Averages of norms and quasi-norms A.E. Litvak
V.D. Milman
Tel Aviv University
G. Schechtman
Tel Aviv University
The Weizmann Institute
Abstract
We compute the number of summands in q -averages of norms needed to approximate an Euclidean norm. It turns out that these numbers depend on the norm involved essentially only through the maximal ratio of the norm and the Euclidean norm. Particular attention is given to the case q = 1 (in which the average is replaced with the maxima). This is closely connected with the behavior of certain families of projective caps on the sphere.
1. Introduction.
The starting point of this paper is a result from [MS1] which we would like to recall. Denote by j j the canonical Euclidean norm on IRn . Given another norm k k on IRn denote by a and b the smallest constants such that
a? jxj kxk bjxj; for all x 2 IRn : R For X = (IRn ; k k), let M = M (X ) = S ?1 kxkd (x) (where is the normalized Haar measure on the Euclidean sphere). Let, as in [MS1], k = k(X ) n be the largest integer such that E ; M2 jxj kxk 2M jxj; for all x 2 E > 1 ? n +k k ; G (1:1)
1
n
n;k
Partially supported by BSF grants
1
where G is the normalized Haar measure on the Grassmanian Gn;k , and let t = t(X ) be the smallest integer such that there are orthogonal transformations u ; : : :; ut 2 O(n) with t M jxj 1 X for all x 2 IRn : (1:2) 2 t i kuixk 2M jxj; For reasons that will become clear shortly, we shall also denote t by t . Combining Theorem 1.1 of [MS1] with the Observation following Theorem 2.2 there, we have s p b M k(nX ) M t ; or let us write this in the form t (b=M ) : (Here and elsewhere in this paper, means c C for some absolute constants 0 < c < C < 1.) We shall see below that this last equivalence contains a lot of information about the set K \ M ? S n? , where K is the unit ball of X . We would like to investigate in a similar manner the sets K \ rS n? for r 2 (b? ; M ? ). For that purpose we need to extend the result above to include `q -averages of the norms kuixk. RFirst extend the de nitions as follows: For 0 < q < 1, let Mq = Mq (X ) = ( S ?1 kxkq d (x)) =q and let tq = tq (X ) be the smallest integer such that there are orthogonal transformations u ; : : : ; ut 2 O(n) with ! Mq jxj 1 Xt ku xkq =q 2M jxj; n: for all x 2 IR i q 2 t n;k
1
=1
1
1
2
1
1
1
1
1
1
n
1
1
1
i=1
As is well known (and follows from the concentration of the function kxk on S n? ), for q not too large, Mq is almost a constant, as a function of q. It is also clear that as q ! 1 Mq ! b. In Statement 3.1 below we show that the behavior of Mq can be described more precisely: (i) Mq M , for 1 q k(X ), q (ii) Mq b nq , for k(X ) q n, 1
1
2
(iii) Mq b, for q > n. The main generalization of the result from [MS1] mentioned above is:
Theorem 3.4. (i) tq t , for 1 q 2, (ii) tq=q t MM , for 2 q. 1
2
1
1
q
Consequently, (iii) tq=q t 2
2
1
b M1
2
, for 2 q k(X ),
(iv) tq=q nq , for k(X ) q n. 2
Moreover, with the appropriate choice of constants (implicit in the notation
), a random choice of the orthogonal transformations u ; : : : ; ut works with 1
high probability. So, essentially, a random choice of orthogonal transformations gives the same result as the best choice. q
We now turn to the case q = 1 in which case the norm max iT ku?i xk corresponds to the body K1;T = K1;T (u ; : : :; uT ) = \Ti ui(K ). Fix r with b? < r M ? and let T (r) = T (r; X ) be the smallest T for which there are T orthogonal transformationsqu ; : : : ; uT with K1;T (u ; : : :; uT ) rD. We shall assume that b=M C n= log n for some absolute constant C . This can be viewed as a condition of non-degeneracy, i.e., K is not an essentially lower dimensional body. Recall also that any symmetric convex body can be transformed, via a linear transformation, to a body satisfying this condition. In fact, it is enough to transform the body in such a manner that for the resulting body the Euclidean ball is the ellipsoid of maximal volume (see, e.g. the proof of Th. 5.8 of [MS2]). Theorem 4.1 below is a re nement of the following Theorem. Under the non-degeneracy condition above, for some universal constants 0 < c < C < 1 and for r in the interval [2b? ; (2M )? ], exp rcnb T (r) exp rCnb : 1
1
1
1
=1
1
1
1
2 2
2 2
3
1
1
The geometric interpretation of this theoremnis? that, for r as above and Cn covers t exp r2b2 , there are t rotations of the set rS n K whose union rS n? (and one can choose them randomly) while, for t exp rcn2b2 no t rotations provide such a covering. It came as a surprise to us that the parameters involved in the Theorem and its geometric interpretation depends on the body K only through b and M . Moreover, the dependence on M is only to determine the range of r's for which the statement holds. Next we would like to observe the relation between this theorem and Theorem 3.4. For r in the range above, pick q such that Mq = r? . Note that, by the formulas for Mq above, q r2nb2 . It then follows from the formulation of Theorem 3.4 (iv), that log(tq ) n r2b2rb . Notice the similarity between the two approximate formulas for t and tq. The results described up to now are contained in sections 3 and 4. In particular, in section 4 we treat the case q = 1. In section 2, we gathered some of the more geometric preparatory results. It contains (Lemma 2.2.1) a separation lemma in a quasi convex setting. It also contains a theorem (2.3.1), which is an adaptation of Lemma 2.1 of [MS1], expressing b of (1.1) in terms of the corresponding quantity for the q-averaged norm (or quasinorm) (1.2). We also give some geometric interpretation of this lemma, pertaining to the behavior of family of caps on the sphere. Let us mention here a curious application of the material in section 2. Application. Let k k ; :::; k kT be norms on IRn . Then 1
1
1
log(
)
1
kxk ::: max kxkT : max ( k x k k x k ::: k x k ) max T ? 1 ? 1 S S S ?1 T T In section 5 we adapt these results to the quasi-normed case and to the range 0 < q < 1. Recall that a body K is said to be quasi-convex if there is a constant C such that K + K CK , and given a p 2 (0; 1), a body K is called p-convex if for any ; > 0 satisfying p + p = 1 and for any points x; y 2 K the point x + y belongs to K . Note that for the gauge k k = k kK associated with the quasi-convex (p-convex) body K the following inequality holds for all x; y 2 IRn n
1
1
2
n
n
kx + yk C maxfkxk; kykg (kx + ykp kxkp + kykp ) 4
and this gauge is called the quasi-norm (p-norm) if K = ?K . In particular, every p-convex body K is also quasi-convex and K + K 2 =pK . A more delicate result is that for every quasi-convex body K , with the gauge k kK satisfying kx + ykK C (kxkK + kykK ) ; there exists a q-convex body K such that K K 2CK , where 2 =q = 2C . This is the Aoki-Rolewicz theorem ([KPR], [R], see also [K], p.47). In this paper by a body we always mean a centrally-symmetric compact star-body, i.e. a body K satis es tK K for any t 2 [?1; 1]. Section 5 deals with averaging of general quasi-convex bodies while section 6, following [MS1], with averaging quasi-convex bodies in special positions. Throughout this paper, c, C always denote absolute constants. These constants might be di erent in di erent instances. We thank the sta of the MSRI where part of this research has been conducted while all three authors visited there. The second named author worked on this project during his stay in IHES also. The rst named author thanks B.M. Makarov for useful conversations concerning the material of this paper. An extended abstract of this work appeared in [LMS]. 1
0
1
0
2. Behavior of family of projective caps on the sphere.
In this section we introduce a few auxiliary statements concerning choices of a special vectors on the sphere possessing certain special properties with respect to a given family of projective caps on the sphere. These statements will serve as a technical tool mainly in the next section, but we also use them in the proof of Theorem 2.3.1. Some geometric interpretation of these statements regarding the behavior of families of caps will be discussed towards the end of this section. 2.1. We begin with a standard inequality. Lemma 2.1.1. Let fxig be a set of vectors in IRn . Then 8 ! > =p for p 2, < supi jxij X p :> P 22? 22? sup?1 jhy; xiij for 1 p < 2. y2S i jxij i 1
p
p
p
n
5
p
Equality holds if and only if the fxig are mutually orthogonal. Moreover, for every p > 0 we have !1 ! =p X X 1 p p ? ? sup?1 c k 2 jxij ; jhy; xiij 1
y2S
0
i
n
p
1
i
p
where k is the dimension of Y = spanfxigi and c = 2e < 2 ( is the Euler constant.) 0
Proof: Assume rst p 1 and let q be such that 1=q + 1=p = 1. Then X
! =p 1
= sup?1 sup hy; y2S y2S kak i X X = sup aixi = sup "max "iaixi kak kak i i s X sup ai jxij
A = sup?1 n
jhy; xiijp
q =1
i=
q =1
i
aixii
1
2
2
kak =1 i
q =1
n
X
q
by the parallelogram equality. Here a = faigi and k kq denotes norm in lq. Equality holds if and only if the fxig are mutually orthogonal. The desired result follows by duality. Assume now p > 0. Let be the normalized rotation invariant measure on the Euclidean sphere S k? = Y \ S n? . Then ! Z X X p p jhy; xiijp d (y): jhy; xiij A = sup?1 1
y2S
n
1
S ?1 i
i
k
There is an absolute constant c such that 1= 1 =p 0 0 Z Z (2:1) c B @ jhy; xiijp d (y)CA B@ jhy; xiij d (y)CA = jxij k? = : 0
1 2
1
0
2
S ?1
S ?1
k
k
Therefore,
1 2
Ap c?pk?p=
2
0
which proves the lemma. 6
X i
jxijp; 2
Remarks. 1. In factpc can be exactly computed through the ?-function and estimated
0
by c = 2e < 2. 2. A straightforward computation gives, for p > 1, 1= 1 =p 0 0 Z Z B@ jhy; xiijp d (y)CA c B@min(p; k) jhy; xiij d (y)CA : 0
1 2
1
2
S ?1
S ?1
k
k
Thus, for p > 1, we have also sup?1
y2S
n
X i
jhy; xiijp
! =p 1
s
X p! 1 min( p; k ) c jxij : k p
i
An immediate corollary is: Corollary 2.1.2. Let fxigTi be a set of vectors on S n? . Then there exists a y 2 S n? such that 8 ? =p >T for p 2, ! =p 1 X jhy; x ijp < T ? = for 1 p < 2, i > ? ? = T i for 0 < p < 1: :c T 1
1
=1
1
1
1 2
0
1
1 2
2.2. The following lemma can be viewed as \non-linear" form of the HahnBanach theorem for p-convex sets. Lemma 2.2.1. Let k k be a p-norm. Let x 2 S n? be vector such that 1
0
kx k = b = xmax kxk: 2S ? 0
Then
n
1
!? p =p h x; x i kxk 2 b jxj 1
0
=p
1+2
jxj
for any x 2 IRn . Proof: Denote the unit ball of kk by K and the unit ball of jj by D. Fix some 2 (0; =2) and let x be a vector in IRn such that hx; x i = jxj sin . Denote r = 1=b then rD K . To prove the claim we are going to nd a lower bound on jxj which will ensure that x 62 K . For that we represent the 0
7
vector x for some as a p-convex combination of x and some y 2 rD. The p-convexity of K and the maximality of b imply then that, if > r; then x 62 K . Without loss of generality we can assume that n = 2, x = (0; 1), x = (x ; x ) = jxj(cos ; sin ). Choose a vector (v; w) = r(cos ; sin ); where 2 (0; =2) will be speci ed later. By maximality of b the point y = (?v; w) 2 K . Thus if x 2 K then =p (x ; x ) + (1 ? ) =p(?v; w) 2 K for any 0 < < 1. Take 0
0
1
1
2
1
1
2
p = xp v+ vp 1
then
! sin ( + ) =p(x ; x ) + (1 ? ) =p(?v; w) = 0; jxj r p p 2 K: (jxj cos + rp cosp ) =p Hence, by the de nition of x we have, if x 2 K , ( + ) r; jxj r p sin (jxj cosp + rp cosp ) =p or, as long as, sinp( + ) > cosp , p p jxjp sinp( r+cos ) ? cosp : Taking = ? , we get p sinp r p jxj 1 ? cosp and, since 1 ? cosp = 1 ? (1 ? sin ) 2 p sin , ! =p ! ?2 2 h x; x i 2 ? =p jxj r(2=p) (sin ) = r p jxj 1
1
1
2
1
0
1
2
p
2
2
2
1
1
1
p
0
1
p
2
and the lemma is proved.
2.3. Corollary 2.1.2 and Lemma 2.2.1 imply the following extension of Lemma 2.1 of [MS1]. 8
Theorem 2.3.1. Let u ; :::; uT be orthogonal operators on IRn . Let k k be 1
a p-norm on IRn and for some q > 0 put
T X jjjxjjj = T1 kuixkq
! =q 1
i=1 IRn
:
Assume jjjxjjj C jxj for every x in and some constant C . Then ( =q T for q ?pp , kxk C (p; q)C jxj T =p? = for q < p ; 2 2 2 2
1
1
1 2
?p
?
where C (p; q) = (C (q)) C (p) with ( ( 1 for q ?p p , p = 1, C (q) = c for q < p , C (p) = 1(2=p) =p for for p < X 2 ? 1 jhy; x ijq T ? =p = for 1 q ?p p < 2, i > ? 2 T i > : c T ? =p = for q ?p p < 1. Hence, using Lemma 2.2.1, 1
1
0
2
1
p
1
1
p
2
+1 2
p
1
p
0
T 1X ku ykq
C jyj = C T i
=1
(C
1
= (C
1
(p))?q b
(p))?1b
q
T
! =q 1
i
X i
jhui y; x
2? ijq
p
0
i
i
?2 (C1(p))?1b(C (q)) p
p
(
! =q 1
p
1 X jhy; x ijq 2?
T
2
+1 2
p
! =q 1
p
? =q TT ? =p 1
1
=
+1 2
for q for q
C" Mb , with = 1= max fq; 2g, then there exist orthogonal operators u ; :::; uT such that ! =q T X 1 q (3:1) (1 ? ")Eq;T jxj T kuixk (1 + ")Eq;T jxj q
1
1
i=1
for all x 2 IRn and Eq;T Mq 4=3 Eq;T . Moreover, one can take q ln (3=") C" = c " :
Proof: Let = "=3 and let N ben a -net in S n? . By Lemma 2.6 of [MS2], N can be chosen with jNj . Using again the concentration inequalities on (S n? )T ([MS2], 6.5.2) we get, for the normalized Haar measure Pr on T 1
3
1
(O(n)) , that 1 0 ! =q T X 1 Pr @ T kUixkq ? Eq;T > Eq;T A c exp ?c Eq;T n T =b i 1
2
2
2
2
=1
for all x 2 S n? . Hence, if 1
T
2
> c ln (1 =) Eb ; 2
2
2
q;T
then, with positive probability fu ; :::; uT g satisfy ! =q T X 1 q (1 + )Eq;T (1 ? )Eq;T T kuixk 1
1
i=1
for all x in N . A standard successive approximation argument gives (3.1) for all x 2 S n? as long as q ln (3=") b 1
T >c
"
18
Eq;T :
Take
9 8 q < ln (3=") = C" = max :2c " ; C ;; 0
where C is the constant from Lemma 3.2. By this lemma if T > C" b=Mq then Mq 4=3 Eq;T and hence q ln (3=") b b 0
C" M c
"
q
Eq;T :
Thus if T > C" b=Mq we get the result.
2
Remark. The case q = 1 is implicitly contained in ([BLM]). The probabilistic estimates there are obtained in a di erent form (using the fact that one can estimate the -norm of sums of independent random variables in term of the -norms of the individual variables). A similar proof can also be used here. One needs to use a generalization, due to Schmuckenschlager ([S]), of the fact concerning -norm above to the setting of general -norms. We turn now to the study of k(X ) and tq (X ), which we introduced in the Introduction. It was proved in [MS1] that for every n-dimensional normed space X the product k(X ) t (X ) is of order n, or, in other words, t (X ) n=k(X ) (b=M ) . We will study now the relation between t (X ) and tq (X ) for any q 1. This will provide a similar asymptotic formula (in n) for tq(X ). Theorem 3.4. There are absolute constants c , c such that for every normed space X = (IRn ; k k) (i) if q > 2 then ! ! M M =q c t (X ) tq (X ) c t (X ) ; 2
2
2
1
2
1
2
1
1
1
1
1
Mq
1
2
2
2
2
1
(ii) if 1 q 2 then
c t (X ) tq (X ) c t (X ): 1
1
2
19
1
1
Mq
2
Proof: By Corollary 2.3.2 we have for every q > 0 that !s b tq (X ) 2A M ; q
where
(
q 1, and s = max f2; qg: A = 1c for for q < 1 Since t (X ) (b=M ) we get the left side inequality. Proposition 3.3 and Lemma 3.2 imply that, if T > C Mb , with = 1= max fq; 2g and C = C = an absolute constant, then there exist orthogonal operators u ; :::; uT such that ! =q T 1 M jxj 2 E jxj 1 X q 4=3 Eq;T jxj 2Mq jxj 2 q 3 q;T T i kuixk for all x 2 IRn . Hence we get that, for q 1, t q(X ) C Mb : q 0
1
2
1
q
1 3
1
1
=1
Thus
tq
(X )
2
Mb q
!
2
! M t (X ) M : q 1
1
2
2
4. 1-averages (intersection of rotated bodies). Let k k be a norm and K be the unit ball of this norm. Let q > 0 and, given orthogonal operators u ; :::; uT , denote ! =q T X 1 and kxk1T = max kuixk: kxkqT = T kuixkq iT 1
1
i=1
To avoid cumbersome notation, we ignore, in this notation, the concrete choice of the operators fuig, which of course in uence the resulted norms. 20
We shall be mostly interested in the dependence of the quantities above on T . Let KqT denote the unit ball of k kqT . Of course, \T ? K1T = ui K 1
i=1
and, for q ln T , K1T KqT eK1T . The question we would like to study is: Given r with M ? r > b? , how many orthogonal operators we need in order to have 1
1
1
K1T rD ? More precisely, what is a correct order of the minimal number T (r) such that there exist T = T (r) orthogonal operators such that K1T rD ? In the following theorem M denotes the median of the function k k on the sphere S n? . A similar statement with almost identical proof holds when M denotes the average of k k (in which case M = M ). Theorem 4.1. There are absolute constants c , C , c , C such that if b > 1=r > M then ! !?n= 1 (1 ? rM ) n = T (r) C n log (1 + n) 1 ? : C exp 2 (rb) (rb) q In particular, if b > 1=r > 2M (say) and Mb > n n , then ! ! c n c n exp (rb) T (r) exp (rb) : 1
1
1
1
2
2
2
2
1
2
2
3 2
2
log
2
1
2
2
2
Moreover, for
2 !?n= 3 1 5 T = 4C n = log (1 + n) 1 ? (rb) q (or T = exp crb2 n2 in the case b > 1=r > 2M and Mb > choice u ; :::; uT satis es, with high probability, K1T rD: 2
2
(
)
3 2
2
2
1
21
log
n
n ),
a random
Proof: Let x 2 S n? be such that kx k = b and let 2 [0; ] be such that cos = rb . For x 2 S n? and " 2 [0; ] denote by S (x; ") the cap 1
1
0
0
1
2
2
n o n o S (x; ") = y 2 S n? ; (y; x) " = y 2 S n? ; hy; xi cos " ; 1
1
where is the geodesic distance on S n? . For to be chosen later, choose = -net, N , on the sphere S n? (with n respect to the geodesic distance ) satisfying jNj (cf. Lemma 2.6 of [MS2]). If, for some set fxigTi of points on the sphere, the union of the caps S (xi; ? ) covers N , then the union of the caps fS (xi; )gTi covers S n? . In this case for any orthogonal operators ui with xi = uix we have max ku? xk b cos = 1 1
1
3 2
=1
=1
1
0
i
iT
1
1
r
for any x 2 S n? , i.e. K1T rD. Let Pr be the normalized Haar measure on (O(n))T . Then ! ! T X [ [T Pr x 2= S (uix ; ? ) Pr 9x 2 N ; x 2= S (uix ; ? ) 1
0
i=1
x2N
0
i=1
= jNj ( Pr (x 2= S (u x ; ? )))T = jNj (1 ? (S (x ; ? )))T 3 exp n ln 2 ? T (S (x ; ? )) : 1
0
0
0
So if
n ln T > B := (S (x ; ? )) ; 3 2
0
then there are ui such that
? xk 1 max k u iT i r 1
1
for all x 2 S n? . To estimate B note that as we saw in the proof of Statement 3.1 s (S (x ; ")) 2(n? 2) (" ? " ) sinn? " ; 1
0
1
22
2
1
for any 0 < " < ". Therefore 1
A := s(S (x ; ? )) 2 (n? 2) (( ? ) ? ( ? ? )) sinn? ( ? ? ) 0
2
as long as + < 1. Set = = n then, since sin sin for
2 (0; 1) and 2 [0; ], s 1 n? 2 ( n ? 2) n? : 1 ? sin A n 2n Hence, n ln n ln + ln (2n) c n = ln (1 + n) B= A sinn? : A 1 2
2
2
3 2
2
3 2
3 2
2
2
Since 1 ? cos for 2 [0; ], we have 2
2
2
= B c n ln (1 +n=n) (1 ? cos ) 3 2
2
2
and we get that, for
2 !?n= 3 1 = 5 T = 4C n log (1 + n) 1 ? (rb) 2
2
3 2
2
with an absolute constant C , there are orthogonal operators u ; :::; uT such that K1T rD. To prove the lower bound let us point out that if \T 2
1
i=1
uiK rD
then, Si = fx 2 S n? ; rx 2= int uiK g, where int K is the interior of K , ST Sforcovers S n? . So, if A := (Si) = (S ) for i i S = x 2 S n? ; kxk 1r 1
=1
1
1
23
then T A 1, i.e. T A : Denote = r M ? 1, i.e. r = M . Recall the concentration inequality (see, for example, ch. 2 of [MS2]. See also Proposition V.4 in the same book for a similar inequality when M denotes the average): r n ? 2 (fx ; kxk > M + b"g) 8 exp ?" 2 for any " > 0. Take " = Mb , then, since b" = r ? M , ! r ! r n ? 2 M n ? 2 (1 ? rM ) A 8 exp ? b : 2 = 8 exp ? 2 (rb) Hence s ! 8 n 1 1 T e exp 2 (1 ? rM ) ; (rb) which proves the theorem. 2 1
1
1 ( +1)
2
1
2
2
2
2
2
2
2
Remarks. 1. Let K be a stripp fx ; jx j 1g (or a bounded approximation of the strip). Then M=b 1= n and we need at least n rotations to get a bounded K1 . This showspthat a condition ensuring that M=b is of order of magnitude larger than 1= n is necessary. In fact a more careful examination 1
q shows that M=b > c nn is the right condition. 2. It may be instructive to notice that, for any ( xed) C > 2, the inequality in the \In particular" part of the theorem for r in the interval [(CM )? ; (2M )? ] can be written as (rM )?c1 k X T (r) (rM )?c2 k X : The left hand side inequality continues to hold also for r close (but smaller than) M ? . More precisely, for 1=r = (1 + )M one has C exp c k(X ) T (r) for C being the constant from Theorem 4.1 and some absolute constant c. Note that in contrast to the exponential behavior of T (r) above, if 1=r = (1 ? )M , 0 < < 1, then ! b ? T (r) C M : ln
1
1
1
1
(
)
(
1
1
2
1
2
2
24
)
This easily follows from the main result of [BLM] (see also [S]) together with the fact that T ? xk 1 X ku? xk: max k u i iT i T 1
1
i=1
1
5. Averages of quasi-norms.
We explained the notion of p-norms in section 2. The results of section 2 can be applied to the case of p-norms in a similar manner as they were applied for norms. However, the use of the p-triangle inequality, leads sometimes to gaps between the upper and the lower estimates. We still get some interesting versions of the convex case. As the proofs are quite similar to the respective ones in the convex case, we do not repeat them here. The real di erence between the proofs here and those in the convex case is the use of the nonlinear separation result, Lemma 2.2.1, in the proof of Theorem 2.3.1 and through it in Corollary 5.4 below. Claim 5.1. Let 1 q n and let X = (IRn ; k k) be a p-normed space. Then ( ( pq ) q =p? = ) b Mq max 2 M ; c pn ; max M ; c b n 1
1
1 2
1
1
and
2
pq ) c b Eq;T Mq max 2Eq;T ; pnT ; where = 1= max fq; 2g and c; c ; c depend on p only. (
1
2
Let us point out, that using ideas of [La] one can get that for every 1 > q > 0 and every p-normed space X
cM Mq M ; where c depends on p but not on q ([L1], chapter 5). In [La] this is proved 1
1
in the normed case. Note also that if k k is p-norm then for every integer T ! =q T X 1 q kxkq T := T kuixk 1
i=1
25
is s-norm for s = min fp; qg. For s-norm it is more convenient to use concentration inequalities not for the function kxkq T but for the function kxksq T . Thus if we de ne =s Lq;T = E kxksq T ; where E is the expectation with respect to the product measure on (O(n))T , which is equivalent to Eq;T by Latala's theorem, then repeating the proof of Proposition 3.3 we obtain Proposition 5.2. Let q > 0, 1 p > 0 and X = (IRn ; k k) be a p-normed space. For 1 > " > 0 denote c =s s 2 log " ; C" = s" 1
1
where s = min fp; qg. If T > C" Mb with = 1= max fq; 2g then there are orthogonal operators u ; :::; uT such that ! =q T X 1 q (1 + ")Lq;T jxj (1 ? ")Lq;T jxj T kuixk q
1
1
i=1
for all x 2 IRn . These statements allow us to extend the results of [MS1] and of the previous section concerning the relation between k(X ), t (X ), and tq(X ) in the following way. Corollary 5.3. Let q > 0. Let X = (IRn ; k k) be a p-normed space for some p 2 (0; 1]. Denote s = min fp; qg then there exists an absolute constant c, such that for q > 2 !q !q=p c b tq(X ) p M 1
q
and for 0 < q 2
c =s b ! tq (X ) s Mq : As we noted after Claim 5.1 in the second case, q 2, the term (b=Mq ) can be estimated by cp (b=M ) , where cp is a constant depending on p only. The proof of this fact is analogous to the proof of Theorem 3.4. 2
2
2
1
2
26
The following fact is a corollary of Theorem 2.3.1. Recall that, for the p-convex part of Theorem 2.3.1 a crucial role was played by the \non-linear" form of Hahn-Banach theorem for p-convex sets (Lemma 2.2.1). Corollary 5.4. Let X = (IRn ; kk) be a p-normed space for some p 2 (0; 1]. Let q > 0. Then ! b tq (X ) 2A M ; where
q
(
= qp 2 2
?p
for q for q
1=r > M then ?n= p) ! (1 ? ( rM ) n = log (1 + n) 1 ? (c rb)? 22? ; T ( r ) C n C exp 2 p (rb) p 1
2
2
1
2
2
27
3 2
p
p
2
where cp = (p=2) =p. In particular, if cpb=2 > 1=r > 2M (say) and Mb > n 22? , then there are constants c0p, c00p depending on p only, such that n 0 2 ? ? p 00 2 ? : exp cpn (rb) T (r) exp cp n (rb) Proof: The proof of this proposition essentially repeats the proof of Theorem 4.1. To prove the upper bound we only need to substitute the inclusion 1 1 n ? n ? x 2 S ; kxk r x 2 S ; jhx; x ij rb with the inclusion ! 2? 9 8 = < 1 1 x 2 S n? ; kxk r :x 2 S n? ; jhx; x ij c rb ; ; p which follows from Lemma 2.2.1. In other words, in the proof of Theorem 4.1 one should chose 2 [0; ] such that cos = c rb 2? . To prove the lower bound we have to apply concentration inequalities to the function k kp. 2 1
log
p
p
p
2
p
1
1
0
p
p
1
1
0
p
1
2
p
p
6. Averages of quasi-convex bodies in M -position. Recall that for any subsets K ; K of IRn the covering number N (K ; K ) is the smallest number N such that there are N points y ; :::; yN in IRn such that N [ K (yi + K ): 1
2
1
2
1
1
2
i=1
De ne the volume radius r of a star-body K by the formula jK j = jrDj, where jK j denotes the n-dimensional volume of K . Let Cp (for p 2 (0; 1]) be the constant from J. Bastero, J. Bernues, and A. Pe~na's extension [BBP] of the second named author's reverse BrunnMinkowski inequality for p-convex bodies, i.e. !c=p 2 C = p
p
(see [L2] for the dependence of the constant on p). Let us recall this result. 28
Theorem 6.1. Let 0 < p 1. For all n 1 and all symmetric p-convex bodies K ; K IRn there exists a linear operator U : IRn ?! IRn with jdet(U )j = 1 and jUK + K j =n Cp(jK j =n + jK j =n) : 1
2
1
2
1
1
1
2
1
In terms of covering numbers this theorem can be formulated in the following way. Theorem 6.10. For every symmetric p-convex body K in IRn with volume radius r there exists a linear operator U : IRn ?! IRn with jdet(U )j = 1 such that 8 p= > exp n ( C =t ) p < for t Cp, =p max fN (UK; trD); N (rD; tUK )g > exp n ln 3 Cp=t for 1 < t < Cp, : Cn for t = 1. 2
1
p
If the operator U in Theorem 6.10 can be taken to be the identity operator then we say that the body K is in M -position. Remark. If the bodies K ; K ; :::; Kl are in M -position then, as in the convex case ([P], pp. 120-121), jK + K + ::: + Kl j =n Cp l =p jK j =n + jK j =n + ::: + jKl j =n : The following theorem has been recently proved in [MS1]. Theorem 6.2. Let k k be a norm on IRn and assume that its unit ball K is in M -position. Assume further that for some T orthogonal operators u ; :::; uT and for some constant C , T 1X 1
1
2
1
2
2
1
1
2
1
1
1
jxj T kuixk C jxj i =1
for all x 2 IRn . Then there are an orthogonal operator u and a constant C , depending on T and C only, such that for some R Rjxj kxk + kuxk C Rjxj for all x 2 IRn . By duality this theorem is equivalent to the following statement. 1
1
29
Theorem 6.20. Let a symmetric convex body K be in M -position. Assume that for some T orthogonal operators u ; :::; uT and for some constant C , T 1X 1
D T uiK CD: i Then there exist an orthogonal operator u and a constant C , depending on T and C only, such that for some R RD K + uK C RD: =1
1
1
In this section we shall extend both theorems to the quasi-convex case. Because duality arguments can not be applied in the non-convex case these two theorems become di erent statements. Lemma 6.3. Let q > 0 and B > 0. Let K be a star body such that for any tB B q ! N (K; tD) exp n t :
Then there exists an orthogonal operator u such that
K \ uK C
1
1+ q
BD:
Remark. An immediate corollary is that if a p-convex body K is in M position then there exists an orthogonal operator u such that for all x 2 IRn kxkK + kuxkK r C1(p) jxj; where r is the volume radius of K . Here and everywhere in this section we denote a function of the type (2=p)c=p by C (p). The absolute constant c may be di erent in di erent places. Therefore the product of two functions of that type is again a function denoted by C (p). Proof of Lemma 6.3: Let the constant C satisfy 0
B q = C < 1: 0
30
By the de nition of covering numbers for N = [e n], there exist fxigN such that N [ K (xi + C D): 1
0
i=1
Consider the normalized rotation invariant measure on the sphere R C S n? , where R > 0 will be speci ed later. Since the measure of the intersection \ (xi + 4C D) R C S n? does not exceed ! r (1 ? 4 =R ) ( n ? 2) A = 8 exp ? 8 for any xi ([MS2], ch. 2), we obtain that if N A < 1 then there exists an orthogonal operator, u, such that R C u(xi=jxij) 62 xj + 2C D for any i and j . But the union of (uxi + C D) covers \ \ u K R C S n? = u(K ) R C S n? : Therefore \ \ K u(K ) R C S n? = ;: Take p C = 5 (1 + q=2) =q B and R = 4p 1 ? 5 q 2 then N A < 1 and RC 4e5 =q q B . This completes the proof. 0
0
1
1
0
2
2
2
0
0
0
1
0
0
0
1
1
2
0
2
0
2+
1
Lemma 6.4. Let B > 0. Let Ki , i = 1; :::; T , be symmetric p-convex bodies such that jKij = jDj and for any t B for all 1 i T . Then
B q ! N (D; tKi ) exp n t \ =n Ki f (T ) jDj =n ; i 1
1
31
where for
? f (T ) = c B 2T=p A =q )) ( ( 2 A = min T; max 2; q(?1 + 1=p) : 1
1
In particular, if all the Ki's are in M -position then 0 ( ) =p1? 4 f (T ) @C (p) 2T=p min T; 1 ? p A : 1
2
This lemma easily follows from the following claim. Claim 6.5. Under the assumptions of Lemma 6.4 B q ! \ T=p Ki) exp n t A : N (D; t 2 i
Proof: Let two star-bodies B and B satisfy 1
2
B 1
N [ i=1
(xi + B ): 2
Then it is not hard to see that there are points yi in B such that N [ (6:1) B (yi + (B ? B )) : 1
1
Indeed,
B 1
and if
i=1
1
2
\
1
(xi + B ) 2
\ B (xi + B ) zi + (B ? B ) : 2
1
Analogously, (6:2)
2
\ (xi + B ) B
N [
zi 2 B
then
2
i=1
B
1
\
2
2
\ (x + B ) x + (B ? B ) B 2
1
32
1
2
for any x 2 B . Hence, using p-convexity and the assumptions of the claim, we get from (6:1) that, for N = N (D; t K ) and for xi 2 D, N1 N1 \ [ \ [ xi + t 2 =p K 2D : xi + t 2 =p K D D 1
1
1
1
i=1
1
1
1
1
1
i=1
1
For N = N (D; t K ) and for yi 2 2D T t 2 =p K , we get by (6:1) and (6:2), that 0 0 11 N N2 [1 @ @ \ =p \ [ D xi + 2D t 2 K yj + 2 2 =p t K AA 2
2
2
1
1
1
i=1
1
1
1
j =1
2
1
2
1
for zi = xj + yk . Continuing in this way we get N1[ :::N \ \ D vi + 2T D t 2T=p K t 2 2 T ? T
1
i=1
2
2
\ \ zi + 4D t 2 =p K t 2 2 =p K
N[ 1 N2 i=1
1
(
2
1
1
=p K
1)
2
2
\ ::: tT 2T ? 2 =p KT ; 1
1
for some vi's, where
B q ! Ni = N (D; ti Ki ) exp n t : i
Setting
ti = t 2 i? (
we obtain that
?
=p) ;
1)( 1+1
T Y \ N (D; t 2T=p Ki) N (D; tiKi )
q X T exp n Bt 2?q i? i (
i=1
?
=p)
1)( 1+1
=1
!
B q ! exp n t A :
2 33
Lemma 6.6. Let Ki, i = 1; :::; T , be symmetric p-convex bodies in M position. Let k ki denote the gauge of Ki . Assume that for any x 2 IRn jxj max kxki : iT Then there exist a number k 2 f1; :::; T g and an orthogonal operator u such that
f (T ) jxj kxk + kuxk ; k k C (p)
where f (T ) was de ned in Lemma 6.4. Proof: Since jxj maxiT kxki ; \
Ki D: Let r ; :::; rT denote the volume radii of K ; :::; KT and rk = min r: iT i 1
1
Then by Lemma 6.4
T =n T K j =n 1 r Ki 1 j f (T ) jDj =n r jDji =n r : k k Hence by Lemma 6.3, there exists an orthogonal operator such that kxkK + kuxkK r Cjx(jp) f (Tj)xCj (p) : k 1
1
1
i
1
1
2
Now we are ready to extend Theorem 6.2 to the quasi-convex case. Theorem 6.7. Let k k be a p-norm on IRn and assume that its unit ball K is in M -position. Assume further that for some T orthogonal operators u ; :::; uT and for some constant C , ! =p X 1 p jxj jjjxjjj = C jxj ku xk 1
1
T
i
i
for all x 2 IRn . Then there exists an orthogonal operator u such that f (T ) jxj kxk + kuxk 2 T =p C jxj : C (p) This theorem is a direct consequence of the previous lemma. To extend Theorem 6.20 we need the following two lemmas. 1
34
Lemma 6.8. Let B > 0. Let K be a symmetric p-convex body such that for any t B B q ! N (rD; tK ) exp n t ;
where r is the volume radius of K . Then there exists an orthogonal operator u such that =q D C (p) rB c (K + uK ) : The proof of this lemma is almost identical to the proof of Theorem 20 of [LMP]. Note also that if the body K is in M -position then B c =q = C (p) and C (p) B c =q can be replaced by C (p). Lemma 6.9. Let Ki, i = 1; :::; T , be symmetric p-convex bodies in M position. Assume that for some constant C > 0 T 1X 1 1
1
1
C D T i Ki D: Then there exist a number k 2 f1; :::; T g and an orthogonal operator u such =1
that
D C C (p) T =p (Kk + uKk ) : Proof: Let r ; :::; rT denote volume radii of K ; :::; KT and rk = max r: iT i 2
1
1
By the assumption of the lemma and the remark after Theorem 6.10 we have PT =n K =T 1 i i C (p) T =p PTi ri C (p) T =p r : k =n C T 1
=1
jDj
2
=1
2
1
Therefore, by Lemma 6.8, we get D Cr(p) (Kk + uKk ) C C (p) T k
=p
2
(Kk + uKk )
for some orthogonal operator u.
This lemma gives us the following extension of Theorem 6.20. 35
2
Theorem 6.10. Let a symmetric convex body K be in M -position. Assume that for some T orthogonal operators u ; :::; uT and for some constant C , T 1X 1
D T uiK CD: i Then there exists an orthogonal operator u such that =1
1 C C (p) T ?
=p D K + uK
1+2
36
2 T D:
References [BBP] J. Bastero, J. Bernues and A. Pe~na, An extension of Milman's reverse Brunn-Minkowski inequality, GAFA 5 (1995), 572{581.
[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), 44{66.
[D] S.J. Dilworth, The dimension of Euclidean subspaces of quasi-normed spaces, Math. Proc. Camb. Phil. Soc. 97 (1985), 311-320.
[G] M. Gromov, Monotonicity of the volume of intersection of balls, Geo-
metrical aspects of functional analysis, Israel Seminar, 1985{86, Lect. Notes in Math. 1267, Springer-Verlag (1987), 1{4. [KPR] N.J. Kalton, N.T. Peck and J.W. Roberts, An F -space sampler, London Mathematical Society Lecture Note Series, 89, Cambridge University Press (1984). [K] H. Konig, Eigenvalue Distribution of Compact Operators, Birkhauser (1986). [La] R. Latala, On the equivalence between geometric and arithmetic means for logconcave measures, to appear in Convex Geometric Analysis at MSRI (Spring 96). [L1] A.E. Litvak, Local Theory of normed and quasi-normed spaces., Ph.D. thesis, Tel-Aviv University, 1997. [L2] A.E. Litvak, On the constant in the reverse Brunn-Minkowski inequality for p-convex balls, to appear in Convex Geometric Analysis at MSRI (Spring 96). [LMP] A.E. Litvak, V.D. Milman and A. Pajor, Covering numbers and \low M -estimate" for quasi-convex bodies, to appear in Proc. AMS.
37
[LMS] A.E. Litvak, V.D. Milman and G. Schechtman, Averages of norms
and behavior of families of projective caps on the sphere, C.R. Acad. Sci. Paris, 325 (1997), 289-294.
[MS1] V.D. Milman and G. Schechtman, Global vs. local asymptotic theories of nite dimensional normed spaces, Duke Math. J., 90 (1997), 73-93.
[MS2] V.D. Milman and G. Schechtman, Asymptotic theory of nite dimen-
sional normed spaces, Lecture Notes in Math. 1200, Springer-Verlag (1986).
[P] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry,
Cambridge University Press (1989). [R] S. Rolewicz, Metric linear spaces, Monogra e Matematyczne, Tom. 56. [Mathematical Monographs, Vol. 56] PWN-Polish Scienti c Publishers, Warsaw (1972). [S] 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, 1989{90, Lect. Notes in Math. 1469, Springer-Verlag (1991), 166-173. A.E. Litvak and V.D. Milman Department of Mathematics Tel Aviv University Ramat Aviv, Israel email:
[email protected] [email protected]
G. Schechtman Department of Theoretical Mathematics The Weizmann Institute of Science Rehovot, Israel
email:
[email protected]
38
