GLOBAL VS. LOCAL ASYMPTOTIC THEORIES OF FINITE ... - CiteSeerX

GLOBAL VS. LOCAL ASYMPTOTIC THEORIES OF FINITE DIMENSIONAL NORMED SPACES V.D. Milman Tel Aviv University and MSRI

G. Schechtman The Weizmann Institute and MSRI

1. Introduction The paper is devoted to the comparison between global properties and local properties of symmetric convex bodies of high dimension. By global properties we refer to properties of the original body in question and its images under linear transformations while the local properties pertain to the structure of lower dimensional sections and projections of the body, i.e., to the linear structure of a normed space in the spirit of functional analysis. In both theories we are interested in the asymptotic behavior, as the dimension grows to in nity, of the relevant quantities. Also, as is common in such questions, the approach we consider does not yield exact isometric results but rather falls into the isomorphic (and for some results the almost isometric) category. Unexpectedly, it appears that there is an exact parallelism between the two theories; the global (geometric) asymptotic theory and the Local Theory. We will demonstrate that several well known facts of Local type have corresponding equivalent geometric results and vice versa. Let us describe, as an example, a well known classical Local Theory fact { Dvoretzky's theorem. A well known version of Dvoretzky's theorem known today states: Let X = (IRn ; k
k) be an n-dimensional normed space. Let j
j denote the canonical Euclidean norm on IRn . R Put M = Sn? kxkd(x) (where  is the normalized Haar measure on the Euclidean sphere); and assume kxk  bjxj for all x 2 IRn . Let " > 0, then for some absolute 1

Supported in part by BSF. Research at MSRI is supported in part by NSF. 1



?




constant c and for k = c"2 n Mb 2 , there exists a subspace E 2 Gn;k (the Grassmannian of k-dimensional subspaces of IRn ) for which

M kxk  jxj  (1 + ")M kxk for all x 2 E : (1) 1+" Moreover, the subset of Gn;k satisfying this has measure tending fast to 1 as n ! 1. (Here, the measure is the normalized Haar measure on Gn;k .) (cf. [Mi1], [FLM], [MS], [Pi], [G], [Sc].) To this statement, and others similar to it pertaining to the behavior of a section (subspace) of convex bodies (normed spaces), we shall refer to as \local statements". In particular, the above is a local form of Dvoretzky's theorem. In [BLM] (see also [Schm]) it was proved (although not stated { we shall return to this in the proof of Theorem 2.2 below) that, under the same conditions, for some integer ?
t  "C Mb 2 , with C an absolute constant, there are t orthogonal transformations ui , i = 1; : : :; t, such that 2

t M jxj  1 X 1+" t i=1 kui xk  (1 + ")M jxj for all x 2 IR :

(2)

Moreover, with high probability a random choice of the ui s will do. (This time the probability measure is the normalized Haar measure on the orthogonal group O(n). An equivalent statement to (2) is that there are t orthogonal transformations such that t M D 1X (3) 1+" t ui (Bk
k )  (1 + ")MD : i=1

Here Bk
k or BX denote the unit ball of X Bk
k = BX  denotes the unit ball of X  and D = B`n . To statements of this nature, pertaining to the structure of the whole convex body or all space (here Bk
k or X ), we shall refer to as \global statements". In particular, the above is a global statement of Dvoretzky's theorem. The known proofs of the statements above use similar tools but, except for that, do not relate the local statements directly to the global ones or vice versa. The purpose of this paper is to remedy this in the particular situation above as well as in several other instances. 2

2

To describe the remedy for the particular case of Dvoretzky's theorem described in detail above, we introduce the following two de nitions:

De nition. Let X be IRn with norm k
k. Let k = k(X )  n be the largest integer such that

Gn;k





E ; M2 jxj  kxk  2M jxj for all x 2 E > 1 ? n +k k :



Let t be the smallest integer such that there are orthogonal transformations u1 ; : : :; ut 2 O(n) with t M jxj  1 X n 2 t i=1 kui xk  2M jxj ; for all x 2 IR : The main result of section 2 below is Theorem 2.2 which is a more precise version of:

Theorem 1.1. For some universal constant C C ?1 n  kt  Cn : In section 3 we deal with symmetric convex bodies which are in M -position (i.e., satisfy the inverse Brunn-Minkowski inequality, see exact de nition in section 3). Recall that every symmetric convex body can be put in such a position by applying an invertible linear transformation. The main result in this section (Theorem 3.3) states that if such a body K has the property that for some t and some orthogonal transformations u1; : : : ; ut t X 1 D  t ui (K )  2D i=1

then there is ONE orthogonal transformation u such that

D  K + u(K )  CD where C depends on t only. Section 4 deals with properties related to nite volume ratio. Corollary 4.5 there gives a global property of a space implied by the (local) random version of the subspace of quotient theorem of the rst named author ([Mi2]). 3

2. The Equivalence of the Local and Global Forms of Dvoretzky's Theorem The new ingredient in the proof is the following simple lemma.

Lemma 2.1. Let u ; u ; : : :; ut be orthogonal transformations of IRn . Let k
k be a norm P on IRn and let j
j denote the canonical Euclidean norm on IRn . Put jjjxjjj = t ti kui xk and assume jjjxjjj  C jxj for all x 2 IRn and some 0 < C < 1. Then p kxk  C tjxj ; for all x 2 IRn : 1

2

1

=1

Proof: Let K denote the unit ball of k
k. Let b = maxfkxk; x 2 S n? g and let x 2 S n? be such that kx k = b. Since xb 2 b S n? \ @K it follows that the functional x supports K , i.e. hx; x i  1=b, for all x 2 K . b 1

1

1

1

1

1

1

1

1

Put xi = u?i 1 x1 , i = 1; 2; : : :; t, and, for i > 0, i = 1; 2; : : :; t, consider the 2t caps    i " n ? 1 Ai = y 2 S ; " hy; xii  b ; i = 1; : : :; t ; " =  : P

Note that if the t sets fA+i [ A?i gti=1 have a point of intersection then t?1 i  C . Indeed, if j hui y; x1i j = j hy; xii j  i =b for some y 2 S n?1 and all i = 1; : : :; t, then kui yk  i , i = 1; : : :; t and thus jjjyjjj  t?1 P i . P Choose "i =  such that "i xi is maximal with respect to all possible choices of P signs. It is easily proved then that, for all i = 1; 2; : : :; t, tj6=i h"i xi ; "j xj i  0. Thus,  P P putting y = "i xi "i xi and i = bj hy; xii j, we get that the caps fA"i i gti=1 have y in P

P  P their intersection. Consequently, C  bt j hy; xii j  bt y; "i xi = bt "i xi  pbt .

Remarks. a. The last part of the proof of the lemma above is similar to the proof of

a special case of a lemma of Bang [Ba] (see also [B]). R R b. Note that M = Sn? kxk = Sn? jjjxjjj and also that in Lemma 2.1 necessarily C  M . Up to a constant factor the converse of Lemma 2.1 also holds: There exists a universal constant K such that if k
k is a norm on IRn satisfying kxk  C jxj for all x 2 IRn , then, given a positive integer t, there exist t orthogonal P transformations u1 ; u2; : : :; ut such that putting jjjxjjj = 1t ti=1 kui xk,   C jjjxjjj  K max p ; M jxj ; for all x 2 IRn : t 1

1

4

The proof of this assertion follows the proof of Theorem 2 in [BLM] (or see Proposition 1 in [Schm]). p c. Note that Lemma 2.1 is informative only if C t is not too large. It is always true p that, for some absolute constant A, kxk  AM njxj. d. One can generalize Lemma 2.1 so as to replace the norms kxki = kui xk with general norms. The proof is similar: Let k
ki , i = 1; : : :; t, be t norms on IRn . Put bi = maxx2Sn? kxki , and jjjxjjj = ? Pt
Pt n 1 1 2 1=2 . i=1 bi t i=1 kxki . Assume jjjxjjj  C jxj for all x 2 IR . Then C  t We are now ready to state the main theorem of this section which is a more precise version of Theorem 1.1. 1

Theorem 2.2. a. If, for some orthogonal transformations u ; u ; : : :; ut and all x 2 IRn , jxj  t? Ptu kui xk  C jxj. Then (IRn ; k
k) contains, for each " > 0, a subspace of 1

1

=1



2



n dimension k = " C t on which the norm is 1+ " equivalent to a multiple of the Euclidean norm j
j.  > 0 is a universal constant. Moreover, the collection of all subspaces of dimension k having this property has probability  1 ? exp(?c(")n=C 2 t). Here c(") > 0 depends only on " and the probability measure is the normalized Haar measure on the relevant Grassmanian. b. Conversely, there exists an absolute constant c > 0 such that if, for some 1  t  n and   some " > 0, the collection of all c n" t +1 -dimensional subspaces V of (IRn ; k
k), satisfying jxj  kxk  2jxj for all x 2 V , has probability larger than 1 ? 1=(2c2"2 t), then there exist P orthogonal transformations u1 ; : : :; ut, such that the norm jjjxjjj = 1t ti=1 kui xk is 1 + " equivalent to a multiple of the Euclidean norm. 2

2

2 2

Proof: Except for the exact dependence on ", a. is a straightforward application of

Lemma 2.1 and [Mi1] (or see [MS] Theorem 4.2 and Remark b. following it). To get the dependence on " to be of order "2 , one needs to use [Go] (or [Sc]) instead of [Mi1]. To prove b., note rst that if, for some absolute constant K and some 1  t  n,   the collection of all nt + 1 -dimensional subspaces V of (IRn ; k
k), satisfying kxk  2jxj p for all x 2 V , has probability larger than 1 ? 1=2t then kxk  2 tjxj for all x 2 IRn . 5

Indeed the assumption easily implies the existence of t orthogonal subspaces, V1 ; : : :; Vt, each of dimension approximately n=t, for which the inequality kxk  2jxj is satis ed for S each x 2 ti=1 Vi . For a general x apply the triangle inequality. p R It follows that kxk  2c" tjxj for all x 2 IRn . Clearly also M = kxkd  1. It follows now from the method of [BLM] (or see [Schm] Proposition 1 for a more precise statement) that there exist orthogonal u1 ; : : :; ut satisfying t X 1 ? 1 kui xk  (1 + ")M jxj for all x 2 IRn : (1 + ") M jxj 

t i=1

We conclude this section with an observation which gives a formula for (the order of magnitude of) the norm of an operator from an Euclidean space to an n-dimensional normed space in terms of parameters of probabilistic nature related to the operator. The proof is basically a repetition of the proof of Theorem 2.2.

Observation. There exist universal constants c and C such that if T : `n ! X = R (IRn ; k
k), M (T ) = Sn? kTxk and k = k(T ) is the largest integer such that 2

1

Gn;k E ; kTxk  2M (T )jxj for all x 2 E > 1 ? ck n : ?

Then,




p

p

C ?1 M (T ) n=k(T )  kT k  CM (T ) n=k(T ):

Note that the de nition of k(T ) here is dierent from the de nition of k = k(X ) given in the introduction even for the formal identity operator because in k(T ) we consider only upper bounds. However, using the by now standard techniques from the Local Theory it is easy to see that, if we change the de nition as to require that kTxk is bounded from below by (1=2M (T ))=jxj, say, the new k(T ) will be universally equivalent to the old one. This observation may have some signi cance in computational geometry. In its dual form, it reduces the problem of estimating the diameter of a convex (symmetric) body to that of deciding whether the diameter is within a factor of 2 of the mean width of the body. 6

3. Behavior of a Convex Body in M -position The results we discussed in section 2 connect an arbitrary norm (convex body) with a xed Euclidean structure (ellipsoid). We did not choose a special Euclidean structure connected with the norm. Note that the Euclidean structure was involved in these results in a \global" form: through the orthogonal groups induced by the Euclidean norm and through the Haar measures on the Grassmannians. However, there is a Euclidean structure which induces the Haar probability measure on Grassmannians which behaves as a \natural" measure. As we described in the Introduction, we always prefer to choose a presentation of a norm k
k in IRn such that the standard Euclidean structure of IRn is this natural structure for the norm k
k. To describe this Euclidean structure, we start with some geometric inequality. Recall that the inverse Brunn-Minkowski inequality of [Mi2] states that, given k symmetric convex bodies B1 ; B2; : : :; Bk in IRn one can nd positions Be1; Be2; : : :; Bek of the bodies (i.e. Bei = Ti Bi for some volume preserving operators Ti ) which satisfy the inequality ?

vol(t1Be1 +


+ tk Bek )


1=n



 Ck (vol(t1Be1)1=n +


+ (vol(tk Bek ))1=n



for all t1 > 0; : : :; tk > 0 and for some constant Ck depending merely on k. The position Bei depends only on the body Bi and not on Bj for j 6= i, if Bi is already in this special position, i.e. if Ti = I , we shall say that Bi is in M -position. The dependence of Ck on k is discussed in [Pi], p. 120: For every  > 1=2 every symmetric convex body B has a position Be (which we shall refer to as -regular M -position) satisfying ?

vol(t1 Be1 +


+ tk Bek )


1=n



 ()k (vol(t1Be1)1=n +


+ vol(tk Bek ))1=n



for all symmetric convex bodies B1; B2; : : : ; Bk in IRn where () depends only on .

? ? Following the proofs in [Pi] one gets ()  exp A= ? 12 1=2 for some absolute constant A. If B is in this position so is its polar, any multiple of B and any image of B under an orthogonal transformation. The main result of this section is

Theorem 3.1. Let k
k be a norm on IRn and assume its unit ball K is in M position.

Assume further that for some t orthogonal transformations u1; : : :; ut and for some 7

0 < r; C < 1,

t X 1 rjxj  t kui xk  Crjxj i=1

(3:1)

r0 jxj  kxk + kuxk  C 0 r0 jxj

(3:2)

for all x 2 IRn . Then there is a C 0 , depending on t and C only and an orthogonal transformation u such that, for some r0,

for all x 2 IRn .

Remarks..

a. In geometric language, the theorem states that if K is in the above position and, for some t, there are t orthogonal rotations fui gt such that d(T; D)  2 for P T = t t ui K , then there is another rotation v 2 O(n) such that d(K + vK; D)  C (t). Here d(A; B ) denotes the smallest ratio R=r such that rA  B  RA. b. The requirement that K be in M -position is crucial. Otherwise, the result cannot be true. Consider the n-dimensional normed space X = `n=  `1 n . Then it is known 1

1

1

1

10

9 10

that there are, say, 20 rotations ui such that (3.1) is satis ed for X . At the same time it is also not hard to show that no 5 rotations in (3.1) can give us the norm uniformly (i.e., independent of the dimension n) equivalent to `n2 . In this particular example an M 10 position can be taken to be the unit ball of the space Y = Ln=  L1 n which is isometric 1 to X . In this realization, Y , of the same space X , it is already impossible to nd a number of rotations, independent of dimension n, satisfying (3.1). We thank Bill Johnson who pointed up this example to us. We prove below, in fact, a stronger statement than Theorem 3.1. By section 2 we know that both (3.1) and (3.2) are equivalent to information about almost Euclidean sections of the space X . The conclusion of Theorem 3.1' is stated in this local form. Clearly, Theorem 3.1 follows from it. Given a subspace Y of the space X = (IRn ; k
k) with unit ball K , we denote by dY = dK the natural distance of Y to the natural Euclidean space (Y equiped with j
j), i.e., dY = d(K; Bj
j). (See remark a. above.) 9 10

Theorem 3.1'. Under the assumptions of Theorem 3.1, for every 0 < < 1, there is a subspace Y of dimension [ n] of (IRn ; k
k) for which dY is bounded by a constant 8

depending on ; t and C only. Moreover, the collection of all [ n]-subspaces satisfying the conclusion has probability tending to one as n tends to 1.

Proof: We may assume r = 1. By Lemma 2.1, we know that p kxk  C tjxj : Let  be the volume radius (v.rad(K )) of K , i.e. jDj = jK j. Then, by the Blaschke-Santalo inequality v.rad(K )  1=. By duality (3.1) implies t X 1 D  t ui K   T  C
D 1

and, by the reverse Brunn-Minkowski inequality 1  v:rad T  C (t) v:rad K   C (t)= : So,   C (t). Finally, p1tC D  K and the canonical volume ratio of K (= (jK j=jDj)1=n for the largest  such that D  K ) satis es 1=n p p j K j = tC  tC (t)
C  C1 (t) : v:r: K  tC jDj If K is in an -regular M -position for some  > 21 , then C (t)  C t. So, for every " > 0

p

there is C (") such that



v:r: K  C (")t1+" :

Now Theorem 3.1' follows from the standard knowledge about spaces with nite volume ratio. This simple proof has an unfortunate disadvantage. The known estimates of the distance of \random" n-dimensional subspaces to Euclidean spaces as a function of  < 1 (and of the v:r: K ) are very poor. We present now another proof, more involved but which implies much better estimates. 9

Theorem 3.1". Let k
k be a norm on IRn and assume its unit ball K is in -regular

M -position for some  > 1=2. Assume further that for some t orthogonal transformations u1 ; : : :; ut and for some 0 < r; C < 1, rjxj  1t

t X i=1

kui xk  Crjxj

for all x 2 IRn . Then for every 0 < < 1, there is a subspace Y of dimension [ n] of (IRn ; k
k) for which dY  KC 2 2 ()(1 ? ? o(1 ? ))1=2?t1=2+ , for some absolute constant K .

Proof. We may and shall assume that vol K = vol D.

We would like to give rst the motivation for the computations below. From Lemma p R 2.1 we know that b = supSn? kxk  Cr t. Also, M = Sn? kxk  r (and M  Cr). It n follows that for each R there is a subspace of dimension k = [minf R C t ; ng] ( an absolute constant) on which the kxk  RM jxj. Actually, most subspaces of this dimension will do. Also, the \M  lower bound theorem" of the rst named author (cf. [MS] or [Pi]), implies that for every 0 <  < 1 most subspaces of dimension n satisfy the lower bound R kxk  Mc jxj, where c > 0 depends only on  and M  = Sn? kxk. We can of course nd a large subspace satisfying both conditions. The problem of course is that we do not know, and it is not always true that MM  is bounded. Our plan is to replace M  with the respective quantity, denoted Ms, for a related norm, show that it is good enough to consider that related norm, when dealing with the lower bound and show that MMs is bounded. We rst estimate r and, since r  M  Cr, also M . The dual ball to that of the P P norm 1t ti=1 kui xk is 1t ti=1 ui K o . We thus get 1

1

2

2

1

t X 1 rD  t ui K o  rCD: i=1

(3:3)

By Brunn-Minkowski inequality and its inverse stated above, (vol K o )1=n

t X 1  (vol( t ui K o ))1=n  ()t(vol K o )1=n : i=1

10

(3:4)

Also, by Blaschke-Santalo's inequality, (vol K o )1=n  (vol D)1=n

(3:5)

for some universal c > 0. (3.3),(3.4) and (3.5) now imply

r  ()t and M  C()t :

(3:6)

We turn now to the lower bound. Fix a  > 1. For 0 < s < 1, let Ks = K \ sD, let R kxks = maxfkxk; s?1jxjg be the norm corresponding to Ks and let Ms = Sn? kxks . Let s be such that Mss = . (Note that Mss = 1 if s is small enough and, since Ms = M  for s large enough, lims!1 Mss = 1). We rst want to estimate Ms . By [BLM] we get that there exist k orthogonal transformations u1 : : : uk , with k = k depending only on  (actually k is bounded by a universal constant times 2 ,because the expression \b=M " for the norm k
ks is ), such that 1

k Ms jxj  1 X   2 k i=1 kui xks  2Ms jxj

(3:7)

i.e.,

k Ms D  1 X  u i Ks  2Ms D: 2 k i=1 By the inverse Brunn-Minkowski inequality as stated above,

Consequently,

(3:8)

k k X 1 Ms (vol D)1=n  (vol( 1 X 1=n 1=n u  (vol( u i Ks )) i K )) 2 k i=1 k i=1  ()k?1 (vol K )1=n = ()k?1(vol D)1=n :

(3:9)

Ms  2()k?1 :

(3:10)

By the M  lower bound theorem ([MS] or [Pi]), for any 0 < < 1 we get for an appropriate  with  ! 1 if ! 1? (actually, one can take  = (1 ? ? o( ))?1=2 see [Mi4], Proposition 2.ii.2, p. 24), that most [ n]-dimensional subspaces X of IRn satisfy 1s jxj = n 1 1 Ms jxj < kxks for all x 2 IR . But s jxj < kxks implies kxks = kxk so (2()k?1 )?1jxj  kxk for all x 2 X: 11

(3:11)

Combining this with (3.6) and the discussion at the beginning of this proof we get that for every 1 < R < 1 there exists a subspace X of IRn of dimension [minf ; R tC gn] on which 2

2

(2()k?1)?1 jxj  kxk  C()t Rjxj:

(3:12)

Where  = (1 ? ? o( ))?1=2 . Choosing R = ( tC 2 =)1=2 we get the conclusion. Another fact in the same spirit we would like to demonstrate here is

Theorem 3.2. Assume the unit ball K of X = (IRn ; k
k) is in M -position. Fix positive

integers t and  . If [n=t]-dimensional subspaces of X are 2-Euclidean with probability at least 1 ? 1t , and [n= ]-dimensional quotient spaces are 2-Euclidean with probability at least 1 ? 1 , then the space X is C (t;  )-Euclidean , i.e. dX  C (t;  ) or equivalently d(K; D)  C (t;  ).

Remark. The global form of this theorem, as follows from the results of section 2, is: Let K be in M -position and jK j = jDj. If for some r and   X 1 D  K =  ui K  2D (for some ui 2 O(n)) i

and

rD  (K )t = 1t

t X

(3:13)

vi K   2rD (for some vi 2 O(n)) :

(3:14)

1

Then d(D; K )  C (t;  ). Moreover, if for some  > 1=2, K is in an -regular M -position, then there is a constant C () such that d(K; D)  C ()t+  + . 1 2

1 2

Proof of the Theorem. Let a and b be the smallest numbers such that a jxj  kxk  bjxj. (3.13) and (3.14) mean that for some orthogonal operators fvi gt ; fuj g  O(n) 1

1

t X 1 rjxj  t kvi xk  2rjxj ; 1  X 1 jzj   kuj xk  2jxj : 1

12

1

p

p

By Lemma 2.1, kxk  2r tjxj and kxk  2  jxj, for all x 2 IRn . This means that p a  2p and b  2r t. Also, by the reverse Brunn-Minkowski inequality (we use here that K is in M -position) we have from (3:13) and (3:14)

r  v:rad(K  )t  c(t) v:rad K   c(t) (Blaschke-Santalo inequality is used here) and

  v:rad K  c( ) :

p

Therefore, d(K; D) = a
b  4c(t)c( ) t . Note that if K is in an -regular M -position for some  > 21 then c(t)  C t and c( )  C   . So, in this case there is C () such that

d(K; D)  C ()t

Remarks.

  21 +

1+ 2

:

a. In the appendix we introduce another ellipsoid that can serve as the

M -ellipsoid in the proof above. Using it one gets another version of the global form of Theorem 3.2. We state it as Corollary A.3 below. b. We would like to make some comments on the proof above. By Theorem 2.2.b and its proof (M=b)2 > (3:15)  1=t and (M =a)2 >  1= : By Theorem 3.2 this implies that K is isomorphic to the Euclidean ball (of course, under 2 2 the condition that K is in M -position). Note, that the space `n=  `n= 1 satis es (3:15) 1 and is not uniformly isomorphic to the Euclidean. So, M -position plays a decisive role here too. From another point of view, (M=b)2 + (M =a)2 =  > 1 is known ([Mi3]) to imply d(K; D)  ?1 1 without any restriction on special position. So, if both quantities M=b and M =a are close to one this always implies isomorphism to   Euclidean space; however, in M -position already MM a
b   > 0 implies dX  C ( ) for some constant C () depending on  > 0 only. 13

4. More Global and Local Equivalences We will present two more examples of parallel statements in the Global and Local Theories. This time the global (geometric) form was known and we develop the Local direction.

Theorem 4.1. Let the space X = (IRn ; k
k; j
j) have the following property: kxk  jxj for all x 2 IRn and for some C > 0 and t > 0, for every integer k, n=2  k < n, there are subspaces Sk 2 Gn;k , such that ?t k (4:1) jxj  C 1 ? n kxk; for all x 2 Sk , and moreover, for every such k, the probability that our element of Gn;k satis es this inequality is at least 1 ? n1 . Then there is a V = V (C; t) such that (jBk
kj=jBj
jj)1=n  V . 

Remark. Recall the fact (discovered in [TS] based on [Ka]. cf. [To]) that if a space X satis es k
k  j
j and (jBk
kj=jBj
jj) =n  V then for every k there is a large measure of k-dimensional subspaces on which the two norms k
k and j
j are equivalent up to some 1

constant f (k=n).

The Proof of the Theorem is standard and we just sketch it. Note rst that the

assumption on measure of subspaces satisfying (4:1) implies that there is a ag of subspaces S n  S n +1


 Sn?1  X (dim Si = i) satisfying condition (4:1). Let a be the smallest such that a1 jxj  kxk  jxj then a  Cnt (by (4:1)). We use Lemma 3.7 from [BM1] (or rather its reformulation as Lemma 4.7 in [BM2] which is a better form for our purposes). It states that for k1 = n ? (lognn) any k1-dimensional subspace, in particular our subspace Sk , has the volume radius of its unit ball almost exactly the same as the volume radius of X : v:rad Sk ' v:rad X : Note that dSk  C (log n)2t. Then, by the same lemma, any k2-dimensional subspace of Sk for n k2  k1 ? (log kd2 )2  n ? (log log n)2 2

2

2

1

1

1

1

Sk1

14

satis es v:rad Sk ' v:rad Sk (' v:rad X ). applying it to our subspace Sk from the chosen

ag we have also dSk  C (log log n)2t. So, continuing along our ag of subspaces we reach, after p steps, where p is the rst to satisfy log(p) n  2, a subspace Skp which is Euclidean up to a constant K (C; t) without signi cant change of v.rad.: v:rad Skp ' v:rad X . (For that conclusion one needs to consult [BM2] to get the right constant of equivalence in each step of the procedure above and check that the product of these constants converge.) Also, (jSkp \ Skp \ Bk
k j=jBj
jj)1=kp  K (C; t) which implies that (jBk
kj=jBj
jj)1=n is bounded by a constant depending on C and t only. In our last result, Proposition 4.3, we describe the global property of a space X being in M -position by local information on some quotient spaces of X . Let X = (IRn ; k
k; j
j). For a subspace q we de ne the normed space qX as the one whose unit ball is the orthogonal projection on q of the unit ball of X . If K = Bk
k we denote by qK the unit ball of qX . For two bodies K and H in IRn we let N (K; H ) denote the minimal number of translates of H needed to cover K . The next lemma is known (part a. is implicitly contained in [Mi4]) although we could not locate a good reference. We include a sketch of a proof. 2

1

2

2

Lemma 4.2. Let K be a convex symmetric body in IRn such that jK j = jDj and N (D; K )  ecn . Then a. With high probability a [(n +1)=2]-dimensional subspace of IRn satisfy that qD  qK

for  > 0 depending only on c. b. N (K; 2D)  ec0 n with c0 depending only on c. Consequently, (jqK j=qDj)1=n is bounded by constant depending only on c for every [(n +1)=2]- dimensional subspace.

Sketch of a proof. Assume n is even. It follows from the assumption that jD \ K j  e?cn this implies, via the Blaschke-Santalo inequality, that the volume ratio of (D \ K ) is bounded by ec . It follows that a random n=2-dimensional section of (D \ K ) is contained in a multiple, depending only on c, of D. Dualizing again we get a. N (D; K )  ecn implies jK + Dj  ecn 2n jK j consequently the maximal cardinality of a 2-separated family in K is bounded by jK + Dj=jK j  ec0 n . This proves b. The next proposition shows that the inverse statement also holds. 15

Proposition 4.3. Let the space X = (IRn ; k
k; j
j) be such that, for K = Bk
k, jK j = jDj and with probability at least 1=2, an [(n + 1)=2]-dimensional quotient qX satis es (jqK j=qDj) =n  C and qD  qK . Then N (D; K )  ecn with c depending on  1

and C only.

Remark. It follows from Lemma 4.2 that also N (K; D) is bounded by ec0 n . For most

purposes this (i.e., both N (K; D) and N (D; K ) are at most exponential in n) can serve as a de nition of \K is in M position". The proof of the proposition follows easily from the following lemma

Lemma 4.4. Let H  K  IRn with H convex symmetric and such that (jK j=jH j) =n  1

C . Then,

N (K; tH )  en log( t?t

2

C)

for all t > 2.

Proof. Let x 2 K then

K \ ( 2tx + H )  t ?t 2 H + 2t (K \ (x + H )):

Indeed, if h1 2 H and h2 2 H with k = x + h2 2 K , then t?t 2 h1 + 2t k 2 K and t?2 h + 2 k 2 K = 2 x + t?2 h + 2 h 2 2 x + H . t 1 t t t 1 t 2 t Since K \ (x + H ) is not empty, jK \ ( 2tx + H )j  j t ?t 2 H j = ( t ?t 2 )n jH j: Let now fxgx2I be a maximal family of points in K such that kx ? ykH > t for all x 6= y, x; y 2 I . Then [x2I (x + tH )  K and f 2tx + H gx2I are disjoint. It follows that 2x + H )j  ( t )n jK j=jH j  en log( t?t C ) : jI j  jK j= min j K \ ( x2I t t?2 2

Proof of Proposition 4.3. Assume n is even. Let q X and q X be two orthogonal 1

2

n=2-dimensional quotient spaces of X satisfying the assumption. By the Lemma, for some C depending only on  and c, N (qi K; qi D)  eCn , i = 1; 2. It follows that N (K; D)  N (K; D)  e2Cn . 16

Remark..

It follows from Lemma 4.4 that, if D is the maximal volume ellipsoid inscribed in K and K has volume ratio v (i.e. (jK j=jDj)1=n = V ), then N (K; D)N (D; K )  ecn with c depending only on V . i.e., for bodies with \ nite volume ratio" the maximal volume ellipsoid can serve also as the M -ellipsoid. A result of the rst named author ([Mi2]) states that when BX is in M position a random subspace of quotient of proportional dimension is Euclidean (cf [MS] or [Pi]). The following is a converse to this statement and is another instance in which we get a global result from local one.

Corollary 4.5. Let the space X = (IRn ; k
k; j
j) satisfy jBk
kj = jBj
jj and, for some C; t > 0 and all k < n=2, a k-dimensional subspace of an [(n + 1)=2]-dimensional quotient of X , Y = sqX , satis es, with probability at least 1 ? n1 , 

C ?1 1 ? 2k n

?t

?t 2 k kxk  jxj  C 1 ? n kxk; 

(4:2)

for all x 2 Y . Then, a. With probability larger than 1=2, an [(n + 1)=2]-dimensional quotient satis es (jqBk
kj=jqDj)1=n  V for some V depending on C and t only. (D = Bj
j) b. X is in M position. i.e., N (Bk
k; D)  N (D; Bk
k)  ecn with c depending on C and t only. Proof: Assume n is even. The assumption implies that, with probability larger than 1=2, an n=2-dimensional quotient, Y , satis es that a k-dimensional subspace of it satis es (4.2) with probability larger than 1 ? n2 . In particular, applying the upper bound in (4.2) for two orthogonal subspaces of Y of dimension approximately n=4 (as in the proof of Theorem 2.2) we get that kxk  ?1 jxj for all x 2 Y ( > 0 depends on C and t only). i.e., qD  BY . Applying Theorem 4.1, we get that, with probability larger than 1=2, (jBY j=jqDj)1=n  V (C; t). Which proves a. Applying Proposition 4.3 and the remark following its statement we also get b.

5. Appendix Here we prove 17

Proposition A.1. Let K be a convex symmetric body and let e <  then there exits an ellipsoid D such that, for all log  <    0

0

N (K; D)  eCn

0

and N (K ; D)  eCn

log 0 2

log 0 2

:

Where K  is the polar to K with respect to D. Before we turn to the proof of the proposition Let us state two corollaries. The rst deals with estimating the volume of sums of few bodies with common ellipsoid satisfying the conclusion above. For simplicity we do it only for bodies which are rotations of one body. The second corollary is the variant of Theorem 3.2 promised in the remark following its proof.

Corollary A.2. Let K be in M position for some  as in Proposition A.1 (meaning that the D in the proposition is the standard Euclidean ball of IRn ). Let log   t   = log  . Then for any fui gti  O(n), 0

2 0

0

0

=1

t

X p jt?1 ui K j1=n  c t log 0 jDj1=n:

i=1

Proof. Let  >  > log  . Let fxij gNj be the set of centers of a minimal covering of P ui K by D, i.e., ui K  [Nj (xij + D) with N  eCn  = . Then T = t? ti ui K is covered by eCtn  = D balls. Therefore jT j =n  eCt  = jDj =n. Now choose p  such that t =  = log  and we get jT j =n  C t log  jDj =n. 0

0

=1

log

=1

log

0

2

2

2

0

1

log

1

0

0

1

0

2

=1

1

1

Corollary A.3. Let K be in M position de ned for a xed  as in Proposition A.1. Let t and  be integers such that log   t;    = log  . If for some orthogonal transformations fui gi and fvi gti 0

2 0

0

=1

=1

 X 1 D  K =  ui K  R1 D i

and

t X 1  vi K   R2rD rD  (K )t =

t

1

18

0

Then d(D; K )  R1R2t log 0 . The proof is the same as the proof of (the Remark after) Theorem 3.2. The only dierence is that we substitute the new bounds for c(t) and c( ) given in Corollary A.2. Proof of Proposition A.1. For a convex body T (not necessarily symmetric) and for a (symmetric) ellipsoid D we put

M  (T ) = D

Z

@D

maxf< x; y >D ; y 2 T gdD (x)

where <
;
>D is the inner product de ned by D and D is the unique probability measure on the boundary of D which is invariant under the orthogonal (w.r.t. D) group. Clearly, MD (T1 + T2 ) = MD (T1) + MD (T2 ) for all ;  0; MD (?T ) = MD (T ) and MD (T1)  MD (T2) if T1  T2 . Sudakov's inequality states that N (T; D)  eCn(MD (T )=) for all  > 0. Below we shall use this in a situation in which T +(2?T )  K for some symmetric convex set K . From the above it follows that MD (T )  MD (K ) and that N (T; D)  eCn(MD (K )=) for all  > 0. For all K symmetric there is a D (which we will call an `-ellipsoid) for which 1  MD (K )MD (K )  C log dK  C log n, C an absolute constant. Here MD (K ) = MD (K ) where K  denotes the polar body to K with respect to D. We can, of course, assume that MD (K ) = 1. Step 1: Given K we let D1 be an `-ellipsoid of K and denote 2

2

MD (K ) = M1; 1

MD (K ) = MD (K  ) = M1 1

1

so that M1M1  C log n. Here K  denotes the polar body to K with respect to D1 . We will also assume without loss of generality that M1 = 1. Then N (K; D1)  eCn(M  =) for all  > 0 and N (K ; D1)  eCn(1=) for all  > 0. Fix a1 > 1 and put   1  K1 = conv (K \ a1M1 D1 ) [ a M D1 : 1 1 Denote by L the polar body to L with respect to an arbitrary ellipsoid E to be speci ed later. Then 1 D ) \ a M D : K1 = conv(K  [ a M 1 1 1  1 1

2

1

19

1

2

Note that d(K1; D1)  a21M1 M1  ca21 log n. Also note that

K

[

i2I

K 

K1i ;

[

i2I

#I  eCn=a

H1i ;

(A:1)

2 1

where K1i = (xi + a1 M1D1 ) \ K; H1i = (yi + a1M1D1 ) \ K ; fxi gi2I  K; fyi gi2I  K . Note that, since (xi + a1 M1D1 ) \ K + (?xi + a1 M1D1 ) \ K  K 1 2 and

(yi + a1M1 D1 ) \ K  + (?yi + a1M1D1 ) \ K   K  ; 1 2 MD (K1i )  MD (K1) and MD (H1i )  MD (K1)

for any ellipsoid D. Step 2: Now let D2 be an `-ellipsoid of K1 and denote

MD (K1) = M2 ;

MD (K1) = M2 = 1 : 2

2

(MD (K1) = MD (K1) where the polarity is with respect to D2 . Without loss of generality, we may assume M2 = 1:) Then M2M2  C log d(K1; D1)  C log(Ca21 log n) and for each i 2 I and  > 0, N (K1i ; D2 )  eCn(M  =) and N (H1i ; D2 )  eCn(1=) . Here L denotes duality w.r.t. the same yet unspeci ed E . It follows from (A.1) that, for  > 0;)   i h 2

2

2

2

2

N (K; D2)  e

Cn

and

N (K ; D2)  e

Cn



M2 

1 +

a2 1

h

()

1 + a2 1

1 2

2

i

:

Let a2 > 1 and put 

K2 = conv (K1 \ a2M2D2 ) [ then



K2 = conv K1 [



1 D a2M2 2



1 D \ a M D 2 2 2 a2M2 2 20

(duality w.r.t. E ) and d(K2; D2 )  a22 M2M2  a22C log(Ca21 log n). Now, for each i,

K1i 

[

j 2Ii

K2i;j ;

[

H1i 

j 2Ii

H2i;j ;

#Ii  eCn=a

2 2

where K2i;j = (xij + a2M2 D2 ) \ K1i ; H2i;j = (yi;j + a2 M2D2 ) \ K1; xij 2 K1i , yij 2 H1i . Again MD (K2i;j )  MD (K2); MD (H2i;j  MD (K2) for every ellipsoid D. Step 3: Letting D3 be an `-ellipsoid of K2 and denoting

M3 = MD (K2);

M3 = MD (K2) = 1 3

3

?




(duality w.r.t. D3 ) we get M3M3  C log a22C log(Ca21 log n) and for each j 2 Ii and >0 N (K2i;j ; D3)  eCn(M  =) (resp: N (H2i;j ; D3 )  eCn(1=) ): 3

2

2

It follows that, for  > 0,

N (K; D3)  e

Cn

and

N (K ; D3 )  e



Cn

1 + 12 + a2 a2 1



  2  M 3



()

1 + 1 + a2 a2 1 2

1

2



:

Step k: Continuing in the same way we get, for any sequence a ;


; ak? > 1, a sequence of convex symmetric bodies K = K ; K ;


; Kk? and ellipsoids D ;


; Dk with MDi (Ki? ) = 1; i = 1;


; k ? 1; Di ; Ki? depending only on a


ai? , satisfying d(Ki ; Di )  ai MD i (Ki? ) (A:2) ?
 ai C log d(Ki? ; Di? ) ; i = 1;


; k ? 1 : 1

0

1

1

1

2

1

1

1

1

2



1

1

1

1



d(K; D0)=d n . Also MD k (Kk?1 )  C log d(Ki?1 ; Di?1 ) and, for all  > 0, 

N (K; Dk )  e

Cn






1 + a2 1

+ 21 a

21

?

i 1

 M

+

 ? 2

Dk (Kk 1 ) 

(A:3)

and

N (K  ; DK )  e

Cn






+ a21

1 + a2 1

?

k 1

+ 12





:

(A:4)

Conclusion: Fix  > 1 and consider the process above with ai = MD i (Ki? ) as long as ai >  and ak? =  for the rst k for which MD k? (Kk? )   . Note that kP ?  C ak? = C . Also MD k (Kk? )  C log  so that we get i ai 0

0

1

1

0

1

1

1

=1

2

1

2

1

1

2 0

resp. N (K ; Dk )  e



1 + 1 2 2 0

Cn



log 0 1 + 2 2 0





for all  > 0 . In particular,

N (K; Dk )  eCn for all log 0 <   0 and

0

0

N (K; Dk )  e Cn

2

log 0 2

N (K ; Dk )  eCn 

1 2

for all   0 . This is true where  denotes duality w.r.t. any ellipsoid E . Take E = Dk to nish the proof.

References [B] K. Ball, The plank problem for symmetric bodies, Inventiones Math. 104 (1991), 535{543. [Ba] T. Bang, A solution to the \Plank problem", Proc. Amer. Math. Soc. 2 (1951), 990{993. [BLM] J. Bourgain, J. Lindenstrauss, and V.D. Milman, Minkowski sums and symmetrizations, Geometrical Aspects of Functional Analysis, Israel Seminar, 1986-87, Lect. Notes in Math. 1317 Springer-Verlag 1988. [BM1] J. Bourgain and V.D. Milman, New volume ratio properties for convex bodies in IRn , Inventiones Math. 88 (1987), 319{340. 22

[BM2] J. Bourgain and V.D. Milman, Sections euclidiennes et volume des corps symetriques convexes dans Rn , C. R. Acad. Sci. Paris Ser. I Math. 300 (1985), 435{438. [FLM] T. Figiel, J. Lindenstrauss, and V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta. Math. 139 (1977), 53{94. [G] Y. Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50 (1985), 265-289. [Ka] B.S Kasin, Orders of the widths of certain classes of smooth functions, Uspehi Mat. Nauk 32 (1977), no. 1(193), 191{192. (Russian) [Mi1] V.D. Milman, A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Func. Anal. Appl. 5 (1971), 28{37 (translated from Russian). [Mi2] V.D. Milman, Inegalite de Brunn - Minkowski inverse et applications a la theorie locale des espaces normes, C. R. Acad. Sci. Paris Ser. I Math. 302i (1986), 25{28. [Mi3] V.D. Milman, Spectrum of a position of a convex body and linear duality relations, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989), Israel Math. Conf. Proc., 3, Weizmann, Jerusalem (1990), 151{161. [Mi4] V.D. Milman, Some applications of duality relations, Geometric aspects of functional analysis (1989{90), Lecture Notes in Math. 1469 Springer-Verlag 1991. [MS] V.D. Milman and G. Schechtman, Asymptotic theory of nite dimensional normed spaces, Lect. Notes in Math. 1200 (1986), Springer-Verlag, Berlin. [Pi] G. Pisier, The volume of convex bodies and Banach space geometry (1989), Cambridge Univ. Press. [ST] S.Szarek and N. Tomczak-Jaegermann, On nearly Euclidean decomposition for some classes of Banach spaces, Compositio Math. 40 (1980), 367{385. [Sc] G. Schechtman, A remark concerning the dependence on " in Dvoretzky's theorem, Geometrical Aspects of Functional Analysis, Israel Seminar, 1987-88, Lect. Notes in Math. Vol. 1376, pp. 274-277, Springer (1989). [SCHM] M. Schmuckenschlager, on the dependence on  in a theorem of J. Bourgain, J. Lindenstrauss and V.D. Milman, Geometrical Aspects of Functional Analysis, Israel Seminar, 23

1989-90, Lect. Notes in Math. 1469 Springer-Verlag 1991. [To] N. Tomczak-Jaegermann, Banach-Mazur distances and nite-dimensional operator ideals (1989), Pitman Monographs and Surveys in Pure and Applied Mathematics. V.D. Milman Department of Mathematics Tel Aviv University Ramat Aviv, Israel email:

[email protected]

G. Schechtman Department of Theoretical Mathematics The Weizmann Institute of Science Rehovot, Israel

email:

24

[email protected]