 8 = c - ¢ then, if 4.11.2. is satisfied, (A;--£)£-4> = At-4> covers Sk and by Lemma 4.10. we get that Ek+l satisfies the conclusion of the theorem. It remains to evaluate the largest k for which 4.11.2. is satisfied. Rearranging 4.11.2. and taking (n - 2) - th root we get that 4.11.2. is satisfied if

with

j.L"

< (-~)

*

for some absolute c.

This is equivalent to

18 or (sin f:)!.? -'------'--:-- ~ JLn (1 - tantPcotf:) cos tP or

1->'

->.-log(sin f:) - log cos tP ~ logJLn

Fix f:( =

i,

+ log(l -

tantPcotf:)

say) and notice that

and

3 n IlogJLnl ~ -2 log-« n c

llyn ~ tP·

Consequently, we get, for small tP ~ ~, that 4.11.2. is satisfied as long as 1~ A

~ etP (c

absolute)

or, as long as

>.


-

l/p

+ l/q =

1,

c. n 1+ 2/ q •

Since clearly k(e;) < n, we get that we actually have equivalence in all the inequalities above. We conclude with three corollaries.

a.

For q > 2, k(e;)/n 2 / q is bounded from above and below by constants depending only on q.

b.

n 2': k(e;) 2': c p • n, p

c.

nll/p-l/21

< 2, cp depends only on

p

2': d(e;,e 2) 2': cp ' nll/p-l/21, 1 < P
2, contains a subspace of dimension'

and "nicely" complemented.

5.5. For the proof of fact 2 we need the notion of the Rademacher functions which will play an important role also later in these notes. Define, for k = 1,2, ... and 0

~

t

~

1

For example,

rICt)

={

1 -1

t < 1/2 { 1 t < 1/4, r2(t) = t > 1/2 ' -1 1/4 < t < 1/2,

1/2 3/4

< t < 3/4

1/2

In particular, for r(x) = max1:5i:5n

O.

IXil ::;

1, A, ~ c'

(log n/n)l/2.

PROOF: Let v be the measure on IR n with density exp( -'If ·2:7=1 tn. This measure is a probability measure and is clearly invariant under isometries of l2' Le., under On. If !:sn-1

-+

IR is any integrable function, define j(t)

JIRR j(t)dv(t)

= Ilt112' !(ntiT;-)

in IRn\{O}. Then

is invariant under the action of On and the uniqueness of the Haar measure

J.L on sn-1 implies the existence of a constant An such that

JIRR j(t)dv(t) =

5.7.1. Using polar coordinates and putting

! ==

An

Jsn-l !(t)dJ.L(t).

1 in 5.7.1. one sees that

24 for some absolute constant C.

fm

n

This reduces the problem to the problem of estimating

maxl~;~m It;ldv(t) from below.

Now, for any a> 0,

Choosing

0:

= C' Vlog m for some absolute c we get that the last quantity is :s 1/2 so that

and

r lm

max It;ldv(t) ~ 1/2. C' Vlog m. l~.~m Combining this with 5.7.1. and the estimate for An we get

r

1sn-l

n

max IX;ldJL(x) =

l~.~m

A~l

r

lm

n

m,ax It;ldv(t)

l~.~m

~ c(log m/n)1/2 o

5.8. We are now in a position to prove Dvoretzky's Theorem.

There exists an absolute constant c such that k(X) n-dimensional normed space X. THEOREM:

~

c . log n for any

PROOF: Let D be the ellipsoid of maximal volume contained in B(X). Let the norm associated with it. Then by F. John's Theorem 3.2. l 2

n- /

In particular, b = 1 and a' b:S

Ixl:s Ilxll :s Ixi

1·1 be

forall x E X.

..;n, so that we may use Lemma 5.1. to evaluate

Mr.

By the Dvoretzky-Rogers Lemma 3.3. there is an orthonormal basis Xl,"" x n

Ilx;11

~

1/4, i

= 1, ... [n/2].

hn-l I ~ a;x;lIdJL(a) hn-l II ~ r;(t)a;x;lIdJL(a).

Ar =

=

Note that, by the triangle inequality, for all {y;} n

1

~

X n

10 II t; r;(t)y;lIdt = Avee,=±lll t;c;y;11 ~ l~fm Ily;ll· So, integrating (*) over [0,1] and using Lemma 5.7., we get,

r 1 II t r;(t)a;x;lldtdJL(a) ~ 1sn-l ~ r max lIa;x;lldJL(a) ~ ~ r max la;ldJL(a) ~ c· (log n/n)1/2 lsn-l l~.~n 4 lsn-l 1~.~[n/2J 1

Ar

=

0

;=1

with

Now, for each fixed t E [0,1],

(*)

25 for some absolute constant c. By Lemma 5.1., we get now that M r 2 c(log n/n)l/2 for some absolute c. Finally, using Theorem 4.2. we get the desired result.

o 5.9.

We conclude this chapter with a theorem stating that every n-dimensional normed

space has a subspace of a quotient space (= a quotient space of a subspace) of dimension proportional to n which is close to being euclidean. THEOREM: Let X 'be an n-dimensional normed spaces and let A < 1. There exists a

quotient space Y of a subspace of X with dim Y

2 An and

where c is an absolute constant. PROOF: We shall need one fact which will be proved only in Chapter 15. Namely, there exists an euclidean norm r(x) =

c (and

a-1lxl :S Ilxll :S blxl

Ixl on X

with a/Mr-

for which MrMr-

:S c 10g(d(X, £2)) for some absolute

:S yn).

By Theorem 4.8. there exists, for any 0 < 4> < 1, a subspace E

~

X with dim E = k 2

(1 - c4»n (c absolute) and

xE E. Moreover, (Note 4.11.1.) this holds for a subset of Gn,k (the Grassmann manifold) of measure larger than or equal to~. Using Lemma 5.1. and averaging over Gn,k we get that for a subset of Gn,k of measure larger than or equal to~, ME,r (the median of r restricted to

BE

= {x E Ej Ixl =

I}) is smaller or equal than c M r for an absolute c. (Actually, for any E

with dim E = k, ME,r

:S

c~ Mr. This follows from 9.5.1. below. We prefer not be use it

here). Consequently, we can find an E ~ X with dim E = k 2 (1 - c4»n on which

4>

M - Ixl :S r

Ilxll :S b[xl

xEE

and

ME,.:S c Mr. Since

b-1lxl :S Ilxll' :S ~-Ixi

xE E

we can find by the same argument a subspace F of E' with

dim F 2 (1 - c4»2 n and

(c absolute)

26

In particular, d(F, ldim F) 2

r- < cA.- 21 < MrM 0g ¢2 'I'

d(X , en) . 2

Note that F* is a subspace of a quotient of X. Since log d(X,lz) Theorem up to a log n factor. To get rid of that log n, define for t f(t)

= inf{d(F,l~n)j

~ log n

=

~,k

this proves the

= 1, ... , n,

F a subspace of a quotient of X with dim F

= tn}

.

Then the proof above shows !CAt)

< -

c

1

(1 -

vxA)2 log f(t)

(for any 0 < A < 1 such that At = ~ for some k = 1, ... ,n). We may extend f to all of (*,1] in such a way that this inequality still holds. In particular, 5.9.1.

We also have 1

~ f(s) ~

n. The result follows by iterating 5.9.1. We indicate briefly how to

perform the iterations. Put g(s) = f(s)

(2 (l_cS~)210g (l_cS~)2)-1

Then, for s close enough to 1 (independently of n), 5.9.2. as long as g(s) 2: e (check).

Iterating 5.9.2. we get

where log(f)x

= log log(t-1)x and log(l)x = log

x.

Given 0 < a < 1, let t be the first such that log(t)n ~ e and take s such that S2' = a. Then, g(a) ~ log(t)g(s) ~ e

and we get the desired result.

o

6. LEVY FAMILIES

6.1. Let (X,P,JL) be a metric space (X,p) equipped with a Borel probability measure JL. For a subset A

~

X and e: > 0 we define, as in Chapter 2,

A e = {x E Xi p(x,A) ~ e:}. Let diam X be the diameter of X and assume diam X 2: 1. The concentration function a(X, e:) is defined for any e: > 0 by

a(X,e:) = 1- inf{JL(Ae)i A DEFINITION: A family (Xn,Pn,JLn), n a Levy family if for every e: > 0,

~X

Borel with JL(A) 2:

~}

= 1,2, ... of metric probability spaces is called

The family is called a normal Levy family with constants

Cll C2,

if

Note that any normal Levy family is a Levy family (since we assume diamXn 2: 1). The omission of the factor diamXn in the definition of a normal Levy family comes because that way most of the examples below become Levy families with their natural metric and natural enumeration. As we have already seen in specific examples in Chapter 2, in a Levy family we necessarily have the phenomenon of concentration of measure around one value of a function. If f : X -+ m is a function with modulus of continuity wf(e:) and with median Mf then we get

6.1.1.

JL(lf - Mfl ~ wf(e:)) 2: 1- 2a(X,e:).

Indeed, put A = {x E Xi f(x) ~ Mf}, B = {x E Xi f(x) 2: Mf} then JL(A), JL(B) 2: 1/2 and

JL(IJ - Mfl ~ wf(e:)) = JL[(J ~ Mf

+ wf(e:)) n (J 2: Mf

- wf(e:))]

2: JL(A e n Be) 2: 1- 2a(X,e:) . That is, f is concentrated close to Mf on most (in the sense of measure) of X, if a(X,e:) is small enough. In 2.2. we have shown that {sn+l }~=l' with the natural metric and probability measure, is a normal Levy family with constants

Cl

=

J1r78 and C2

= 1/2. We applied this fact in

28 Chapters 2 and 4. In this Chapter we shall give a lot of other examples of Levy families. We shall see that a number of natural families of metric probability spaces are Levy families. Most of these examples found applications in the Local Theory of normed spaces. A few of them play a central role in this theory and for them we present a full detailed proof here or in later Chapters; the rest are only briefly sketched. 6.2. Let E2: = {-I, 1}n with the (product) measure

and with the normalized Hamming metric l I n

dn(x,y)

= -1{i;Xi t- y;}1 ="2 L n

n

IXi - Yil,

x,y E

E;.

i=1

THEOREM: (E2:, d n , P n ) is a normal Levy family with constants

Cl

= 1/2 and

The theorem follows from an isoperimetric inequality (i.e. identifying the sets A

C2

~

= 2.

E2: for

which the infimum inf{P(A~); P(A) 2 1/2} is attained) of [HarJ. We shall present a different proof (giving somewhat worse constants) in Chapter 7 below. The same proof will prove also the next result of [Maj. 6.3. Let TIn be the group of all permutations of the set {I, ... ,n}. Let P n be the normalized counting measure;

Pn(A) = IAI/n!,

A

~

TIn

and let d n be the normalized Hamming metric;

THEOREM: (TIn, d n , P n ) is a normal Levy family with constants

Cl

= 2,

C2

= 1/64.

6.4. The following few examples of Levy families are consequences of a general isoperimetric inequality for connected riemannian manifolds due to Gromov [Gr1J. Appendix I, written by Gromov, contains the proof together with the necessary definitions. Let J.lx be the normalized riemannian volume element on a connected riemannian manifold without boundary X and let R(X) be the Ricci curvature of X. THEOREM: Let A

~

X be measureable and let

where B is a ball on the sphere

and J.lx(A)

= J.l(B),

T •

sn

E:

> 0 then

with n = dimX, and

T

such that

J.l being the normalized Haar measure on T·

sn.

29 The value of R(X), known in some examples ([C.E.]), together with the computation for the measure of a cap in 2.2. leads to the following examples. THEOREM: The family SOn = {T EOn; detT = 1}, n = 1,2 ..., with the metric discribed in 1.5.b and the normalized Haar measure is a normal Levy family with constants

6.5.1.

Cl

= V1T/8,C2 = 1/8.

6.5.2. Similarly for each m the family X n = sn the product measure and the metric

X

sn

X •••

Sn(m - times), n = 1,2, ... , with

m

d(x, y)

=

(L p(x;, y;)2)1/2, X = (Xl,""

X m ),

Y

=

(Yl,"" Ym) E X n

i=l

(p-the geodesic metric in sn), is a normal Levy family with constants 6.6.

Cl

= V1T/8,

C2 = 1/2.

Next we show that homogeneous spaces of SOn inherit the property of being Levy

family. Let G be a subgroup of SOn and let V = SOn/G. Let II- be the Haar measure on V and let d n be the metric introduced in 1.5.d, Le. dn(t,s)

= inf{p(g,h);tpg = t,tph =

s}

where tp is the quotient map. Clearly II-(A ~ V) = lI-(tp-l(A) ~ SOn). By the definition of d n , tp-l(A~) :2 (tp-lAk Therefore, if II-(A ~ V) ~ 1/2, then lI-(tp-l(A) ~ SOn) ~ 1/2 and II-(A~) ~ II-((tp-l A)~). We conclude THEOREM: Let, for n = 1,2, ... , G n be a subgroup of SOn with the metric described above and with the normalized Haar measure II-n. Then (SOn/Gn,dn,lI-n),n normal Levy family with constants

6.7.

cl

= y'7rf8

and C2

=

1,2, ... ,18 a

= 1/8.

Theorem 6.6. together with Examples 1.5.b and 1.5.c implies immediately that the

following families are normal Levy families with constants

Cl

=

y'7rf8,

C2

6.7.1. Any family of Stiefel manifolds {Wn,k n };:C=l with 1 ~ k n ~ n, n

= 1/8:

= 1,2, ... ,

6.7.2. Any family of Grassman manifolds {Gn'k n };:C=l with 1 ~ k n ~ n, n = 1,2, ....

6.7.3. Exercise: Define a natural metric and probability measure on Vn,k = {€ E Gn,k;X E S(€)} and show that it is a normal Levy family (here, as before, S( €) is the unit euclidean sphere of the subspace

€).

6.8. We consider in this section a compact connected riemannian manifold M with II- being the normalized riemannian volume element of M. Then the Laplacian -D.. on M has its spectrum

consisting of eigenvalues 0

= Ao < Ar(M)

~

A2(M) .... The first non zero eigenvalue Al may

be represented as the largest constant such that

30

for every "sufficiently smooth" function

I

fM I

on M such that

= O. We refer the reader to

[B.G.M.l for more details. For the reader who is not familiar or does not feel comfortable with the notions above, we give a somewhat more detailed explanation on a model situation. We hope that this degression will help the reader to develop some intuition. Let X be a smooth n-dimensional compact connected C 2-smooth submanifold without boundary of the euclidean space (IR N

,

I . I).

Then norm

I . I induces

a metric on X. We,

however, consider a different metric, called the length metric, p: for x, y E X, p( x, y) is the length (with respect to

1·1 of the shortest curve in X

joining x and y. The Lebesgue measure

in IR n induces a measure on X. We normalize it so that the measure of X equals 1 and denote the normalized measure by J.l. We now have a metric probability space (X, p, J.l) For every x E X let T" denote the tanget (n-dimensional) plane to X at x. We write T"

= x + ~"

where

~"

is the parallel plane through the origin. Similarly the normal plane (of

dimension N - n) is N" = x + C:" where C:" E IR N • The space of pairs

GN,N-n

is the orthogonal complement of ~" in

T(X) = {(x,y); x E X, yET,,} is called the tangent bundle of X. We shall also use the dual bundle

T*(X) = {(x,z); x E X, z E T; = x + ~;} where ~; is the dual space to ~" (which we do not identify with ~,,). For e > 0 let U~(X) denote an

(1'1-)

e- neighbourhood of X in IR N

.

There exists an

e > 0 such that for every y E U~(X) there exists a unique x E X with yEN" (see [Mil). Therefore, given any I E C(X) we may extend it to j E C(U~(X)) by j(y) = I(x) for yEN". If

I-

is smooth enough we may consider

that the derivatives of

j

111-

N

(where standardly in IR ,11

N = Li=l Ix:). 2

Note

in the directions of N" are zero and therefore 11 may be written in sn ~ IR n +1, then, rewritting

terms of derivatives in the direction of T". For example, if X = 11 in spherical coordinates, we obtain the Laplacian on

sn,

(see [Vi], p. 493).

It is sometimes more usuful to see the Laplacian as constructed in two steps. In the

I on X the "il I(x) = "il II" E ~".

first step we introduce the gradient operation "il. For every Lipschitz function gradient is defined for almost all x and for such x is an element of ~'"

Indeed, it is more convenient to work with the dual space ~; and to consider "il I(x) as a linear functional (an element of G) we denote it then by dl(x)(E ~;) of course Idl(x)I' = l"ill(x)l. Then dl: X

-+

T'(X) or alternatively dl: T(X)

-+

IR by dl(x,y) = (dl(x),y) for y E ~". The

operator d from a dense subspace of L 2 (X) into L 2 (X

-+

T·(X),J.l) (the space of vector valued

functions F:X -+ T'(X) with IIFII = (J(IF(x)I;)2dJ.l)1/2,1·I;being the norm in ~;) can be extended to become a closed operator (also denoted by d)with the domain D(d). Let d' be the adjoint operator, then the Laplacian is given by

31 By construction, -A is non-negative. It is known that for a compact connected manifold

-A has a discrete spectrum, {A;}~O' Ai ~ 0, and that -AI = 0 only for I == const. It follows that AO = 0 and AI> O. The min-max principle implies now the inequality (*). 6.9. THEOREM: Let (M, p, J.L) be a compact connected riemannian manilold (J.L the normal-

ized riemannian volume). Let A

~

M with a = J.L(A) > O. Then, for all e > 0,

PROOF: Let A,B be two open subsets of M, J.L(A)

= a,J.L(B) = band p(A,B) = p > O.

Consider the function

I(x) = II a - (II p)(lla + 1/b)min(p(x, A), p), Let

fM IdJ.L

= a and apply

x E M.

(*) of 6.8. to get

Now, I is a constant on each of A and B so that the integration is on a set of measure

= 1 - a-b. I is a Lipschitz function with Lipchitz IV'II ~ (l/p)(l/ a + lib). Consequently we get

constant ~ (1 I p)( 1I a

+ lib)

so that

On the other hand

A1111 - aliL ~ Ad{ (f -

a) 2 dJ.L +

L

(f - a)2dJ.L)

= Ad(l/a - a)2 . a + (lib

~ A1(1/ a

+ a)2 . b)

+ lib).

Therefore,

1-a-b 2 AI' p ~ (l/ a + 1/b)(1 - a - b) ~ a. b or

b
0 and consider the sequence of pairs (Ai, Bi), i

= 0,1, ... , of subsets

defined inductively by:

A o = A, Bo =" ((Ao)s)C Ai+! Let ai

= J.L(Ai),

bi

= (Ai)s,

= J.L(Bi),

Bi+! = ((Ai+ds)C,

then

1-ab- < • • - 1 + Al .8 2 • ai

i = 0,1 ....

of M

32

Since ai

~

a for all

t,

bi

=

1 - ai+l ~

1+

1 - ai A 82 l'

.

a

Take 8 = 1/";;:;, then

l - a i+l

1 - ai

< -- l+a

and, by induction I-a 1 - ai ~ (1 + a)i'

If E: = i . 8 for some i, then i = E:~ and

1- J.L(A~)

= 1- ai

~ (1 - a)exp(-nj5:;log(1

+ a)).

< (i + 1)8. Then

In the general case pick the i such that i8 ~ E:

1 - J.L(A~) ~ (1- J.L(Ai.6)) ~ (1- a)exp(-E:~ log(1

+ a) + log(1 + a))

~ (1- a2)exp(-E:~log(1+ a)). EXAMPLES: 1. Al (sn) = n (this is the so called Wirtinger's inequality see [Gr2] and [Vi] p. 494) and we get: if A

. L dj )IJi-d) = E(exp(>. L dj) . E(exp>.diIJi-d j=l

j=l

i-I

:S E exp(>. L

dj) . exp(>.2I1dill~).

j=l Now, for all >. > 0, n

P(/ - EI ~ c)

= P(Ldj

n

~ c)

= P(exp(>. Ldj - >.c)

j=l n

n

:S E exp(>. Ldj - >'c):S exp(>.2. L

j=l Putting>.

= c/

~ 1)

j=l Ildjll~

- >.c).

j=l

2Lj=1I1djll~, we get n

P(/ - EI ~ c) :S exp(-c 2/4 L IIdjll~)· j=l Similarly, n

P(EI - I ~ c) :S exp(-c 2/4 L Ildjll~) j=l and n

P(II - Ell ~ c) :S P(/ - EI ~ c)

+ P(EI - I

~ c) :S 2exp( _c 2 /4 L IIdjll~). j=l

o 7.5. To illustrate the way one uses the lemma, we will prove the following theorem, already stated in 6.4. THEOREM: The family TIn of permutation of {I, ... n} with the metric d(ll", e) = ~1{ijll"(i) t= e(i)} I and the uniform measure P (assigning mass tion) is a normal Levy family with constants C1 = 2 and C2 = 1/64.

;h

to each permuta-

Later on we will state a more general theorem (7.8.) 7.6.

PROOF: Let 1j, 0 ::; j ::; n, be the algebra of subsets of TIn generated by the atoms

{Ai, ,.. o,ij j 1:S i 1, .. . , i j :S n distinct} where Ail ,o .. ,ij

Then {rEC 1(7r) = TiT L>rEB l(ep(7r)) !/i+lI B - IH 1 1c I~

I~I

L

TiT L>rEB f(7r)

and

so that

1/(7r) - I (ep (7r))I

>rEB

~ l~l

L

d(7r,ep(7r))

>rEB

ItfollowsthatforanyA,Basin(*) Ili+lIB-liIAI ~~. Indeed, lilA

~ ~.

= n~i LC~Ali+1IC

so that Ili+lIB - lilAI ~ n~i LC~A Ili+11B - IH11cI ~ ~. Since this holds for all A, B as in (*) we get that Ildi+llloo ~ ~ for i = 0, ... , n - l. Applying the lemma we get that if I is a function with Lipschitz constant 1 then

(**) If A ~ II n then d(·, A) is a Lipschitz function with constant one so we can use (**). Choose c such that 2exp(-c::) =~, i.e., c = 4..)(log 4)/n. Then 1

P(ld(·,A) - Ed(·,A)1 < 4..)(log 4)/n) . > -2 On the other hand if P(A) ~ ~ then P(d(·,A) = 0) ~

!.

So there exists a 7r E II n such that

Id(1r,A) - Ed(·,A)1 < 4..)(log 4)/n and d(7r,A)

= 0,

i.e., Ed(·,A) < 4..)(log 4)/n.

Plugging this back into (**) we get

P(d(., A) ~ c + 4..)(log 4) /n ) ~ 2 exp( -c 2 n/16) or, for all

E:

> S..)(log 4)/n,

It is easily checked that this holds also for

E:

~

S..)(log 4)/n.

To prove (*) let p be the permutation which changes

T

with s and leaves the rest at

= po 1r then ep(7r)(i) = 7r(i) for all i except possibly for i = j + 1 (where 1r(i) = T,ep(1r)(i) = s) and for i = 7r- 1 (s) (where 7r(i) = sand ep(7r)(i) = po 7r(i) = p(s) = r), place. Define ep(1r)

so d(7r,ep(1r)) ~ ~.

0

36

7.7. The property (*) of II n suggests the following definition: DEFINITION: Given a finite metric space (n, d). We say that (n, d) is of length at most

£ if there exist positive numbers a1,' .. ,an with £ = (I:7=1 an 1/2 and a sequence {nk}~=o' nk = {An:,\, of partitions of n with the following four properties: 7.7.1.

mo = 1 i.e. nO = {n}

7.7.2.

mn =

7.7.3.

n

7.7.4.

for all k = 1, ... ,n,r = 1, ... ,mk-1 and i,j such that A~,Aj ~ A~-l there exists a

k

Inl

i.e.

nn =

{{X}}XEIl

is a refinement ofn k- 1,k = 1, ... ,n

one to one and onto function

Aj with d(x,

R be a function satisfying If(x) - f(y) I :S d(x, y) for all x, yEn. Then for all

c~o

P{lf - Efl (ii) Let B ~ n, P(B) ~

7.9.

c2

~ c}:S 2 exp(- 4£2)

!' then for all c ~ 0

Another family of probability metric spaces which can be shown to be a normal Levy

family by the same approach is the sequence E!J: = {-I, l}n, n = 1,2, ..., with the normalized Hamming metric d((Ci)f=l' (oi)f=l) = ~I{i;ci # oi}1 and the normalized counting measure. With the natural choice of partitions one gets that the length of {-I, 1}n is at most 1/ Vii so that by 7.8. for all A ~ {o, l}n with P(A) ~

! and all c ~ 0,

This proves 6.2. (with different constants).

7.10.

REMARK: Note the difference in the order of deduction between this Chapter and

Chapter 2. There we used an inequality similar to the one in Theorem 7.8. (it) to deduce an inequality similar to the one in Theorem 7.8. (i). Here the order is reversed. As in Chapters 2 and 4 the inequalities we use in most applications to Banach space theory are for estimating large deviations of nice functions, like the inequalities in Lemma 7.4. and Theorem 7.8. (i).

7.11.

Our next theorem is a further abstractization of the method of the proof (7.6.) that

II n is a Levy family.

37 Given a compact metric group G with a translation invariant metric d (i.e. d(g, h)

=

d(rg,rh) = d(gr,hr) for all g,h,r E G) and a closed subgroup H. One can define a natural

metric d on G/ H by d(rH,sH) = d(r,sH) = d(s-lr,H).

The translation invariance of d implies that this is actually a metric and that d(r, sH) does not depend on the representative r of r H. 7.12. THEOREM: Let G be a group, compact with respect to a translation invariant metric d. Let G

= Go ;2 G l

;2 ... ;2 G n

= {I}

be a decreasing sequence 01 closed subgroups 01 G. Let

ak be the diameter 01 Gk-dGk,k = 1, ... ,n. Then

(i) II I:G c

-+

R is a lunction satislying I/(x) - l(y)l:S: d(x,y) lor all x,y E G, then lor all

> 0, n

JL(II - Ell ~ c)

:s: 2 exp( _c 2 /4 L

a%)

k=l

(il) II B ~ G with JL(B) ~

!' then, lor all c ~ 0, n

JL(B c ) ~ 1 - 2 exp( _c 2/16

L a%) k=l

(JL is the normalized Haar measure).

PROOF:

The implication (i) ~ (ii) is obtained as before using the function I(x)

d(x,B).

To prove (i), let l k , k

=

0,1, ... , n, be the a-algebra generated by the sets {gGd9EG.

Note that if gG k - l ;2 hGk then g-lh E Gk-l. If both gG k - l ;2 hlGk and gG k - l ;2 h 2G k , let s E G k - l be such that diam(Gk_dGk) = d(g-lhl,g-lh2Gk) = d(g-lh l ,g-lh 2s) and define rp:hlGk -+ h 2Gk by

Then

If follows that if we define

then the oscillation of

!k

on each atom of l k -

l

is at most ak, so that II d k II 00

:s: ak and using

Lemma 7.4. we get

o

38

'1.13. The two previous examples (II" and {-I, I}") are instances of this theorem. Another

example is 'lI''' with the normalized product measure I-' and the l1 metric

d(l,s)

" Lit; - s;l·

=

;=1 Taking 'lI'k, k

= 0, ... , n,

as subgroups, one gets ak :S 1 for all k. Then one can apply the

theorem to get e.g. that if B ~ 'lI''' with I-'(B) ~ ~ then

this should be compared with 6.5. where we used the euclidean distance. '1.14. We bring now two more applications of the method developed here. The first is a proof

of Khinchine's inequality used in 5.5. Recall that {T;}~1 are the Rademacher functions. THEOREM:

FOT

aliI :S p
0 one can find c > 0 and 0 > 0 such that 1 +c - < 1 + ~ and Al-1 -

1-0"

C-

1 - Co A-I --0 > 1

1-

~

- ...

2

_ c n). 4

41 Then, for k


0 4

- -)

and one can find e: E {-1, 1}kn such that

o 7.16.

REMARK: It is known ([Sz1], [HaaJ) that Al =

Theorem b = 1 + €, a = ylfi tends to zero as € tends to zero.

€ for

any

€>

V2 so

that one can choose in the

O. In that case a seems to depend on

E and

8. EMBEDDING

e;

INTO

e~

This chapter is devoted to a more complicated application of the martingale method developed in Chapter 7. The proofs and the style here are more technical. The reader who is interested mainly in the outline of the local theory of normed spaces is advised to skip to the next chapter in first reading. In this chapter we shall estimate the dimension m of into

e~

e;, 1 < p < 2, which embed nicely

and see that m can be chosen to be proportional to n. This is the first application, in

these notes, of the method developed here to normed spaces which does not involve euclidean spaces. In Chapter 10 we shall see some more applications which do not involve euclidean spaces.

8.1.

p-Stable Random Variables. The proof of the main theorem in this chapter uses the notion of p-stable random variables.

Since these random variables are an important tool in the local theory of normed spaces, we take this opportunity to discuss some of their properties.

Proofs of facts about p-stable

variables which are not proved here (8.1.1 and 8.1.2 below) may be found e.g. in [Lo]. DEFINITION: A random variable g on a probability space (O,.1,P) is called symmetric p-stable, for some 0 < p

~

2, if Ee itg

for some c

> 0 and every

-00

=

1

eitog(W)dP(w) = e- clW

< t < 00.

We shall need two facts about these random variables: There actually are such variables for each 0 < p ~ 2 (for p > 2 there are no such

8.1.1.

variables!) and

8.1.2. The tail distribution of a symmetric p-stable variable satisfies the inequality

P(lgl 2: t)

~

Ct- P

,

t

>0

,

with C depending on c and p only.

< p (but

It follows easily that a symmetric p-stable variable belongs to Lr(O) for all r not for r

= p).

p-stables can be used to isometrically embed

ep

into L r for 1 ~ r < p ~

00.

Indeed, if

gl, g2, •.. is a sequence of independent symmetric p-stables with the same distribution (i.e.,

the c in the Definition 8.1 is the same for all the gi-S) and {ai}~=l are scalars, then,

E ",,,

(itt.a; t) = P(lgl > t) for all 0 < t < 00) and

= g"C£), i = 1, .. :,n. Then there exists an a = a(e,p) such that for all m,n E IN with an, if Yl, ... ,Ym is a sequence of independent, symmetric random variables with each of IYil, i = 1, ... , m, having the same distribution as that of

let ai m ::;

n

L aiX('~' .1.J ' i=l

Y=

then

(1 - e) for all scalars bl

, ... ,

~ Ibjl (

lip P

m

::;

)

LbjYj

j=l

bm. (XA is the indicator function of the set A).

PROOF: Draw a picture to check that

Ilg" - ylll

=

{I {lin 10 (g" - y)dA ::; 1 g"dA. 0

Let C = C (p) be such that

P(g" > t) Then

=

P(lgl > t) ::; C . t- P

•

44 and

JIgO -

Let

yld>' ~

l

l/n

o

ll/n

gOd>' ~ Clip.

0

t-l/Pd>. = Cl/p_P_n(l-p)/p . p-1

gl, ... ,gm be independent symmetric p-stable variables with Ilgj III Zj = Eaisign gjX{a,:$lgjl2

-

exp

[-(2 -

p)]

8p

2: 2 e- l / 8 > 1 ,

and we get the desired inequality trivially in this case. D

8.5. Consider now the probability space 0 (e,1f) where

e is a matrix

= {-1, 1 }m

o "

x II;:', Le., the space of all couples

{ei,j}i~\,j~l with ±1 entries, and 1f is a vector (1rl, ... ,1rm) of

permutations of {1, ... , n}, with the normalized counting measure P(e,1f) = 2monl("I)m for all (l,1f) E 0. Fix 1 < P < 2, e > 0 and let m ~ an (a from Lemma 6.2). For each (e,1f) EO we define a sequence

Xl

(e, 1f), ... , Xm(l, 1f) of vectors in xi(e,1f)

=~L 1 "

f.~

by

ei,jaje",(j),

i

= 1, ... , m

j=l

(aj were defined above, (ej )j'=l is the unit vector basis in f.~). Our purpose is to find (l,1f) E

such that

n

47 for all scalars b1 , ••• , bm , with m as large as possible. We shall do it in two steps. First we show (Lemma 8.6 below) that for every fixed bi>" . , bm this is true on the average over

(g, w) E 0, then we show (Proposition 8.7) that for each b1 , • •• , bm

is very close to its average on most of 0. Now we use the new distributional inequality for martingales (Lemma 8.4). The estimates are such that they allow us to combine the estimates for different b1 , • •• , bm and still get good estimates for the deviation of II E~ 1 biXi(g, w) II from its mean, holding simultaneously for a lot of different (b 1 , •.• , bm

).

We choose the (b 1 , .•• , bm ) to form a o-net in the unit sphere of l; and conclude by approximating a general vector in the sphere of 8.6.

l;

by a member of this o-net.

We begin with an estimate of the mean.

m

l-E::;E :L.>jXj(g,w) j=1

PROOF: n

m

L L biEi,jaje..,(;) j=1 j=1

Fix 1 ::; k ::; n and consider the random variables Ui(g, w) = Ei, .., ' (k) .

a..,

1

(k)'

U1, . •• , U m

on 0 defined by

i = 1, ... , m .

These variables are symmetric and independent (check) and have the same distribution as Y1,' .. , Ym

in Lemma 8.2. So, by Lemma 8.2, for each k = 1, ... , n, 1 - E ::;

Elf

biEi''''l (k)a",' (k)

I::; 1 + E

1=1

and the same estimates hold when taking the average over k.

o 8.7.

We are now ready for the main part of the proof. For later use we state the next

proposition for a general norm replacing

l~.

48

PROPOSITION: a1

2 ···2

20

an

p

< 2, m, n

positive integers,

II . II

a norm on JR n ,

= ~ Ei=l ci,jaje".,(j))

and let

Ii = (b 1 , ••• , b m )

satisfy Ez:,llbil P

= 1.

Let,

n,

Then, for all c

where ~P


0

>0

depends only on p,

PROOF: Introduce the lexicographic order on 0 U {(i,j)}i~l,j~l' i.e.,

0< (1,1) < (1,2) < ... < (l,n) < (2,1) < ... < (2,n) < (3,1) < ... Let J o = {

-

C1 •

2 . n 2/ q n . M r2 > c . Cq

•

o REMARK: The proof actually shows that the factor C q can be replaced by Cq(X,n). 9.7. THEOREM: Let X be a normed space whose gaussian type 2 constant on n vectors is

= o:(X,n).

o:(n)

Then k(X) ~ c . 0:(n)2

for some absolute constant c.

PROOF: We actually prove a somewhat stronger theorem: If

for some {Xj}i=l ~ X, not all of which are zero, then k(span{x;}f=l) ~ c· For

0: 2 •

Fix m, k E :IN and consider the probability space {-I, 1}k.m.n with the uniform measure. E {-I, l}k.m.n define

"t

1

u,("t)

n

k

= Yk L

L e,,j,txj j=l t=l

and, given independent, normalized gaussian variables {g"j} ,~\,j~l' define k

v,(w)

= Lg"j(w)Xj

.

j=l

Note that the central limit theorem implies that the distribution of (U1, ... ,um) tends, as k

-> 00,

Let

to that of (VI,'" ,v m ).

a = (a1,"" am)

be such that E~ 1 a:

= 1 and consider the function m

f("t)

= fa:("t) = II L

,=1

a,u,("t) II

.

We form the obvious sequence of k· m' n (f fields fs,j,t (Le. an atom of fs,j,t is a set where all of eu,v,w are specified for all (u, V, w) ~ (i,j, l) in some pre-specified, say the lexicographic, order) and we get a martingale Is,j,t

= EUlfs,j,t)

with martingale differences (d"j,t}':'l,j~l,t~l' We leave it to the reader to check that

56

By Lemma 7.4 we get

Let k

-+ 00

to obtain

Note that since E~l a: = 1, (E~l aigi,l,"" E~l aigi,n) is again a sequence of independent gaussian variables normalized in L 2 so that n

m

Ell Laivi(')11 =EIILgjxjll.

i=l Choosing

C

= E:' EIIEj=lgixill

j=l

we get

The rest of the proof is standard (see the proof of Lemma 4.1).

o 9.8. Given a measure space (11, Y,p,) and a Banach space X, we define the space L 2 (X) = L 2 (X,I1) as the space of all measurable functions from (11, Y) to X with 1/2

IIIII =

(


1/..;n), we get

10.3.3. Using the antipodal property of 'P and the cotype inequality we get,

Fix (J

> 0 and let e

= (J{(K + 1) . L· Cq}-l

1

1

. ml1-~. Then

and we get 10.3.4. J.t{Xj

lfa(x) -lllallll < (J • III aIII}

CJ;;

~ 1 - V 1r /2 exp

Now pick a 6-net N in the sphere of the norm

_(J2 • n

(

[m1-(2/q).

III . III .

2 . {(K

+ 1) . L. CqF]

One can do it with

(see Lemma 2.6). Therefore, if 10.3.5.

CJ;;

V 1r /2 exp

then one can find

xE

(1- (J)

(2m - 6

(J2 • m 1 -(2/q)

n

·2· {(K

+ 1) . L . Cq }

2

) 0 such that, given any n and a

sequence Xl, .•. , Xn of linearly independent vectors in a normed space E, with cotype q constant C q, there exists a block sequence {YI."" Ym} of x}, ... , Xn which is (1 + c)-unconditional and m ~ c(c)nl/q/Cq.

PROOF: We give only the beginning of the proof leaving the rest to the reader. Divide

{l, ... ,n} into m almost equal parts find, using Theorem 9.6, a (1

{Ai}~l'

IAil

+ ~)-isomorphic copy

~

k ~ n/m. In each Ei = span{xihEAi

Fi of e~ for t ~ c(c)k 2 / q/C q. Define now

o 10.5.

Finite dimensional version of Krivine's Theorem. In Chapter 12 below we shall state

and prove a theorem of Krivine about embedability of

e; spaces in general Banach spaces.

Here we present a special case of this theorem with a proof which, unlike the proof of the general theorem given in Chapter 12, allows one to get reasonable estimates on the dimension of the embedded

e; space. lip be the ep norm n be another norm on m and assume

Let {e;}i=l be the unit vector basis in IR n and let II·

IIEi=laieillp

= (Ei=daiIP)I/P).

Let 11·11

on

mn

(i.e.,

THEOREM: For any c > 0 and C < 00 there is a constant a = a(c, C) such that for mn satisfying (*) there exists a block sequence UI ... ,U m of el ... , en which

any norm 11·11 on satisfies

forallal, ... ,a m

I (~/36C)P

andm~a'n3

.

64 The proof consists of two parts. In the first we find a symmetric block sequence g > 0 there exists a constant C(e) > 0 such that if II ·11 is a norm (*) then there exists a block sequence {Yl, ... ,yd of {el, ... ,e n } which is 21' (1 + e)- symmetric and k ~ C(e) . C-T . n l / 3.

10.6.

LEMMA: For any

on /Rn satisfying

PROOF: Consider the family {Xn = E'2 x IIn}~=l with the metric

= ~n I{i;

p{(t,7r), (s,ip)}

ti

#

Si aT 7r(i)

#

ip(i)}l.

Using the natural sequence of u-fields (an atom in the k-th u-field is specified by the values of the first k components of t E E'2 and the first k values, 7r( 1), ... , 7r (k), of 7r E lIn) it is easily seen that the length (see 7.7) of X n is S 2/.,fii. By Theorem 7.8 we get that X n is a normal Levy family (with. constants

Cl

= 2 and C2 = 1/64).

= k .m

(k to be chosen later). Divide {I, ... , n} into k disjoint sets AI, ... , Ak each of cardinality m. For (t, 7r) E X n let Without loss of generality, we may assume that n

Yj

= Yj(t, 7r) = m- l / p

L

tie ..(i),

J'

= 1, ... , k.

iEAj

For

a = (al"

.. ,ak) E IRk, consider the function k

fll(t,7r) =

L ajYj j=l

on X n • We now estimate the modulus of continuity of fll

Wa(e)

= m-

l p / •

sup {

t

aj

L

tie ..(i)

iEAj

j=l

k

L aj L (tie ..(i) j=l

sie'l'(i»)

iEAj

For each j the maximal number of i - s such that tie ..(i)

#

sie'l'(i) is e . n.

Thus, we get,

Clearly, E(fa) now yields

= IlIalll is a I-symmetric norm on IRk.

The same computation as in 10.3.2

65 Now

plugging this into the inequality above yields, as in 10.3.4,

1L{(t,1r); (I-(J)lllalll::;

k [ 2 ( 2C (J ) 2P] ~a;Y;(t,1r) ::;(I+(J)llIalll}2:1-2exp~: .

The rest of the proof follows steps 10.3.5 - 10.3.6 in the proof of Theorem 10.3 and, we hope, can be completed by the reader.

o 10.7.

When proving Theorem 10.5 we may now assume, using Lemma 10.6, that we are

dealing with a I-symmetric sequence. The next theorem will then complete the proof of Theorem 10.5. THEOREM: If Y1,' .. ,Yk is a l-symmetric sequence in a normed space satisfying k

C1

1

11 allp::; II LaiYil1 ::;

C 2 11 allp

i=1

for all a = (a1,"" ak) E IRk. Then, for all ~ > 0, there is a block sequence U1, . .. ,U m of Y1, ... ,Yk satisfying m

(1 - ~) ·llall p ::; II

L aiuill ::; (1 +~) .lIall p i=l

foralla=(a1, ....am)EIRm wherem 2:

r 3P k r ,

r=(~/36C)P, C=C 1 ·C2 •

PROOF: After nine chapters of results in which we found nice substructures of given structures, using probabilistic method, it is an interesting feature of the following proof that it is deterministic. One can actually write a formula (depending on

~,p

and k) for the block

sequence {Ui}~1' We begin with a description of this formula. Let

"~"

mean "is a rearrangement of" and "$" mean "sum with disjoint supports". Let

Yr stand for the ~ equivalence class of the sum of r distinct elements of {Y1," . ,Yk}. Let s = 1, ... ,m, be disjoint with

z.,

N-1

z. ~

L

$p(N-j)/p.

Yip;]

;=0 where p = (a

+ I)/a

and the integers a, N, depending on ~, p and k, will be chosen later.

The integer m is determined by

k~m'

N-l

L

j=O

[pi] ~m·

N-l

L

j=o

pi

$;mpN/(p_I) =ma«a+l)/a)N).

66

Finally we set U

8 = z8/llz811,

s

= 1, ... ,m.

The dependence of a and N on c, p and k may be traced through the following formulas which we bring in detail in order that the dependence of the different parameters in the proof will be clear:

co

= ~(c/2C)P, 6 = co' m- 1 / p , a = [1/6pJ = [m 1 / P/pco], (a/(a + 1))[N~oJ ~ c~/m. 12

The substitution of all the relations above in (*) will determine m. We now give a formal proof. Fix T < N, M ::; m and t s e {a, 1, ... , T}, s = 1, ... , M, with M

L:>-t

8

= 1 + 77,

1771::;

1.

8=1

Then M

N-l+T

L

Lp-t 8 / P Z8 ~ 8=1

S=l,

j=O

L

EEl p(N-j-t 8 )/PlI".ij ~ ,M

EEl p(N-i)/PYai ,

i=O

N-l

where ai

L

=

[pi- t 8 l·

.E{I .....M}

i-t.E{O•...• N-l}

Observe that (for ie{O, ... ,N-l})

[pi] _ ai

=

(f

p-to - TJ) [pi] - ai

8=1

=

L

i-t. 0 (to be chosen later) and let f: Ni~l be the function which assigns to ill"" in' satisfying i l < ... < in' the number of the interval to which II Ei=lajXij II belongs. By Ramsey's Theorem 11.1 one can find a subsequence M l such that IIEi=lajxijll always belongs to the same interval for il, ... ,in E MI' If S > 0 is small enough, this ensures that

for any two sequences i 1 < ... < in' fl < ... < fn in MI' We now repeat the process for a different sequence aI, ... , an to find a subsequence M 2 of M 1 good also for these scalars. We repeat the process finitely many times to take care of all scalars (al" .. , an) in a fine enough net of B(li). The last sequence will be N n . We leave the details to the reader.

o 11.5. COROLLARY: Let M be the sequence given in Theorem 11.4. Then the sequence Yi=Xm~i-Xm:2i_l

satisfies

,i=1,2, ...

71 for any finite sets A ~

B

with min

B 2 IBI.

PROOF: Fix a finite set B and scalars {ai};EB. Fix n E :IN and let {Ui};EB be subsets of M with the following properties: (i)

(ii)

if i > ;" then min Ui > max Uj min Ui

>

(iii)

for i E A

(iv)

for i E B

mIBI.(n+l) for any i E B

IUil = 2. Say, Ui = {J(i, I),J'(i,2)} - A IUil = n + 1. Say, Ui = {J(i, 1), ... ,;"(i,n + I)}.

By the property of M (almost invariance under spreading) we get for any 1

L

Il .EA

ai(Xj(i,2) - Xj(i,l))

ai(Xj(i,k+l) - Xj(i'k»)11

.EB-A

Averaging over 1

(1

+ .L

+ en) I L

'EB

~

k

~

aiYil12

~ lin L ai(Xj(i,2) -

Xj(i,l»)

'EA

.EA

-+ 00

IlL

k

~

n

aiYil1

'EB

n, we get

= II?: ai(Xj(i,2) Letting n

~ (1 + en)

~

Xj(i,l))

+

t .L

k=l

+ ~ .L

ai(Xj(i,k+l) - Xj(i,k»)

'EB-A ai(Xj(i,n+l) - Xj(i,l»)

.EB-A

I

I

we get the desired result.

o 11.6.

DEFINITION: We say that a Banach space Y is K-finitely representable in a Ba-

nach space X if for every finite dimensional subspace E of Y there exists an isomorphism

~X with IIT1i·IIT-111 ~ K. Y is said to be crudely finitely representable in X if it is K-finitely representable in X for some K. It is said to be finitely representable in X if it is K

T: E

finitely representable in X for all K > 1. Using this definition one can state, for example, Dvoretzky's theorem as stating that £2 is finitely representable in any infinite dimensional normed space. DEFINITION: We say that a sequence {Yn}~=l in a Banach space Y is K-block finitely representable on a sequence {Xn}~=l in a Banach space X if for each N there exists N vectors {Zn}~=l in X of the form Zn = EiEO'Raixi, (]n disjoint, such that

for all choices of {an}~=l. We define now the notions of crudely block finite representability and block finite representability in an analogous way to that of the previous definition.

72

11.7.

COROLLARY:

Let

{Xi}~l

be any sequence satisfying the assumption of Theorem

11.4. Then there exists a sequence {Y;}~l (of nonzero vectors) in some Banach space Y which is block finitely representable on {Xi}~l and have the two properties. (i) It is invariant under spreading, i.e.

for all n, k l < ... < k n and all scalars

al,' ..

,an'

(ii) It satisfies

for all A ~ B, B finite, and all scalars {ai},EB. PROOF: Let Y be the completion of the space of all finite sequences endowed with the norm

where {Y;}~l is as in Corollary 11.5 (with en ....... 0) and let {Y;}~l be the unit vector basis in this space.

o 11.8. Note that (ii) of Corollary 11.7 implies

for all n, aI, ... ,an and all choices of signs. One can actually strengthen this to get the constant 1. THEOREM: Let {Xi}~l be a sequence as in Theorem 11.4. Then there exists a sequence {Zi}~l {x;}~l

of nonzero vectors in some Banach space Y which is block finitely representable on and has the two properties.

(a) It is 1 - spreading

for all n, k l < ... < k n and all scalars

(il) It is 1 - unconditional;

for all n and

al, ... ,

an'

all"

.,

an and

73 PROOF: Since block finite representability is a transitive property it is enough to prove the theorem for {Xi} = {yJ of the previous corollary. We may also assume IIYil1 = 1. We shall distinguish between two cases: Case 1:

Fix k and n. Let AI." ., An be intervals of integers each of length 2k,

(ji,ji +

Ai =

1, ... ,ji + 2k - 1) with

ji +

2k

< ji+1 for all i

= 1, ... ,n. Let

Then

Put, for i

= 1, . .. , n, Vi =

udluil .

Given a1,"" an and signs ~I."" ~n let Te be the isometry on span {Yi}~l given by

TeYj TeYj No~ice

= Yj = Yj+l

that for i such that

~i

= 1,

if if

j E Ai

and and

j E Ai

TeVi

= Vi

~i ~i

=1 = -1

and for i such that

So

If k is large enough with respect to n we get

I

Te(taiVi) - taivill

I

and in particular

Note also that

.=1

.=1

.

~ ~ IltaiVil1 .=1

~i

= -1

74

for all B

~

{1, ... , n} (since this holds for the y;'s).

The v,'s depend on n, denote v,

= vf.

For any sequence at, ... , am of scalars the sequence

(IIE~la,vfll)~=1 is bounded. We may pass to a subsequence M such that

exists for a dense set of all finite sequences of scalars

{a'}~1

(and thus for all the finite

sequences). The limit is necessarily a norm (note II{a;}~111 ~ maxla,\). The sequence {z;}~1 can be taken to be the unit vector basis in the completion of the space of finite sequences with respect to this norm.

Case 2: There exists an M such that IIEi::d/,1I :::; M for all n. (Note:

II Ei= dli II is a

nondecreasing sequence so that the two cases are inclusive). In this case we shall show that the unit vector basis of Co, the space of all sequences tending to zero, with the sup norm, is block finitely representable on {Y'}~I' Le., that for each e > 0 and k > 0 there are disjoint blocks ZI, • •• , Zk of {Y'}~1 such that

t

li ,=1

a,z,ll < (1 + e) max ja,J.lI z ,1I 1~,~k

(note that the property (i) of the {Y,} immediately implies IIE~=1 a,z,ll ;::: maxl$i~k la,I·lIz,II). First note that

Il

ta,Y,l1 :::; 2M m!lX la,l . 1~,~n

,=1

Indeed, let f be a norm one functional such that IIEi=la,y,1I = !(Ei=la,y,), then,

II~ a,y, I = ~ la,I!((sign a,)y;) n

:::; max la,l· !("(sign a,)y,) 1 0 and integer m there exist m elements Yl, ... ,Ym of X with

(i) 1 - 0 ::; IIYil1 ::; 1, i = 1, ... , m (ii) (Yl"'" Ym) is an unconditional basic sequence with constant::; C and is (1 + E:) -invariant to spreading. 1

(iii)

IIEi=lYjll ::;,' S'X

,

for all s ::; m.

Indeed, suppose that the claim is true. In the case qx =

00

the inequality (iii) implies

IIEl'ajYjll ::; , maxl$j$m lajl and span{Yj}.i=l is (C ·,)-isomorphic to l~. Use now the lemma in 11.8 to finish the proof in this case. If qx < 00, then we are in a position to apply Krivine's theorem. For each k, let E:k

= t,

say. Define

zj = Yj/IIYjll,

Yt, ... ,YZ J'

be k elements of X satisfying the claim with

= 1, ... ,k.

>

Then, for any qk

qx,

we shall have, for

every s ::; k,

•

1

sO« =

I>jrj(t) j=l

Thus, for any r, and each s ::; k,

L j=l Now it follows that if r

z]·k




0 depends on p only.

13.14. PROPOSITION: For an appropriately chosen 8 = 8(c,p), and for any a in the sphere

of

l; 1 - P ((1 - c)EIISall ~

IISal1

+ c)EIISaID

~ (1

~ 2 exp( -8(STp (XW).

13.15. For the proofs of the two propositions we need a representation theorem for p-stable random variables from [L.W.Z.]. We refer the reader to [Mar.P.] for a proof. Let {A;}~1 be independent copies of an exponential random variable, Le.,

P(A; > t) = e- t , t > 0, i = 1,2, ... and let

;

r; = LA;. ;=1

PROPOSITION: For each 1 < p < 2 there exists a constant 00 > cp > 0 such that, if y is any symmetric random variable with ElylP = 1 and {y;} are independent copies of y which

are independent of the A; 's, then 00

1

X=L r7'y; ;=1

is a symmetric p-stable random variable with Let {r;,j} ;'::1' i

EIXI =

cpo

= 1,2, ..., be independent copies of {r;};'::1

of {Yi} ;'::1, ;~. Form 00

S;

=

1

LraY;,; , ;=1

k

Sa = La;S;. ;=1

which are also independent

95 It follows from the proposition above that f(Si) is a symmetric p-stable with Elf(Si)1 cp • (~Ei=1If(XiW) ~ and thus also f(Sa) is a symmetric p-stable with the same expectationcp • (~Ei=llf(Xi) JP) ~. The variable f(Ei=18iX;) is also a symmetric p-stable random variable.

Its expectation is Elf(E~=18ixi)1 = (Elxi!P)~. It follows that the two processes

have the same distribution (compare distributions of identical linear combinations of the two processes). In particular for each a in the sphere of t~.

13.15.1

To compare

- I~

EllSal1 with EIISa!l, one first shows EEf=l Iri~ r~

=


c)

P(IIISall -

:S 2 exp (

-8 . c

q

11{2. a,'. J'-1;P}.k Il q ,=1,J=1 P,OO

)

00

q

8 .c ) =2exp ( -~ and Proposition 13.14 follows now from Proposition 13.13.b.

o 13.16. One of the main problems in the local theory of normed spaces is to find the structure of the nicely complemented finite dimensional subspaces of a general normed space. In this connection one may wonder whether one can choose the copies of the e;x's and

e~x's

in X

to be uniformly complemented. The examples of L oo and L 1 show that this is not the case. Moreover, a recent example of Pisier [Pis3] shows the existence of a Banach space X which admits uniformly complemented

e; - s for no 1 :S p :S

00.

We conclude this chapter with a

proposition showing that under some additional assumptions one gets a positive result. We refer the reader to Chapter 9 for the definition of the projections Rad n and recall that if sUPnllRadnl1 < 00 on X then for each 2 :S q :S 00 1

1 q

(-+-=1). p

In particular ...!... Px

+ ...!... qX

= 1.

PROPOSITION: Let X be an infinite dimensional Banach space. Assume

sUPnllRadnl1 < 00

on X and assume in addition that X· is of type Px'

definition of Px' is attained). Then, for some K
0 and positive integer

+ ~)-isomorphic

to

e;x'

PROOF: Let p = Px, q = qx. By Theorem 13.2, X contains for each n and ~ > 0 a sequence e1, ... , en with

It follows from the Hahn-Banach theorem that one can find

and xi(ej)

= 8;j,

1:S i, J':S n. In particular for i

xi, ... ,x~

in X· with

implies now that for each k, if n is large enough, there exists a subsequence that the sequence

II x: II

:S 1

=f i, 1 :S IIxi - xiII :S 2. Proposition 11.10

xi,,· .. ,Xi,.

such

97 is 3-unconditional. Let d j = ei,j j = I, ... ,k, then yi(d j ) = Oij 1 projection P from X onto spun{dj }j=l by

:s;

i, j

:s;

n. Define a

k

Px

= Lyj(x)dj . j=l

Then,

IIPxIl

~ (1+ e) (t,IY;(X)I') \ k

=

(1 + c)sup{1

L

Ujyj (x) I;

L

IUjlP = I}

j=l 1

:s; 3(1 + c)llx ll sup{l

k

L Tj(t)Ujyj

dtj

o j=l

L \UjIP = I}

k

:s; 3(1 + c)llxIITp(X*) sup{(L IUjIPllyjIlP)l/Pj

L

IUjlP = I}

j=l

:s; 6(1 + c)Tp(X*)llxll and

IIPII :s; 6(1 + c)Tp(X*) .

o

14. THE RADEMACHER PROJECTION In this chapter we return to the subject of estimating the norm of Rad n projection onto the subspace {E~1Ti(t)Xi;

-

the natural

E X} of L 2 (X). The main theorem here (14.5),

Xi

due to Pisier, states that sUPn, a + ib E V

for all

-11" ::;

b ::;

11".

Also, it is easily checked that

It follows that

It remains to check that if £1 is not finitely representable in X then it is not finitely representable in L 2 ({-1,1}n,X) uniformly in n, Le., that it is not finitely representable in

L 2 ([0, 1], X). This follows from the fact (Theorem 13.2) that if £1 is not finitely representable in X thim X has some type p > 1. Then, by 9.12, L 2 ([0, 1], X) has the same type and consequently £1 is not finitely representable in L 2 ([0, 1], X).

o 14.6. We now state a theorem estimating IIRadnll for a general finite dimensional space.

104 THEOREM: Let X be a finite dimensional normed space 01 dimension k. Then, lor all

n,m

IIRadnIIL.({-l,l}~.X)~ (e + 1)log(d(X,l~)

+ 1)

.

In particular,

lor some absolute constant K. PROOF: Note that Radnl = 1* 9 where 9 = E~l rio For any g, III * gIlL.(X) ~ II/IIL.(X) . IIgIIL If X is a Hilbert space and 9 = III * gilL. (X) ~ 1I/IIL.(x) . maxlcAI· It follows that, in any space X 1 •

ECAWA we also have

For -1 ~ 0: ~ 1 let g,,(c) = TIi=l(1 + O:ci) then g" ~ 0, f g" = 1, g" = Ek'=oo:kEIAI=kWA. For J.L E (C[-I, 1])* let 9 = f~l g"dJ.L(O:). Then

III * gIIL.(x) That is, the operator Tgi = we get that if J.L satisfies

~ i l l III * g"IIL.(x)dJ.L(O:) ~ 1I/IIL.(x)IIJ.L11 .

1* 9 has norm ~ IIJ.LII· Since 9 = Ek=o(f~l o:kdJ.L(a))EIAI=kWA

then

Bernstein's inequality (see e.g. [Zy]) states that for a degree l polynomial p on [-1,1]

Ip'(O)1 ~ l max Ip(t)l. -l~t~l

It follows from the Hahn-Banach theorem that there exists a J.L E (C[-l,l])* with

IIJ.LII

and with f~l tdJ.L(t) = 1 and f~l tkdJ.L(t) = 0 for k = 0,2,3, ... ,l. Let a> 1 and define

I/(A) = aJ.L(aA). Then

III/II

~ a· e,

For k > l we have

1 1

-1

tdl/(t)

=1

and

1 1

-1

tkdl/(t)

=0

for

k

= 0,2, ... ,i .

~

l

105 so that

and we get

IIRadnll = II T (Ei= 1 r;) II S a.e+a-l.e.d(X,e~). Taking a = e and e = [log d(X,e~)l

+ 1 we get the desired result. o

14.7. We conclude the chapter with a corollary to Theorem 14.5 which shows that the

eI,eoo

example in 9.11 is essentially the only example to the case where duality between type of a space and cotype of its dual does not hold. COROLLARY: Let X be a Banach space with px > 1. Then p:x 1 PROOF: If Px > 1 then

sUPnIIRadnIIL,(X)
= trace BA. THEOREM: Let X and Y be n-dimensional normed spaces. Let a be a norm on B(X, Y) and let a* be the dual norm. Then there exists an operator W: X ~ Y such that a(W) . a*(W-l) = n.

15.2.

REMARKS: 1)

Note that for any W, n = trace W· W-l :s; a(W) . a*(W-l).

2)

One can prove F. John's Theorem 3.3 as a corol1ary to Theorem 15.2, see [Lew] and

[Pel· PROOF OF THEOREM 15.2: Let W be the operator in B(X, Y) for which the maximum max{ldet Ulj is attained (the set U: X ~ Y, a(U) is clearly invertible). Let U: X

~

U: X ~ Y,

= 1 is compact).

a{U) = I} We shal1 show that a*(W- 1)

=n

(W

Y be arbitrary and consider {3

which is wel1 defined for smal1 {3

= Idet(a(W + tU)-l . (W + tU)) I ItI·

By the choice of W, {3 :s; Idet WI. On the other hand

= a(W + tU)-n ·Idet WI·ldet(I + tW-1U)1

.

Hence Idet(I + tW-1U)11/n :s; a(W

+ tU) :s;

a(W)

+ ta(U) = 1 + ta(U)

.

Using the Jordan canonical form it is easily seen that 1

-tr n

(W-1U) _ I' det(I + tW-1U)1/n -1 - 1m < a (U) . t-O t -

o

108 r~

15.3. The next le=a relates the LEMMA: Let T: X Then

-+ l~

norm of T to the r u norm of its adjoint.

and let

U

=

(Ul,"" un) be any orthonormal basis in l~.

PROOF: By the orthogonality of the Rademacher functions, for any S: n

l~ -+

X

n

< TSUi,Ui >= L

trace TS = L i=l 1

< SUi,T*Ui >

i=l

1

< tr,(t)Sui' tri(t)T*Ui > o i=l ,=1 :S ru(S)ru(T*) ,

=

and we conclude that

To prove the other inequality recall the duality between L 2 (X) and L 2 (X*). It follows that for some {Y;}i=l' {y..d

with

IAI'"

A~{l •...• ft}

n

Ly,ri(t) i=l

ru(T*)

=

lit

+

L

Define S:

l~ -+

X by SUi

= Yi.

L.(X)

=

ri(t)TOUill

.=1

L.(X)

n

< T*Ui, Yi >

.=1

n

< ui,TSUi >= L

i=l

ru(S) =

t

Then

ru(S)r~(T) 2:: trace TS = L and

:S 1 ,

YAWA(t)

IAl;o!l

lit •=1

< T*Ui,Yi >= ru(T*)

i=l

:S IIRadnIIL.(x) .

ri(t)Yill L.(X)

o 15.4. As a corollary to Theorem 15.2 and Le=a 15.3 we get that there exists aT: with r u (T)r u (T-l*) :S IIRad n ll' n. The result of Appendix II implies that for any q > 2 and

Xl, • •• ,X n

EX,

l~ -+

X

109

where K depends on q and Cq(X) - the cotype q constant of X - only. A similar inequality holds for X*. We conclude that 15.4.1. where K depends only on q, q*, Cq(X) and Cq• (X*). Moreover, using Exercise 15.1.2, one can easily get 15.4.1 with a universal constant K. We now introduce an euclidean norm on X by taking its unit ball to be the image by T of the unit ball of l~. Using the method of Chapter 4 and in particular Proposition 4.6, we would like to find a large dimensional almost euclidean subspace of X which is nicely complemented. The quantities M r and ~l(T) are closely related (the first is the median and the second in the quadratic mean of IITxl1 on sn-l). Instead of relating everything to M r and using Chapter 4 we prefer to work with l(T) and imitate the method of Chapter 4. The production of the complemented euclidean spaces will be done in three stages. Let

=n

X C Y, dim X show that T- 1 the

r: norm.

:

X

and let T: -t

l2

-t

X be as above. In the first step (Lemma 15.5) we

l~ can be extended to S: Y

-t

l~ in such a manner as to preserve

Then one gets the same estimate for l(T)l(S*) as in 15.4.1. The second step

(Corollary 15.8) is to pass to a large dimensional subspace E of

l~

on which we have good

estimates for II TIE II and IISi~1I in terms of the l norms of T and S* respectively. The last step (Le=a 15.9) is an imitation of Proposition 4.6 which shows that one can produce a good projection onto T E. 15.5.

LEMMA: Let a be a norm defined on B(X, Y) for some finite dimensional normed

space X and all finite dimensional normed spaces Y, which is injective in the sense that if Y c Z, with i the inclusion map, then for all T: X - t Y, a(T) = a(iT). Then for any Y C Z and any R: Y - t X, R admits an extension R: Z - t X with a*(R) = a*(R). Note that r u and l are injective. PROOF OF LEMMA 15.5: The injectivity of a implies that (B(X, Y), a) is naturally a subspace of (B(X,Z),a). Consider the functional f on (B(X,Y),a) defined by f(T) =

trace RTwhose norm is a*(R). By the Hahn-Banach theorem one can extend f to a functional Ion (B(X, Z), a) with the same norm. Any such functional is given by an operator R: Z - t X acting by i(T) = trace (RT). It is easy to check that R extends R. Finally, a* (R) = 11111 = 11111 = a*(R).

o The second step (the passage to a subspace on which we have good estimates for both

IITIFII

and IISI~II) will be done in two propositions. The proof of the first one is deterministic and of the second one probabilistic. We recall that fh(X, k) is the gaussian cotype 2 constant of X on k vectors (see 9.4). Clearly (see 9.12), .B2(L 2(X),k) = .B2(X,k) so one gets easily that .B2((B(l~,X),l),k) =

.B2(X,k) .

110

15.6. PROPOSITION: Let U: i~ -+ X be any operator. Then for every 1 ::; k ::; n there exists an E ~ i~ with dim E ~ n - k and

PROOF: First notice that if {Pi};=l are pairwise orthogonal projections in i~ then 15.6.1.

min i(UPi)::;

l~i~k

~.B2(X,k)i(U).

yk

Indeed, k

1

k

\~}~ki(UPi)2::; ~i(UPi)2::; .B2((B(i~,X),i),k)2li(~ri(t)UPi)2dt ::; .B2(X,k)2i(U)

llll~ri(t)PiI12dt::; .B2(X,k)2i(U) .

Let now {P;}~l be a maximal set of pairwise orthogonal projections with rank Pi such that, for all 1 ::; i ::; m,

By 15.6.1, m < k. Let F projection P with ImP

= (ImE~lPi).L. ~

=1

Then, by the maximality of {Pi}~l' any rank one

F satisfies

Finally,

11U1F11 = sup{11U Pili P:

i~ -+ F, P a rank one orthogonal projection} .

For rank one operators the operator and the i-norm coincide, hence,

and F is the desired subspace.

o We apply the proposition for both T and S' (see 15.4. S is the extension of T-l to Y in a manner which preserves the that dim F l , dim F 2 ~ ~n and

r: norm) with k = in to obtain two subspaces F

l ,

F 2 such

111 The subspace F = F 1 n F 2 has dim F ~ ~n and

PROPOSITION: Let U: l2' -> X be any operator. Let Jl, be the normalized Haar measure on sm-l. Then for any C ~ 1

15.7.

Jl,{x E S

2

m-l

IlUxll >

j

PROOF: The function x

->

3C { C l(U)2} y'ffll(Un:s 2 exp - 11U11 2

IIUxll is a Lipschitz function with constant :S

11U11.

Let M

denote the median of this function. Then

So ~l(U) ~ M and by Chapter 2,

o We apply this proposition to F = l2', m Jl,{x E Fj

~ 1-

IITxl1 :S

2 exp

~ ~

and to the operators TI F and SIF to get

3 y'ffll(TIF) and

(4(32(;

IIS'xll :S

n)2 ) - 2 exp

3 y'ffll(SIFn

(4(32(;:,

n)2 )

The method of Chapter 2 now implies 15.8. COROLLARY: with T and S as in 15.4, there exists a subspace E

for some absolute constant c

~

l2 with

> 0, on which

This concludes the second step. The third step consists of a standard lemma resembling Proposition 4.6.

112

LEMMA: Let X a 1 - 1 operator, and let

15.9.

~

Y be Banach spaces with dim X = n


£2 be an extension of T-1

Then there exists a projection P from Y onto TF with

IIPII

:

X be

Let F S;;; £2 be any subspace.

•

IITIFII

IISi~.II.

PROOF: Let Q be the orthogonal projection from £2 onto F and let P = TQS. clearly a projection onto T F. Now, for any x E Y,

IIQSxl1 2 =

Pis

(QSx, QSx) = (Sx, QSx)

IIxll IIS·QSxll IlxIlIISI~IIIIQSxll ,

=< x, S·QSx >: is the Euclidean scalar product in

(using ~(v,t)

=v

for t

IR n. Since

< V,OiF >= 0 for t = 0, we have

= 0 and operating with OJ) d < dtojF,oiF >= - < v,oiojF > at t = 0

and so

g~=O(Oi,Oj) =

:t < OiF,OjF >

=-2 .

118 Thus the operator A ... is uniquely defined by go(A ...a"aj) = - < v,a,ajF > at t = O.

o 1

REMARK. Fix a point Vo E Yo, let Vo be a unit normal to Vo at Vo and let IR C IR be the I-dimensional subspace parallel to vo. Take the orthogonal hyperplane IR R - 1 ~ L C IR R which is parallel to T vo (Vo) C IR R and observe that the orthogonal projection P: Vo -+ L R

diffeomorphically sends a small neighbourhood Uo C Vo of Vo into a domain U C L. Denote by PI the orthogonal projection IR R -+ IR 1 and observe that Uo C IR R equals the graph of the function

f

= PI p- l

:

U

-+

IR I. Furthermore,

< vo, a,aj F >= a,a;! ,

u, in this U and, hence, - ~g~o equals the second differential

for the Euclidean coordinates 2

d f of fat

Uo

= P(vo)

E U.

Next consider a hypersurface V o C IR R which is an inside tangent (relative to vo) to Vo at vo. That is V o contains vo, the tangent hyperplane Tvo(Vo ) C IR R equals Tvo(Vo) and the corresponding function 1 satisfies 1 ~ f near uo. Then d 21 ~ d 2 f and we conclude to the following (simple but useful) inequality

See the first chapter in [Mil for additional information. Next we turn to the following normal geodesic deformation F: Vo x IR -+ IR R which isometrically sends each line R = v x IR c Vo x IR onto the straight line in IR R normal to

Vo at v. Such a map exists if and only if Vo is normally orientable in IR R (unlike the Mobius band in IR3 ) which is always the case for small neighbourhoods of points in Yo. We want to study the second derivative for normal geodesic maps F. We denote by A(t) the operators

g;'

A ... assigned to the hypersurface F(Vo x t). Again, our consideration is local near a fixed point v E Vo and we only allow those values t for which the map F diffeomorphically sends U' x t for a small U' around v onto a smooth hypersurface in IR R. This is the case (by the implicit function theorem) if and only if the differential of F is injective on the tangent space Tv,t(Vo , t). The following theorem shows this property to fail exactly for those t which equal the reciprocals of the eigenvalues of the operator A(O) at v and which are called the principal curvatures of Vo at v relative to the unit normal v THEOREM (Gauss-Weil).

= ~(v,O).

The derivative 01 A (t) in t satisfies A'(t) = -A 2 (t) ,

where A 2 is the ordinary square 01 the operator A. PROOF. Since
= 0, we have

d

d

Id

dt

t

Z t

I,

< -a,F,ajF >=< a,F, -d ajF >= --d < a,F,ajF >= -zgt(a"aj) = gt(A(t)a"aj) .

119 Hence,

2

d

d

d2

d

< dt o.F, dt ojF >= dt 2 < o.F, ojF >= 2 dtgt(A(t)o., OJ)

= 2g~(A(t)o.,Oj) + 2gt (A'(t)o., OJ)

= 4g t (A 2 (t)o.,Oj)

+ 2gt (A'(t)o., OJ)

,

which equals, by the above, 2g t (A 2 (t)o.,Oj).

o EXAMPLES. (a) Let Vo be the unit sphere sn-l C

m. n and let F(s, t) = s(l- t) be the

geodesic deformation corresponding to the interior normals. Then, clearly, g~ = -2(1 - t)go and therefore A(t) = -(1- t)-l Id. Thus all principal curvatures of the sphere of radius 1- t equal -(1 - t)-l for the interior normal direction. They become infinite at t F collapses sn-l to a single point.

= 1 as the map

m. n which bounds a compact region c m. • Consider the interior normal geodesic deformation F: Vo x m.+ -+ m. n and let (b) Let Vo be an arbitrary closed C 2 -hypersurface in

V+

n

r(v) E R+ = m.+ x v for v E Vo be the first singular point on m.+ x v where the map F fails to be a local diffeomorphism. (We know by the above that -(r(v))-l equals the smallest

= {v,tlv E Vo,t < r(v)} C Vo x m.. of the difference V +\F(W+) c mn equals

eigenvalue of A(v,O)). Consider the open subset W+ PROPOSITION. The n-dimensional measure zero.

PROOF. Take a point x E V+ and let r., be the radius of the greatest ball B., in V+ around x. The boundary oB., is an inside tangent to Vo at some point v = v(x) E Vo. Hence,

x = F(v,r.,) and r(v) ~ r.,. The set F{v,r(v)} c almost all points x E V+ are contained in F(W+).

m.n

obviously has measure zero, and so

o EXERCISE. Let Vo be a closed convex hypersurface whose principle curvatures relative to the exterior normal are :S 1. Show the normal geodesic map F(v,t), where ~ is the

exterior normal field, to be injective on Vo x (-1,00) c Vo x m.. Show that the convex region bounded by Vo contains a unit ball. Next consider a convex subset V+ with smooth convex

boundary Vo = OV+ which has the principal curvatures> 1. Define Vt C V+ as the set of those v E V+ for which dist(v, Vo) ~ t. Show that each set Vt is convex. Prove, moreover, that Vt is the intersection of convex subsets with smooth boundaries whose principal curvatures > (1 - t)-l. In particular, the minimal non-empty Vt is a single point, say, v in V+. Show that V+ is contained in some ball of radius 1 in m. n . 1.4. Normal deformation of the Riemannian volume. Each Riemannian manifold of dimension n carries a canonical measure (volume) which is uniquely defined by the following two axioms:

120 (1) (2)

The volume of the unit cube in IR n equals 1. IT I: VI -+ V2 is a distance decreasing map of VI onto V2 , then Vol V2 where dim VI

:::;

Vol Vb

= dim V 2 = n.

This is obvious since each continuous Riemannian structure gu in U C IR n can be approximated near each point Uo E U by the Euclidean structure guo (which is constant guo in

U). The following computational formula for this volume is equally obvious. Let

11-0 be the

Euclidean Haar measure in IR n and let l1-u be the Haar measure associated to the form gu on IRn (for which the measure of the gu-unit cube is one). Then the ratio r(u)

= l1-u/l1-o

is

a continuous function in u and Vol U = I r(u)dl1-o. Moreover, one defines the Jacobian of a smooth map f: VI -+ V2 between Riemannian manifolds, say J(v), v E Vb as the absolute value of the determinant of the differential D,: T,,(VI) -+ T",(V2 ) for

Vi

= f(v),

relative to

the Euclidean structures g" in T,,(VI) and g~, in T",(V2 ), where g" and g~, are the Riemannian structures in VI and V2 respectively (which by definition are Euclidean structures in the tangent spaces). IT f is a bijective map, then Vol V 2

= Iv,

J(v)dv where dv is the Riemannian

measure in VI. EXERCISES. (a) Show every compact manifold V to have finite total volume Vol V. (b) Show the hyperbolic plane

H2

to have Vol

H2

= 00.

(c) Let go be a Euclidean structure in IR 2 and let a continuous function h(u) on IR2 equal (lIulilogllull)-2 outside a compact subset in IR2 for lIuli = distgo(u,O). Show that the Riemannian metric 9 = hgo on IR2 is complete and Vol(R 2 ,g) < 00. (d) Show every non-compact Lie group with a left invariant metric to be complete of infinite volume. (e) Let SLnIR be the group of unimodular (n x n)-matrices with a left invariant metric and let S L n 1l be the subgroup of the matrices with integer entries. Show the quotient manifold SLnIR/SLn'lL to be complete of finite volume (and noncompact for n ~ 2).

REMARK. The double coset space SOn\SLnIR/ SLn'lL is naturally homeomorphic to the space of flat tori yn of unit volume. Now, with the Gauss formula gHX, Y) = 2go(A v X, Y) for normal deformations we immediately see the following formula for the derivative in t of the volume Volt = V oI g, (Uo) for all domains Uo C Vo V ol~=o =

r trace Avdvo

Juo

for the volume element dvo of the metric go (which is the usual volume of a submanifold in n

IR n ). The trace of A v relative to a unit normal is called the mean curvature of Vo C IR and it clearly equals the sum of principal curvatures. For example, the mean curvature of the unit sphere sn-l C IR n equals n - 1 for the exterior normal and it is -(n - 1) for the interior normal.

121

°

In order to apply this formula to t iwe express the volume element of gt on n F(Vo x t) C IR by dVt = J(vo, t)dvo for the Jacobian of the map F. Then for all t E IR,

Vol~ = ~ {

dtl uo

J(vo,t)dvo

= {

(trace Av(vo,t))J(vo,t)dvo . Juo

This is equivalent to the relation dJ(vo, t) dt = J(vo,t)Trace Av(vo,t) which holds on the greatest interval a(vo) ~ t ~ b(vo) around zero where the Jacobian does not vanish. Then one exresses the above by d -log J(vo,t) = Trace Av(vo,t) , dt

and the Gauss-Weil formula implies d2 l i d -d2log J(vo,t) = -trace A~(vo,t) ~ ---(trace A v (vo,t))2 = ---(-log J(vo,t))2 . t n-l n-l dt

Now, a straightforward computation shows that JVo, () t

~

1

+(

t Trace Av(Vo,O))n-1

EXERCISE. Show that J(vo,t) = Det(1

n-l

+ tAv(vo,O)).

1.4.A. THEOREM. (Paul Levy [Lev]) Let Vo be a closed hypersurface in IR n whose mean curvature relative to the interior normal everywhere is ~ li for some number li < 0. then the

region V+ bounded by Vo has V ol V+ ~ - nnl/ Vol Vo . (In fact the equality holds if and only of radius -Jil( n - 1).

If Vo is a round sphere PROOF.

Vo x IR+ -+ IR n be the interior normal geodesic map and let

Let F:

°

W+ = {va E Vo, ~ t < r(vo)} C Vo x IR+ be the maximal open subset on which F is locally diffeomorphic. (This is equivalent to J(vo,t) > for (vo,t) E W+). Then J(vo,t) ~ (1 + ~l)n-l which implies r(vo) ~ f = - nil for all (vo,t) E W+. Hence, Vol F(W+) ~

°

J1

r("o)

Vo

0

Since meas(V+\F(W+))

J(Vo,t)dvodt ~ Vol Vo

1" 0

(1 +

lit n- 1 __ )n-l = --=-Vol Vo . n - 1

nlJ

= 0, the proof follows. o

1.5.

Normal deformations in Riemannian manifolds.

Let V be a COO-smooth Rie-

mannian manifold of dimension n which is complete as a metric space. A Cl-map f: IR -+ V is called a geodesic if dist(f(to),f(t)) = Ito - tl for all to E IR and for all t E IR close to to. Let Vo be a C 2 -smooth normally oriented hypersurface in V. Then there exists a unique

122 (with the given normal orientation) normal geodesic map F: Va x IR

-+

V for which the

curve .l'I"oxIR: R = va x IR -+ V is a geodesic normal to Va at va for all va E Va - (see [Mil. [G.K.M.], [C.E.]). We define as above the family gt of the induced Riemannian metrics on Va = Va x t and we study g~ and g; at t = 0 as earlier. First we define the operators A v = A(va, t). where the normal direction v in V is understood as the image of the field on Va x IR under the differential of the map F, by setting gt(A v , X, Y)

=

ft

~gaX, Y) for all

pairs of tangent vectors X and Y in Va. PROPOSITION-DEFINITION (Riemann).

There exists a smooth map v

-+

K v which

for each va E Va C V assigns to the unit vector v E T"o(Vo), normal to T"o(Vo) C T"o(V) for all va E V, an operator K v : T"o(Vo) -+ T"o(Vo), called normal curvature operators, such that ftA v = -A~ - K v at t = 0 for all normal vectors. Furthermore, the operator K v depends onlll on v (and on the Riemannian metric in V, of course) but not on Va. That is, if Va and Ve: have a common tangent space at some point va, that is va E Va n Ve: and T = T"o(Vo) = T"o(Ve:) C T"o(V), then the operator K v : T -+ T defined bll K v = -(ftAv+A~) for vonve:, automaticalillsatisfll the same relation for Ve:. (Observe that the operator A v , unlike K v , does depend on Ve:.) The proof (by a straightforward computation) can be found in any textbook on Riemannian geometry (e.g. [G.K.M.], [C.E.], where an equivalent language of Jacoby fields is employed). EXAMPLES. (a) Let V be the round sphere sft in IR ft +1 of radius R and let Va be a round sphere Sft-l in Sft whose points are all within (geodesic in Sft) distance ro from a fixed point v E V = sft. then, clearly, the metric gt on Va = Va x t satisfies gt = P sin 2 [(t + ro) / R]go for p = sin 2(ro/R). Then g~ = (i[sint)[Ocos!1P-)gO and so A v = (R-l ctg t)[O)Id. Next, A~ = (_R-2 - R-2ctg(tij?)2)Id and so K v = R- 2Id. Thus the sphere Sft ofradius R has (constant) curvature R- 2 • (b) The exponential map. There exists a unique map T,,(V) -+ V for each point v E V, called exp: T" (V) -+ V, which isometrically sends each straight line 1.. in (T" (V), g,,) ~ IR ft through the origin onto a geodesic 'Y in V through v which is tangent at v to t. (See [Mi], [G.K.M.], [C.E.].) It follows that each ball Be C T,,(V) around the origin of small radius e > 0 is diffeomorphically sent onto the e-ball Be C V around v. Hence, the boundary Se of Be for small e is a COO-smooth hypersurface in V, whose interior normal geodesic map

F: Se x IR+ -+ V diffeomorphically sends Se ~ [O,e) onto Be\{v}, while the map F(s,e) collapses Se to v. This implies, like in the Euclidean case, the following property of the normal geodesic map F of an arbitrary hypersurface Va in V.

If for a fixed point Xo E V the function dist(xo, v), v E Vo, assumes a local minimum at some point va E Va then F(vo, r) = Xo for r = dist(xo, va) (and for the obvious choice of the normal Vo at vo) and the map F isiocalill diffeomorphic at (vo,t) for 0 ~ t < r. COROLLARY.

Let Va be a closed hllpersurface in V which bounds a compact re-

123 gion V+ let W +

C

V, let F:

= {vo E Yo,

0 ~ t

Vo x m+ -+ V be the interior normal geodesic map and C Vo x m+ be the maximal open subset on which F is locally

< r( t)}

diffeomorphic. Then meas,,(V+ \F(W+)) =

o.

REMARK. We shall need below a simple generalization of this fact to compact subsets

V+ C V with possibly non-smooth boundary. Namely, let Vo be the topological boundary of V+, let V& C Vo be the maximal open subset which is a COO-smooth hypersurface and let E = Vo\ V& be the complementary singular part of Yo. The singularity E is called negligible for V+ if no compact subset V~ C V+ with COO-smooth (n -I)-dimensional boundary ever meets

E. EXAMPLE.

Take a domain D in S2 bounded by a smooth curve C in S2 and let

V+ C m. :) S2 be the cone over D from the origin. The boundary Vo = av+ is singular at the origin unless C is the great circle and this singularity is negligible if and only if D contains 3

no hemisphere. Now, for Vo with a negligible singularity we define the map F outside E, that is

F: V& x

m.+

-+

V and the image of the corresponding W~ C V& x

m.+

obviously covers

almost all of V+ . EXERCISES. (a) Show that the curvature of the hyperbolic plane everywhere is -Id. (b) A hypersurface Vo is called convex (concave) for a given normal orientation if the operators A... are positive (negative). Show that normal geodesic deformations preserve convexity if the (symmetric) operator K ... is negative for all

Similarly, these deformations

11.

preserve the concavity if K ... ::::: O. RICCI CURVATURE.

Define Ricci

11

for all unit vectors

11

in T(V) byRicci

Trace K .... For example, the round n-dimensional sphere of radius R has Ricci

11

11

=

= const =

(n - I)R- 2 • The following (by now obvious but important) formula generalizes the volume

deformation property from

m."

to all V.

l.S.A. LEMMA. The Jacobian of the map F satisfies

d dt'og J(vo,t) = Trace A ... (vo,t) ,

and d22'og J(vo,t) = -trace

&

A~(vo,t) -

Ricci

11

~

Observe that the equality here holds for spheres Vo

__ I_(dd

log J(vo,t))2 - Ricci

n-l

t

= S,,-1

C V

11.

= Sri.

I.S.B. THEOREM. (Paul Levy). Let a closed submanifold Vo with a negligible singularity in V bounds a compact region V+ C V such that the mean curvature (that is Trace .4... for the interior normal) of the non-singular locus V& ~ Vo everywhere is ~ Ji, for a given Ji E m (of any sign now) and let the Ricci curvature of V everywhere::::: (n -1)R- 2 for some R > O. Let be the ball of the (geodesic) radius ro in the round sphere Sri C m,,+1 of radius R, where

V;

124

= -ll/(n - 1). (The boundary Vo* V';)Vol Va T'h en, V 0 I V+ $ (VolVol V. .

0$ ro $ 1rR such that R-1ctg!j

mean curvature

= -) p, •

01

this ball has constant

a

PROOF. The Jacobian J(vo,t) of the interior normal map on w~

= {va E vri,

0 $ t < r(t)} C

vri x IR+

is majorized by the Jacobian of the corresponding map for Yo*, that is J(vo,t) $ J*(va,t), where the Jacobian J* (obviously) depends only on t but not on va' This majorization is obtained by a direct computation with the above formulae. It follows that r(vo) $ ro for all Vo E Vb and that

Vol V+ $ Vol F(W~) $

ii

i ira .()

V 01 Vb $ V 0I 0 v.'V o · o

v~

r(vo)

J(vo,t)dvodt

0

* _ Vol Vb Vol t dvodt V 0 I v.'

J

v;

0

This is obvious for those who have confidence in their Riemannian geometry. A novice is invited to check all the details step by step by comparing with the Euclidean case. EXERCISES.

(a)

Let V+ be a compact region in V with possibly non-smooth

boundary. Say that the mean curvature of the boundary is ~ II if V+ is the intersection of a decreasing sequence of regions with smooth boundaries which have mean cur-

ll. Take the e:-neighbourhood of the boundary, say Ue(aV) C V and define Vol av = limin!e.....oe:-1Vol(Ue(aV) n V+), where Vol on the right-hand- side denotes the n-dimensional Riemannian volume (or rather the measure) which is obviously defined for all Borel subsets in V. Generalize the above theorem to these regions V+.

vatures

~

(b) Let V have Ricci 1/ ~ 0 for all unit vectors and B 2 of radii R 1 and R 2 ~ R 1 have

1/

E T(V). Show that concentric balls B 1

(n - 1)R- 1, for R > 0, and let Bi and B 2 be the balls of radii R 1 and R 2 in the sphere SR of radius R. Show that (b') Let Ricci

1/

~

Vol B

Vol Bi

1 -< --Vol B - Vol'B 2

2

and prove as a corollary that

Vol V $ Vol

SR

and

diam V $ diam

SR = 1rR .

(c) Generalize the above (*) to the case Ricci 1/ ~ (n - 1)R- 2 for R < 0, by replacing the sphere SR by a pertinent hyperbolic space (or, alternatively, by writing explicit formulae in place of Vol Bi/Vol B 2).

125 (d) Show all balls in a complete simply connected manifold of negative curvature (that is, the operators K v are negative) to be convex (i.e. to have the convex boundary for the exterior normal). (e)

Let a complete manifold V have K II ~ R-2Id for some R > O. Show that the

complement to each ball in V of radius ~ 1rR is a convex subset in V(i.e. it is an intersection of regions with smooth convex boundaries).

2. ISOPERIMETRIC INEQUALITIES 2.1. Inequalities for Banach spaces. The classical isoperimetric inequality in rn. n claims that among all domains V+ c rn. n of a given volume the round ball B n has the minimal

(n - I)-dimensional volume of the boundary, (1) where C n = Vol Bn/(Vol sn-l)n/n-l for the unit sphere sn-l = aBn. This can be equally expressed with the characteristic function !+ of V+ (which is 1 on V+ and 0 outside) by Sobolev's inequality which relates the Lp-norm of functions on rn. n for p = n~1 to the L 1norm of the differentials df,

r (Jrn. rn. n

n

Ifl n / n -

1

) (n-l)/n :S c~n-l)/n Jr

Rn

lidflldu

(2)

for all functions

f on

measure in

If f is not differentiable then df is understood in the sense of distributions.

rn. n.

with a compact support and where du is the (Euclidean) Haar

For example,

r

Jrn. n

Ild!+lldu =

Vol av+ for compact domains V+

c

rn. n with C 1 -

boundary.

This can be seen with the approximation f. -> f +, E: -> 0, where f. ('U) = f + (u) for u E V+ and f.(u) = max(O, 1 - dist(u, V+)) outside V+. Hence the inequality (2) for !+ reduces to (1). On the other hand, the inequality (2) can be obtained by applying (1) to the regions Vt = {u E rn.nllf(u)1 :S t}, t ~ 0, and by a (clever) integration in t (see [Maz], [Bu.Ma.]).

The inequality (2) is a member of a large family of inequalities between various L p norms of f and df. For example, the Poincare inequality for functions f in a compact domain U

c

rn. n

with a smooth boundary claims

(3) where we assume Ju f du = 0 and where C(U) = (AdU))-1 for the eigenvalue Al of the Laplace operator -~ on U (with Neumann's boundary condition). The non-trivial content of

126

(3) is the implied inequality C(U) < 00 which amounts to '\1(U) > O. In fact, the inequality (3) can be obtained like (2) from an appropriate isoperimetric inequality for U. Namely, consider hypersurfaces Vo C U which have avo c au and which divide U into two regions U+ and U_ in U. Let I(Vo) = min(Vol U+, Vol U_)jVolVo and let Is(U) be the supremum of I(Vo) over all Vo C U. Then Cheeger's inequality (see [Bus]) claims >.t{U) ~ (2 I s(U)) -2 . (4)

£

which means

f2(u)du

~ 2 Is(U) fu lidfll 2du

(5)

for all functions f in U which have fu f(u)du = O. Thus (5) is proven like (2) by applying the isoperimetric inequality to the levels of the function f (see [eh]). The inequalities (1) and (2) were generalized by Brunn back in 1888 to an arbitrary ndimensional normed (Banach) space X = (X, II II). Namely, let the Haar measure dx in X be normalized to have the unit ball B = {x E X IlIxll ~ 1} of volume one (which disagrees with the Euclidean convention but has an advantage of simpler formulae) and let II II" denote the norm in the dual space X". THEOREM. (Brunn [Br]).

An arbitrary Cl-function f on X with a compact support

satisfies [

(

1 If(x)ln/(n-l) x

) (n-l)/n

dx ~

[

n- l 1x II df(x)lI"dx .

(6)

PROOF. (Knothe [Kn]) Fix a linear coordinate system x}, ••• , X n in X such that dx}, dx 2, ... , dX n = dx and let J,'(x) be a continuous function on X with a compact support SeX, whose interior is denoted by SO c S. LEMMA. There exists a Cl-map Y of SO into the cube C

= {O < Xi < 1} C

X with the

followt'ng two properties:

(1) The map Y is triangular: the i-th coordinate function of Y depends on

Xl, .. " Xi

only. That is

(e) The partial derivatives ~ are non-negative on SO and the Jacobian J(x)

= Det(;Y~) =

x,

PROOF. For s Ai(S)

IT ;Yi.

i=l

satisfies J(x)

X.

= J,'(x)j

[ J,'(x)dx,

1s

for all X

= (xt{s), ... ,xn(s)) E SO set

= {x = (Xl,."

,xn )

I Xj = Xj(s)

for

i

and

Xi

~ Xi(S)}

ESO .

127 and

B;(s) = {x = (Xl,'" ,Xn) I Xj = Xj(s) for j < i} . Then the map Y(s)

y;(s) =

= {y;(s)} with

!

Ai(S)

lL(s)dx;, ... , dx n /

J

Bi(s)

lL(s)dx;, ... , dXn , i = 1, ... , n

clearly satisfies (1) and (2).

o REMARK. The above formula equally applies to the characteristic function of an arbitrary open convex subset U in X which gives a triangular Cl-diffeomorphism of U onto C whose Jacobian identically equals (Vol U)-l. We are especially interested in the inverse of this diffeomorphism in case U is the open unit ball B = {llxll < 1} C X. This inverse map, called Y': C -+ B, clearly is triangular with the Jacobian ;: 1. We now compose the above map Y

= Yj
dx ,

for the canonical bilinear pairing < , > between X' and X. Therefore,

j I!(x) In/n-ldx

=j

I!(x) IIL(x) l/ndx

~ n -1 (j I!(x) Idiv

Z(x)dxHj lL(x)dx) lin

~ n-l(j II d!(x)lI dxHj lL(x)dx)l/n Since the inequality (6) is homogeneous, we may normalize to

.

f lL(x)dx = 1, and then

(6)

follows from the above.

o REMARKS. (a) The inequality (6) generalizes with an obvious approximation argument = IR n , amounts to Sobolev's

to all functions! with (generalized) derivatives in L l which, for X inequality (2).

128 (b) The above proof shows that the equality in (6) holds if and only if f is a scalar multiple of the characteristic function of a metric ball in X. In fact, the proof gives an integral formula for the isoperimetric deficiency

(c) The inequality (6) (and its proof as well) obviously generalizes to the spaces X with non-symmetric norms. This generalization (in a slightly different but equivalent form) is called the Brunn-Minkowski inequality (see [Had]). (d) Brunn's inequality generalizes, up to a certain extent, to differential forms f on X of degree> 1, as was discovered (in the dual isoperimetric language by Federer and Fleming (see [F.F], [Gr2]). 2.2. Levy's Inequality. Let V be a closed n-dimensional Riemannian CCX>-manifold whose Ricci curvature is everywhere 2 (n - 1)R- 2 , R > 0, (which is the Ricci curvature of the

sn C rn. n+ 1 of radius R). Let V+ C V be a compact region with smooth boundary Va = av+ and let V'; be a round ball in the sphere sn of radius R in rn. n+ 1 such that Vol V';/Vol sn = Vol V+/Vol V.

round sphere

THEOREM. (Levy [Lev]).

the sphere Vo·

= av.;

The (n - l)-dimensional volume of Va is related to that of

by the inequality Vol Vo > Vol Vo· Vol V - Vol Sn

PROOF.

Consider the functional Vol ao+ on all domains 0+

C

V with a fixed n-

dimensional volume Vol 0+ = Vol V+. Then, by the global calculus of variations, there exists an extremal domain, say

n+

Unfortunately, the boundary

in V for which the (n - l)-volume

an+

an+

is the least possible.

is not necessarily smooth. However, a deep theorem of

F. Almgren [AI] claims the singularity to be negligible both from inside and outside (Notice that

n~ c

no

=

an+ an+

is smooth for n :S 7, see [Law].) of this

°

n+.

Furthermore, the non-singular locus

has constant mean curvature (this is obvious since the mean

curvature equals the normal derivative of the (n - l)-dimensional volume of n~). If this curvature, called Il, does not exceed that of Vo·, then the proof follows from lo5.B. Otherwise, we go to the complement

n_ = V\n+, in which (interior) direction the mean curvature of n~

equals -p. which is necessarily less than the mean curvature of Vo· in the direction of the ball

V~

= sn\v.;.

Then lo5.B applies to n~ and the proof is concluded.

o REMARKS AND COROLLARIES. (a) If V the classical isoperimetric inequality on

Among all domains in assumed by a round ball.

sn

sn:

= sn, then Levy's

inequality amounts to

with a fixed volume the minimal volume of the boundary is

129 A similar inequality holds true in the n-dimensional hyperbolic space Hn (see [Schm]), but for no space except m. n , sn and Hn one knows the exact solution of the isoperimetric problem. (b) Let Ve C V denote the e-neighbourhood of Vo in V. Then an obvious integration in e shows that Vol Ve Vol V/ ->--V 01 V - Vol sn . (c) Levy's inequality (and his proof as well) generalizes to all Riemannian manifold with a given (possibly negative) lower bound on the Ricci curvature (see [Grl]). This leads to sharp estimates on the eigenvalues Al ~ A2 ~ ... of the Laplace operator on V (see [B.G.M.J, [Ber]). (d) In order to apply Levy's inequality to a specific manifold V one needs some information on the Ricci curvature. In fact, the Ricci curvature is an easily computable invariant. EXAMPLES. (1) Let vn C m.n+I be a smooth hypersurface whose principal curvatures at some point Vo satisfy

XI, ••• ,X n

Then Ricci(v, v) ~ a for all v E Tv. (V). This follows from the famous theorema egregium of Gauss which expresses the intrinsic curvature of V in terms of the principal curvatures (see [G.K.M] or prove it yourself). (2) Let Riemannian manifolds Vi,"" Vk have Ricci(Vj ) ~ a, j = 1, ... , k. Then the Riemannian product V = VI X ••• X Vk also have Ricci(V) ~ a. This is immediate from the definition of the Riemannian structure g in V, which is g = gl EEl ... EEl gk. (3) Let V be the orthogonal group O(n) and let g be the (natural) left invariant metric on O(n) which is invariant under conjugations (which is equivalent to being right invariant as well as left invariant) and such that the circle consisting of the rotations around a fixed subspace m. n - 2 C m. n has length 271". (With these conditions g obviously is unique)then Ricci(O(n),g) ~ 1 everywhere (see [C.E.] for an explicit computation of the curvature of Lie groups).

APPENDIX II GAUSSIAN AND RADEMACHER AVERAGES

11.1. THEOREM: For all C < 00 and 2::; q < 00 there exists a constant K that if I3q(X) ::; C then for all n and Xl>-. _ ,X n E X

I I Lg,x, ,=1 n

L.{X)

::; K I L r,x, I n

,=1

= K(C,q)

such

L.{X)

In particular, Cq(X) ::; Kl3q(X) ((g')~=1 are independent symmetric gaussian variables normalized in L2' (r')~=1 are the Rademacher functions). Given a I-unconditional basis ZI, Z2, ••• in some Banach space X and a function m. -+ m., we may view f as acting on the finitely supported elements in X by applying f to each of the coefficients f:

11.2. DEFINITION: The p-convexity constant A p (resp. q-concavity constant B q) of the basis (z,) is the smallest A (resp. B) satisfying

A(Ellxn IIP)I/ p (EllxnllqP/q::; BII(EJxnJqP/qll)

II(ElxnJP)I/ PII ::; (resp.

for all finite sequences (x n ) of finitely supported elements in X. 11.3. DEFINITION: The upper p-estimate constant a p (resp. lower q-estimate constant bq ) of the basis (z,) is the smallest a (resp. b) satisfying

(resp.

IIEXnll ::; a(EllxnII P)I/p (EllxnIJqP/q::; bllExnlD

for all finite sequences (x n ) of finitely and disjointly supported (with respect to (z,)) elements inX. Clearly, ap

::;

A p , bp

::;

B p • Also for 1 ::; P ::;

00,

~

+ ~ = 1,

131

ap(Zi)

= bq(zi)

,

ap(zi)

= bq(Zi)

where (zi) are the biorthogonal functionals to (Zi) (see [L.T2]). Note also that a p :S ap(X), bq :S ,Bq(X). (a p , ,Bq are the gaussian type and cotype constants). 11.4. PROPOSITION: For aliI < r < q < B q :S K(r,q)b r

00

there exists a constant K(r,q) such that

for

1

. < 1. Indeed

V ol(>.A + (1 - >')B) lin 2: >.(V 01 A) lin

+ (1 -

If K is any convex body of finite volume in

>.)(Vol B) lin 2: {(Vol A)>' (V 01 B) 1->' pin.

rn. n , put JLK(A) =

V~~1~K). Then JLK clearly

satisfies the same kind of inequality

Ill.3.

JL(>.A JL(A)

Let JL be any Borel probability measure on mn satisfying (1 - >')B) 2: JL(A)>' JL(B) 1->'. Then for all symmetric convex sets A ~ mn with

THEOREM (C. Borell).

+

= 11 > !

JL((tA)C)~I1(I~I1)(1+t)/2

for all t>I.

PROOF. We have the inclusion

(check!). Consequently 1 - 11

= JL(A C) 2: JL((tA)c)2 / (t+ l )I1(t-l)/(t+l)

.

o 111.4. Let now (X,

II II)

be any normed space and let

where JL is the Lebesgue measure on the cube K Let

Xl, ••• , X n

= [-I,Ijn

normalized as to give JL(K)

n

A

= {a E Kj Ill: a;x;11 ;=1

E X. Assume

~ 3} .

= 1.

136 Then A is convex symmetric and JL(A) ~ Theorem 111.3 we get, for all t > 1,

111.4.1.

j, (since 3JL(A

JL{IIE?::laixili > 3t}

C

)

~

JK

IIEaixilldJL(a)

= 1).

Applying

~ ~(~)(1+t)/2

and consequently, for all p > 1,

for some constant K p depending only on p. It follows by homogeneity that

111.4.2. Kahane's inequality now easily follows from 111.4.2. Indeed, by the I-unconditionality of the Rademacher functions,

By the fact that (ad?::l and (!ai!ri(t))f=l have the same distribution and by the triangle inequality

We can now conclude from 111.4.2 that, for p > 1,

which proves Kahane's inequality. (The other side inequality is trivial.) We remark that this proof does not give the right order of magnitude for the growth rate of the constant. It only gives K p ~ Kp with K < 00 absolute. See [L.T2] for a proof which gives the right order of magnitude for the constant, K p

~

yP.

APPENDIX IV PROOF OF THE BEURLING-KATO THEOREM 14.4

Define

· StX - x A x= Il m - - t .... o t

whenever the limit exists. Then A is a linear (usually unbounded) closed operator with dense domain D(A). For any x E D(A) dStx dt

= lim

St+s x - St X

s.... o

s

exists and StX

= AStx = St Ax .

Let SUPt III - Stll = Po and let Po < p < 2, say p = STEP 1: For all f3 ~ p,

M

= M(po).

Indeed, since E~=o(

II 1-/, II :::;

e:)n = M. Also,

2+.p.

((f3 - 1)1 + St}-l exists and 11((f3 - 1)1 + St)-lll :::;

e:,

T

=

!ff-

with

E~=o( I-/,)n converges and its norm is at most

Thus,

STEP 2: There exists 0 < ()

larg

zl ~ I - (),

= (}(Po)

:::;

I

and M'

zl + A has a bounded inverse and

PROOF: Not~ that for.all t ~ 0 and x E D(A),

..

= M'(po)

such that for z E 0 fixed), then it was proved in [M.W.] that X necessarily contains a (1 + c)-isomorphic copy of with k = k(e, c, n) -+ 00 as n -+ 00.

!'

! !

et

3.3 is taken from F. John's [Jo]. 3.4 This is a weak form of the Dvoretzky-Rogers Lemma [D.R.] (or see [Da]). The estimate

can be improved to 1 Johnson.

~ IIxili ~ Jj.~t~):;:J"

The proof given here was shown to us by W.

4.2, 4.3 were proved in [Ml]. The estimates on the constant e(c) in 4.2 Remark a, and 4.3,

145 can be improved-to C(e) ~ Ce 2. This follows from a recent proof of Dvoretzky's Theorem due to Y. Gordon [Go]. 4;.4-4.7 are proved in [F.L.M.]. 4.8-4.11 are taken from [M3]. The best constant in 4.8 is

";f! Ixl ~ IIxll (see P.T-J.]).

5.1-5.4 are taken from [F.L.M.] except for Fact 2 of 5.4 which was proved in [B.D.G.J.N.].

The condition a· b ~

vn in 5.1 is not needed (T. Figiel, private communication).

Actually

all of Lemma 5.1 is not needed, instead one can use Proposition VA. 5.3 admits a lot of generalizations, we indicate here only one direction: proved that for any A < 1, t~ contains a C(A)-isomorphic copy of t~, for k

Kashin [Kas] has

= [An].

The proof

uses a different approach involving the following property of t~: Let 0 be the ellipsoid of maximal volume inscribed in the unit ball, K, of t~, then vr(K) = (Vol KIVol O)l/n is bounded independently of n (see also [Sz2]). The quantity vr(K), called the volume ratio of K, was introduced and studied for various K's in [S.T-J.], where a generalization of Kashin's theorem is proved for spaces whose unit balls have bounded volume ratio. For recent developments on this subject (in particular its relation with cotype 2), see [B.M.2] and [MLP.]. For the original proof of Khinchine's inequality see [L.T.1]. The best constants = 1, by S. Szarek [Szl] (AI = ~), and for all the p's by U. Haagerup [Haa]. 5.5

Ap,Bp were computed, for p

5.7 was first proved in [M1]. There is a disturbing gap between k(t~) ~ log n and k(t~) ~

n 2 / q (which becomes a constant for q = log n for which d(t~,t~) = e). This gap can be bridged - one can prove k(t~) ~ cqn 2 / q for all q ~ log n with an absolute c > O. This estimate is best possible up to numerical constants. 5.8 The original proof of Dvoretzky's Theorem only gives k(X) ~ cy'log nllog log n. As stated here, the theorem was first proved in [M1l (with the proof given here). 5.9 The theorem (with a different dependence on A) was proved in [M2]. We follow [M3]. 6.1 The notion of Levy family was introduced in [G.M.] which also contains a list of examples

of Levy families and applications of the concentration phenomenon of such families to topology and fixed point theorems. 6.2 was observed and used in [A.M.]. 6.3 was proved and used for the Local Theory by Maurey [Ma]. 6.5-6.7 These examples were observed in [G.M.]. Concentration properties for Wn,k and

Gn,k were also noted and applied in [M4], [M5]. 6.9 was proved in [G.M.]. This method was later extended in [Al.M.] to discrete cases.

146

1.4 is due to K. Azuma (see [St] for this and related inequalities). For other related inequalities, see Lemma 8.4 below and Proposition 3.1 in [J.S.Z.]. 1.5 was proved by B. Maurey [Maj. The proof here follows [Sell. We remark that unlike the

similar case of

E'2

no precise isoperimetric inequality for JI n is known.

1.1-1.8 is taken from [SCI]. 1.9 For a proof through an isoperimetric inequality, see L.H. Harper's [Har] (and [A.M.1] for

the form in which it appears here). [Fr.F] contains a simpler proof of Harper's isoperimetric inequality. 1.15 is taken from [Sc2], which contains also generalizations to spaces other than l~. 8.1 p-stables were introduced by P. Levy. Their relevance to the problem of embedding lp in L r was noticed by M.1. Kadec. 8.2-8.8 is taken from [J.S.] in which a more general case is also considered (in the statement

of Theorem 8.8 one can replace l~ by l~ for any 0 < r :5 1 and also by 1 < r < p provided 1 + e is replaced with Kr,p and f3 = f3(r,p)). Theorem 8.8 was further generalized, replacing l~ by a wide family of spaces, by G. Pisier [Pis2], see Chapter 13. For generalizations in a different direction, replacing l;' by a general finite dimensional subspace of L p see [Sc3], [Sc4]. 8.4 This inequality is due to G. Pisier. 9.1 The type and cotype inequalities were first considered by J. Hoffmann-JorgenseI]- [H-J)

in connection with limit theorems for independent Banach space valued random vatiables. [M.P.] contains the first major relations between these inequalities and geometrical properties of Banach spaces. 9.2 is proved in [Kah]. The original proof as well as the proof in Appendix III do not give the right order of the constant K p • For a proof, due to Kwapien, which gives the right order

Kp

~

yIP see [Kw2] or [L.T.2J.

9.6 is taken from [F.L.M.]. 9.1 appears here for the first time. It should be compared with Theorem 13.12 below. 9.10 The Rademacher projection, Rad n , and its relevance to duality between type and cotype

was noticed in [M.P.]. 10.

is based on [A.M.1] and [A.M.2J. However, the main idea of the proof of Theorem 10.7

is an adaptation of Krivine's original proof [Kr] to the finite dimensional case. 11.

A good reference for combinatorial treatment of Ramsey's Theorem is [G.R.S.]. For

a review of applications of Ramsey's Theorem to Banach space theory see Odell's rOd]. A.

147 Brunei and L. Sucheston were the first to notice the relevance of Ramsey's Theorem to Banach space theory; they proved Theorem 11.4 (cf.[B.S.l1, [B.S.2]). 11.8 The Lemma here is due to James [Ja]. 12.3-12.4 are due to J.L. Krivine [Kr]. We follow here a simpler proof due to H. Lemberg

[Lem]. The proof given for 12.5 appears here for the first time. It simplifies the proof of the Maurey-Pisier Theorem 13.2. 13.2 is due to Maurey and Pisier [M.P.]. We follow [M.S.] (for the cotype case, 13.8-13.9) and [Pis2] (for the type case, 13.10-13.12). [Sc3] and [Sc4] contain some extensions of 13.12: It is proved there that any k-dimensional subspace of L p , 1 < p < 2, (1 + c) imbeds into X provided k1+l/P(log k)-l ~ S(p,c)(STp(X))q. It is not known if the conclusion holds with

k

~

S(p,c)(STp(X))q.

14.5 is contained in [Pisl]. 14.6 is contained in [Pis4] and [PisS] with a different proof. The proof here was shown to us

by T. Figiel. The estimate IIRad n XIl ~ K log n for X with dim X = n is best possible, up to the choice of K. This was shown by J. Bourgain [Bou]. But in special cases (notably, spaces with I-unconditional bases) one gets IIRad n X11 ~ 15.

Kvrogn,

see [PisS].

The material in this chapter is due mainly to T. Figiel and N. Tomczak-Jaegermann

[F.T-J.]. 15.2 is a generalization, due to D. Lewis [Lew], of a theorem of F. John [Jo].

l.l.5.B first appear in that form in [Grl]. It is more commonly referred to as the LevyGromov Theorem. 11.1 The theorem is due to Maurey and Pisier [M.P.]. The proof here is different. 11.2-11.3 For more information about p-convexity, q-concavity, upper p-estimate and lower

q-estimate, see [L.T.2]. 111.3 is from [Bo]. 111.4 G. Pisier showed us how to deduce Kahane's inequality from Borell's theorem.

IV. The proof is taken from [R.S.].

148 V.I See [Pis7] for this and some generalizations.

V.4 is probably known to several people but we don't know of any reference.

INDEX (Roman numerals refer to appendices) Hyperbolic plane 1.1.1

Approximate eigenvector, eigenvalue 12.1; 12.2, 12.3, 12.5 Banach limit 12.2 Banach-Mazur distance 3.1 Bernstein's inequality 14.6 Beurling-Kato theorem 14.4, IV Block sequence 10.4-10.7 Borell's theorem 111.3 Brunn-Minkowski inequality 1.2.1, III.1 Brunn's theorem 1.2.1 Cheeger's inequality 1.2.1 Concavity (q-) 11.2, 11.7, 11.8 Concentration function 6.1 Concentration of measure 2.3 Convexity (p-) 11.2 Conditional expectation 7.1, 7.2 Cotype 9.1, 9.3, 9.6, 9.10, 9.11, 10.3, 10.4, 12.5, 13.1, 13.2, 13.3, 13.5, 14.7, 15.4, Dual norm 3.1.1, 15.1, 15.2 Dvoretzky's theorem 4.1, 5.8, 13.8, 13.12 Dvoretzky-Rogers theorem 3.4, 5.8 E~ 6.2, 7.9 Ellipsoid of maximal volume 3.2, 3.3, 3.4, 5.8,9.6 Exponential map 1.1.5

Finitely representable (K, crudely, block) 11.6-11.8, 12.3-12.6, 13.2, 13.12, 14.2, 14.3, 14.5 g-length 1.1.1 Gauss' theorema egregium 1.2.2 Gauss-Wei! formula 1.1.3, 1.1.4 Gaussian cotype 9.4 Gaussian type 9.4, 9.7 Gaussian variable 9.4,'9'.7, 15.1,11.1; 11.8, V~

Grassman manifold 1.5, 6.7.2 Haar measure 1.1 Homogeneous space 1.2, 1.3, 6.6

.

Ideal norm 15.1 Invariant riemannian metric 1.1.2 Isomorphic (A-) 3.1 Isoperimetric inequality 2.1, 1.2 F. John's Theorem 3.3, 15.2

k(X), k(X, e) 4.3-4.7,5.8,9.6,9.7 k(l~)

k(l~)

5.3 5.4

k(lex» 5.7 Kahane's inequality 9.2, 9.4.3, III Khinchine's inequality 5.5, 7.14 Krivine's theorem 10.5, Ch. 12, 13.8 i-norm 15.1

l; spaces 5.2, 10.5-10.7, Ch.

12, 13.2, 13.12, 13.13, 13.16 L 2 (X) 9.8,9.10 Laplacian 6.8 Length (of a metric probability space) 7.7,7.8, 10.6 Levy family 10.0 Levy family, normal 6.1-6.8, 7.5, 7.9, 10.3 Levy's lemma 2.3, V.1 Levy mean 2.3 Levy's theorem 1.1.4.A, 1.1.5.B Lower q-estimate 11.3 Marriage theorem 1.1 Martingale 7.3, 7.6, 8.4, 9.7 Martingale difference 9.7 Maurey-Khinchine inequality 11.7 Maurey-Pisier theorem Ch. 13 Median 2.3, 4.1, 4.8, 5.1, 10.1, 10.3.1, V.3, V.4 Mean curvature 1.1.4 OrthogonaLgroup 0;.. 1.2, 1.5, 6.3 lIn 6.3, 7.5, 7.7 p-stable (symmmetric) 8.1, 8.2, 13.10, 13.11, 13.15 Plane, tangent, normal 6.8

150 Poincare inequality 1.2.1 Poincare metric 1.1.1 r-norm 15.1 Rad n , RadnX 9.10, 9.11, 13.16, Ch. 14, 15.3, 15.4 Rademacher functions 5.5, 7.14, 9.0, 9.4, 9.9, 9.10, 15.1, ILl, II.8 Ramsey's theorem 11.1, 11.2 Ricci curvature 6.4, 1.1.5, 1.2.2 Riemannian manifold 6.3, 6.8, 6.9, App. Riemannian metric 1.1.1 Riemannian metric, induced 1.1.2 Riemannian structure 1.1.1 1.5 SOn 6.5.1, 6.6

sn-l

Semigroup, analytic 14.1, 14.4, IV Sobolev's inequality 1.2.1 Spreading 11.7, 11.8, 11.9, 12.3, 12.5, 12.6, 13.8 Stable type 13.10, 13.12 Stiefel manifold 1.5,6.7.1 Strongly continuous (semigroup) 14.4 Symmetric sequence 10.2, 10.3, 10.7 Type 9.1, 9.3, 9.4, 9.10, 9.11, 13.1, 13.2, 13.5, 13.10, 14.7, 15.10 Unconditional sequence 10.2, 10.4, 11.8, 11.10, 12.3, 12.5, 12.6, 13.8 Upper p-estimate II.3 Walsh system 9.9, 14.1 Weak lp-norm 8.3

REFERENCES

[AI]

F. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. A.M.S., 4 (1976).

[Al.M.]

N. Alon and V.D. Milman, Ai> isoperimetric inequalities for graphs, and superconcentrators, J. Comb. Theory, Ser. B, 38 (1985), 73-88.

[A.M.1]

D. Amir and V.D. Milman, Unconditional and symmetric sets in n-dimensional normed spaces, Israel J. Math., 37 (1980),3-20.

[A.M.2]

D. Amir and V.D. Milman, A quantitative finite-dimensional Krivine theorem, Israel J. Math., 50 (1985), 1-12.

[B.D.G.J.N.j G. Bennett, L.E. Dor, V. Goodman, W.B. Johnson and C.M. Newman, On uncomplemented subspaces of L p ,l < p < 2, Israel J. Math., 26 (1977), 178187. [B.G.]

Y. Benyamini and Y. Gordon, Random factorization of operators between Banach spaces, J. d'Analyse Math., 39 (1981),45-74.

[B.G.M.]

M. Berger, P. Gauduchon and E. Mazet, Le Spectre d'une Varillte Riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer 1971.

[Ber]

P.M. Berard, Lectures on spectral geometry, IMPA 1985, Rio de Janeiro. Lecture Notes in Mathematics, Vol. 1207, Springer 1986.

[Be]

A. Beurling, On analytic extension of semigroups of operators, J. Funct. Anal., 6 (1970),387-400.

[Bo]

C. Borell, The Brunn-Minkowski inequality in Gauss spaces, Inventiones Math., 30 (1975), 207-216.

[Bou]

J. Bourgain, On martingales transforms in finite dimensional lattices with an appendix on the K-convexity constant, Math. Nachrichten, 119 (1984), 41-53.

[B.M.1]

J. Bourgain and V.D. Milman, Distances between normed spaces, their subspaces and quotient spaces, Integral Equations & Operator Theory, 9 (1986), 31-46.

[B.M.2]

J. Bourgain and V.D. Milman, On Mahler's conjecture on the volume of a convex symmetric body and its polar, Preprint, IRES 1985.

[B.S.1]

A. Brunei and L. Sucheston, On B-convex Banach spaces, Math. Systems Th., 7 (1974), 294-299.

[B.S.2]

A. Brunei and L. Sucheston, On J-convexity and some ergodic super-properties of Banach spaces, Trans. A.M.S., 204 (1975),79-90.

[Br]

R. Brunn, Uber Ovale und Eiflachen, Inaug. Diss., Miinchen, 1887.

[Bu.Ma.]

Yu. Burago, and V. Mazya, Some problems of the potential theory and the function theory for domains with non-regular boundary, Zapiski LOMI, Leningrad, 1967 (Russian).

h:,

[Bus]

P. Buser, On Cheeger's inequality Al ~

[Ch]

J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis, Symp. in honor of Bochner, Princeton, 1970, 195-199.

Preprint.

152 [C.E.]

J. Cheeger and E.G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975.

[D.M.T-J.]

W.J. Davis, V.D. Milman and N. Tomczak-Jaegermann, The distance between certain n-dimensional spaces, Israel J. Math., 39 (1981), 1-15.

[Da]

M.M. Day, Normed Linear Spaces, Springer, 1973.

[Do]

W.F. Donoghue Jr., Distributions and Fourier Transforms, Academic Press, 1969.

[D.S.]

N. Dunford and J.T. Schwartz, Linear Operators, Part I, Wiley, 1957.

[Dv.]

A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Symp. on Linear Spaces, Jerusalem 1961, 123-160.

[D.R.]

A. Dvoretzky and C.A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci., U.S.A., 36 (1950), 192-197.

[F.F.]

H. Federer and W. Fleming, Normal and integral currents, Ann. of Math., 72 (1960),458-520.

[Fe]

W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Wiley, 1966.

[Fi]

T. Figiel, A short proof of Dvoretzky's theorem on almost spherical sections, Compositio Math., 33 (1976), 297-301.

[F.L.M.]

T. Figiel, J. Lindenstrauss and V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math., 139 (1977), 53-94.

[F.T-J.]

T. Figiel and N. Tomczak-Jaegermann, Projections onto Hilbertian subspaces of Banach spaces, Israel J. Math., 33 (1979), 155-171.

[Fr.F.]

P. Frankl and Z. Fiiredi, A short proof for a theorem of Harper about Hammingspheres, Discrete Math., 34 (1981), 311-313.

[Gi]

D.P. Giesy, On a convexity condition in normed linear spaces, Trans. A.M.S., 125 (1966), 114-146.

[Gil

E.D. Gluskin, The diameter of the Minkowski compactum is roughly equal to n, Funct. Anal. Appl., 15 (1981), 72-73.

[Go]

Y. Gordon, Some inequalities for gaussian processes and applications, Israel J. Math., 50 (1985),265-289.

[G.R.S.]

R.L. Graham, B.L. Rothschild and J.H. Spencer, Ramsey Theory, Wiley, New York,1980.

[G.K.M.]

D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geometrie in Gruppen, Springer, 1968.

[Grl]

M. Gromov, Paul Levy's isoperimetric inequality, Prepri-nt, 1980.

[Gr2]

M. Gromov, Filling Riemannian manifolds, J. Diff. Geom., 18 (1983), 1-147.

[Gr3]

M. Gromov, Partial Differential Relations, Springer Verlag, to appear.

[G.M.]

M. Gromov and YD. Milman, A topological application of the isoperimetric inequality, Amer. J. Math., 105 (1983), 843-854.

[Haa]

Uffe Haagerup, The best constants in the Khinchine inequality, Studia Math., 70 (1982), 231-283.

153 [Had]

H. Hadwiger, Vorlesungen iiber Inhalt, OberfHi.che und Isoperimetrie, Springer Verlag, 1957.

[Har]

L.H. Harper, Optimal numberings and isoperimetric problems on graphs, J. Comb. Theory, 1 (1966), 385-393.

[H-J]

J. Hoffmann-Jorgensen, Sums of independent Banach space valued random variables, Aarhus Universitat, 1972/73.

[Ja]

R.C. James, Uniformly non-square Banach spaces, Ann. of Math., 80 (1964), 542-550.

[Jo]

F. John, Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume, Interscience, New York, 1948, 187-204.'

[J.S.]

W.B. Johnson and C. Schechtman, Embedding (1982), 71-85.

[J.S.z.]

W.B. Johnson, G. Schechtman and J. Zinn, Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Prob., 13 (1985),234-253.

[Rah]

J.P. Kahane, Series of Random Functions, Heath Math. Monographs, Lexington, Mass., Heath & Co., 1968.

[Kas]

B.S. Kashin, Sections of some finite dimensional sets and classes of smooth functions, Izv. ANSSSR, ser. mat. 41 (1977), 334-351 (Russian).

[Ka]

T. Kato, A characterization of holomorphic semi-groups, Proc. A.M.S., 25 (1970), 495-498.

[Kn]

H. Knothe, Contributions to the theory of convex bodies, Michigan Math. J., 4 (1957),39-52.

[Kr]

J.1. Krivine, Sous-espaces de dimension finie des espaces de Banach reticules, Ann. of Math., 104 (1976), 1-29.

[Kwl]

S. Kwapien, Isomorphic characterizations of inner product spaces by orthogonal series with vector coefficients, Studia Math., 44 (1972),583-595.

[Kw2]

S. Kwapien, A theorem on the Rademacher series with vector coefficients, Proc. Int. Conf. on Prob. in Banach Spaces, Lecture Notes in Math. Vol. 526, Springer, 1976.

[Law]

B. Lawson, Minimal varieties, Proc. of Symp. in Pure Math., A.M.S., 27, Part I, (1975), 143-177.

[Lem]

H. Lemberg, Nouvelle demonstration d'un theoreme de J.L. Krivine sur la finie representation de t p dans un espace de Banach, Israel J. Math., 39 (1981), 341-348.

[L.W.Z.]

R. Lepage, M. Woodroofe and J. Zinn, Convergence to a stable distribution via order statistics, Ann. Prob.,9 (1981),624-632.

[Le]

P. Levy, Problemes concrets d'analyse fonctionnelle, Gauthier Villars, Paris 1951.

[Lew]

D.R. Lewis, Ellipsoids defined by Banach ideal norms, Mathematika 26 (1979), 18-29.

[L.S.]

J. Lindenstrauss and A. Szankowski, On the Banach-Mazur distance between

t;'

into

tl ,

Acta Math., 149

154 spaces having an unconditional basis, Aspects of Positivity in Functional Analysis, North Holland, to appear.

[L.T.]

J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics Vol. 338, Springer-Verlag, Berlin, 1973.

[L.T.1.]

J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, Berlin, 1977.

[L.T.2.]

J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Spaces, Springer-Verlag, Berlin, 1979.

[Lo]

M. Loewe, Probability Theory I, Springer-Verlag, 1977.

[Mar.P.]

M.B. Marcus and G. Pisier, Characterizations of almost surely continuous pstable random Fourier series and strongly stationary processes, Acta Math., 152 (1984),245-301.

[Mal

B. Maurey, Construction de suites symetriques, C.R.A.S., Paris, 288 (1979), 679-681.

[M.P.]

B. Maurey and G. Pisier, Series de variables aleatoires ve