Averages of norms and quasinorms

AE Litvak VD Milman G Schechtman

Tel Aviv University Tel Aviv University The Weizmann Institute

Abstract

We compute the numb er of summands in q averages of norms

needed to approximate an Euclidean norm It turns out that these

numb ers dep end on the norm involved essentially only through the

maximal ratio of the norm and the Euclidean norm Particular atten

tion is given to the case q in which the average is replaced with

the maxima This is closely connected with the b ehavior of certain

families of pro jective caps on the sphere

Intro duction

The starting p oint of this pap er is a result from MS whichwewould

n

like to recall Denote by jj the canonical Euclidean norm on IR Given

n

another norm kk on IR denote by a and b the smallest constants suchthat

n

x IR a jxjkxkbjxj for all

R

n

For X IR kk let M M X kxkd x where is the

n

S

normalized Haar measure on the Euclidean sphere Let as in MS k

k X n b e the largest integer suchthat

M k

E jxjkxkM jxj for all x E

G

nk

n k

Partially supp orted by BSF grants

where is the normalized Haar measure on the Grassmanian G and

G nk

nk

let t tX b e the smallest integer such that there are orthogonal transfor

mations u u O n with

t

t

X

M

n

ku xk M jxj for all x IR jxj

i

t

i

For reasons that will b ecome clear shortlywe shall also denote t by t

Combining Theorem of MS with the Observation following Theorem

there wehave

s

p

n

t M b M

k X

or let us write this in the form

t bM

Here and elsewhere in this pap er means c C for some

absolute constants cC We shall see b elow that this last

n

equivalence contains a lot of information ab out the set K M S where

K is the unit ball of X

n

Wewould liketoinvestigate in a similar manner the sets K rS

for r b M For that purp ose we need to extend the result ab ove

to include averages of the norms ku xk First extend the denitions as

q i

R

q q

follows For q letM M X kxk d x and let t

n

q q q

S

t X b e the smallest integer such that there are orthogonal transformations

q

u u O n with

t

q

t

X

M

q

q

n

ku xk jxj M jxj for all x IR

i q

t

i

As is well known and follows from the concentration of the function kxk

n

on S for q not to o large M is almost a constant as a function of q It

q

is also clear that as q M b In Statementbelowwe showthat

q

the b ehavior of M can b e describ ed more precisely

q

i M M for q k X

q

q

q

ii M b for k X q n

q

n

iii M bfor qn

q

The main generalization of the result from MS mentioned ab oveis

Theorem

i t t for q

q

M

q

for q ii t t

q

M

q

Consequently

b

q

iii t t for q k X

q

M

n

q

for k X q n iv t

q

q

Moreover with the appropriate choice of constants implicit in the notation

a random choice of the orthogonal transformations u u works with

t

q

high probability So essentially a random choice of orthogonal transforma

tions gives the same result as the b est choice

Wenow turn to the case q in which case the norm max ku xk

iT

i

T

corresp onds to the b o dy K K u u u K Fix r with

T T T i

i

b r M and let T r T rX b e the smallest T for whic h there are

T orthogonal transformations u u with K u u rDWe

T T T

q

shall assume that bM C n log n for some absolute constant C This

can b e viewed as a condition of nondegeneracyieK is not an essentially

lower dimensional b o dy Recall also that any symmetric convex body can be

transformed via a linear transformation to a b o dy satisfying this condition

In fact it is enough to transform the b o dy in such a manner that for the

resulting b o dy the Euclidean ball is the ellipsoid of maximal volume see

eg the pro of of Th of MS Theorem b elow is a renementofthe

following

Theorem Under the nondegeneracy condition ab ove for some universal

constants cC and for r in the interval b M

Cn cn

T r exp exp

r b r b

The geometric interpretation of this theorem is that for r as ab oveand

Cn

n

there are t rotations of the set rS n K whose union covers t exp

 

r b

cn

n

rS and one can cho ose them randomly while for t exp no

 

r b

t rotations provide suchacovering It came as a surprise to us that the

parameters involved in the Theorem and its geometric interpretation dep ends

on the b o dy K only through b and M Moreover the dep endence on M is

only to determine the range of r s for which the statement holds

Next wewould like to observe the relation b etween this theorem and

Theorem For r in the range ab ove pick q such that M r Note

q

n

It then follows from the that by the formulas for M ab ove q

q

 

r b

n logrb

formulation of Theorem iv that log t Notice the similarity

q

 

r b

between the twoapproximate formulas for t and t

q

The results describ ed up to now are contained in sections and In

particular in section we treat the case q In section we gathered

some of the more geometric preparatory results It contains Lemma

a separation lemma in a quasi convex setting It also contains a theorem

which is an adaptation of Lemma of MS expressing b of

in terms of the corresp onding quantity for the q averaged norm or quasi

norm We also give some geometric interpretation of this lemma

p ertaining to the b ehavior of family of caps on the sphere Let us mention

here a curious application of the material in section

n

Application Let kk k k b e norms on IR Then

T

kxk kxk

T

max kxk kxk kxk max max

T

n n n

S S S

T T

In section we adapt these results to the quasinormed case and to the

range q

Recall that a b o dy K is said to b e quasiconvex if there is a constant C

such that K K CKandgiven a p a body K is called pconvex

p p

if for any satisfying and for any p oints x y K the

p oint x y b elongs to K Note that for the gauge kk kk asso ciated

K

with the quasiconvex pconvex b o dy K the following inequality holds for

n

all x y IR

p p p

kxk ky k kx y kC maxfkxk ky kg kx y k

and this gauge is called the quasinorm pnorm if K K In particular

p

every pconvex body K is also quasiconvex and K K K A more

delicate result is that for every quasiconvex body K with the gauge k k

K

satisfying

kx y k C kxk ky k

K K K

q

there exists a q convex body K such that K K CKwhere C

This is the AokiRolewicz theorem KPR R see also K p In this

pap er by a b o dy wealways mean a centrallysymmetric compact starb o dy

ie a b o dy K satises tK K for any t

Section deals with averaging of general quasiconvex b o dies while section

following MS with averaging quasiconvex b o dies in sp ecial p ositions

Throughout this pap er c C always denote absolute constants These

constants might b e dierent in dierent instances

We thank the sta of the MSRI where part of this research has b een

conducted while all three authors visited there The second named author

worked on this pro ject during his stay in IHES also The rst named author

thanks BM Makarov for useful conversations concerning the material of this

pap er

An extended abstract of this work app eared in LMS

Behavior of family of pro jective caps on the sphere

In this section weintro duce a few auxiliary statements concerning choices

of a sp ecial vectors on the sphere p ossessing certain sp ecial prop erties with

resp ect to a given family of pro jective caps on the sphere These statements

will serveasatechnical to ol mainly in the next section but we also use them

in the pro of of Theorem Some geometric interpretation of these state

ments regarding the b ehavior of families of caps will b e discussed towards

the end of this section

We b egin with a standard inequality

n

Lemma Let fx g b e a set of vectors in IR Then

i

p

sup jx j for p

i

i

X

p

p

sup jhy x ij

p

p

P

i

p

n

for p jx j

y S

i

i i

Equality holds if and only if the fx g are mutually orthogonal Moreover for

i

every p wehave

p

p

X X

p

p



jhy x ij c k jx j sup

i i

n

y S

i i

p

where k is the dimension of Y span fx g and c e is the

i i

Euler constant

Pro of Assume rst p andletq b e such that q p Then

p

X X

p

sup hy a x i jhy x ij sup A sup

i i i

n n

kak

y S y S

q

i i

X X

sup a x sup max a x

i i i i i

i

kak kak

q q

i i

s

X

jx j sup a

i

i

kak

q

i

by the parallelogram equality Here a fa g and k k denotes norm in l

i i q q

Equality holds if and only if the fx g are mutually orthogonal The desired

i

result follows by duality

Assume now p Let b e the normalized rotation invariant measure

k n

on the Euclidean sphere S Y S Then

Z

X X

p p

p

jhy x ij jhy x ij A sup d y

i i

n

y S

i i

k

S

There is an absolute constant c suchthat

p

Z Z

B p C B C

c jhy x ij d y jhy x ij d y jx jk

A A

i i i

k k

S S

Therefore

X

p

p p p

A c k jx j

i

i

whichproves the lemma

Remarks

In fact c can b e exactly computed through the function and estimated

p

by c e

A straightforward computation gives for p

p

Z Z

C p C B B

jhy x ij d y jhy x ij d y c min p k

A A

i i

k k

S S

Thus for p wehave also

s

p

p

X X

min p k

p

p

jhy x ij sup c jx j

i i

n

k

y S

i i

An immediate corollary is

n T

b e a set of vectors on S Then there exists Corollary Let fx g

i

i

n

a y S suchthat

p

T for p

p

X

p

T for p

jhy x ij

i

T

T for p c

i

The following lemma can b e viewed as nonlinear form of the Hahn

Banach theorem for pconvex sets

n

Lemma Let kk be a pnorm Let x S be vector suchthat

kxk kx k b max

n

xS

Then

p

p

p hx x i

kxk jxj b

jxj

n

for any x IR

Pro of Denote the unit ball of kk by K and the unit ball of jj by D Fix

n

some and let x be a vector in IR such that hx x i jxj sin

Denote r b then rD K Toprove the claim we are going to nd a

lower b ound on jxj which will ensure that x K For that we representthe

vector x for some as a pconvex combination of x and some y rDThe

pconvexityof K and the maximalityof b imply then that if r then

x K

Without loss of generalitywe can assume that n x x

x x jxjcos sin Cho ose a vector v w r cos sin where

will b e sp ecied later

By maximalityof b the p oint y v w K Thus if x K then

p p

x v w K for any Take x

p

v

p

p

x v

then

sin

p p

K x x v w jxj r

p

p p p p

jxj cos r cos

Hence by the denition of x wehave if x K

sin

r jxj r

p

p p p p

jxj cos r cos

p

p

or as long as sin cos

p p

r cos

p

jxj

p

p

sin cos

we get Taking

p

p

r sin

p

jxj

p

cos

p

p

p



sin and since cos sin



p

p



hx x i

p

p

jxj r p sin r

p jxj

and the lemma is proved

Corollary and Lemma imply the following extension of

Lemma of MS

n

Theorem Let u u b e orthogonal op erators on IR Letkk be

T

n

a pnorm on IR and for some q put

q

T

X

q

jjjxjjj ku xk

i

T

i

n

Assume jjjxjjj C jxj for every x in IR and some constant C Then

p

q

T for q

p

kxk C p q C jxj

p

p

T for q

p

p

p

C p with where C p q C q

p

for q

for p

p

C p C q

p

p

c for q

p for p

p

and c is the same numb er as in Lemma

The following two corollaries follow immediately from the statementof

the theorem the second one by sending q to zero

Corollary Under the condition of Theorem

i if p q then

q

T for q

n

kxk C q C jxj for all x IR

T for q

ii if p q ie if

T

X

ku xk C jxj jjjxjjj

i

T

i

n



then kxk C pCT jxj for all x IR

n

Corollary Let u u b e orthogonal op erators on IR and let kk

T

n

be a norm on IR then

T

T

p

Y

n

kxk c jxj for all x IR T max ku xk

i

xD

i

n

Pro of of Theorem Let x S be a vector such that

kx k b max kxk

xD

n

and set x u x By Corollary there is a y S such that

i

i

p

q

T for q

p

q

X

p

p

p

q

T for q

p

jhy x ij

p

i

p

T

i

p

p

p

T for q c

p

Hence using Lemma

q

T

X

q

ku y k C jy j C

i

T

i

q

q

X

p

b

q

q

p

C p jhu y x ij

i

T

i

q

X

p

q

p

C p b jhy x ij

i

T

i

p

q

p

T for q

p

p

C p bC q

p

p

T for q

p

which implies the theorem

Some of the results ab ovehave what seems to b e an interesting

geometric interpretation

n

Fix a set of p oints fx g i T on the Euclidean sphere S let

i

b and dene the family of seminorms

p x bjhx xij i T

i i

Cho ose such that the Euclidean norm of

i

T

X

z x

i i

i

n

is maximal Denote jz j and let y z S

Lemma

i

p

T T

p

T if and only if the p oints fx g are mutually orthogonal and

i

T if and only if x x for every i

i

ii

T

X

hy x i for every i and p y b

i i i

i

iii

b

p y b for all i

i

Pro of i Since

T

X

x T jz j Ave

i i

i

i

we get the lower b ound The upp er b ound is obvious

ii By the maximalityof z hz x x i Hence hz x ifor

i i i i i i

every i Clearly

T T

X X

p y b hy x i b hy zi b

i i i

i i

iii follows from ii and the denitions

The following claim gives some information concerning the b ehavior of

family of pro jective caps on the Euclidean sphere

Claim Denote

n

A t tS fx j p x g

i i

and

n

A A S fx j p x g i T

i i i

Then

i the pro jectivecaps A b i T have a common p oint

i

ii for b at least

b T

k

b

of the pro jective caps A have a common p oint

i

y A b for all i T Let Pro of Lemma iii implies that

i

b

k jfi j p y gj Then by Lemma ii and iii

i

T

X

p y T k bk b

i

i

from which ii follows easily

p

Remark The case T is easier One may directly checkthat A b

p

b Weleave this easy exercise to the reader A

n

Corollary Let kk k k b e norms on IR Then

T

kxk kxk

T

kxk kxk kxk max max max

T

n n n

S S S

T T

Pro of Without loss of generalitywemay assume that

kxk b max

i i

n

S

n

for all i T Let x S for i T besuch that kx k By Claim

i i i

T

there exist an x A Note that x A implies that kxk

i i i

so

T

Y

T

ky k ky k kxk max

T i

n

S

i

Lemma i no w gives the result

For Twehave a b etter result

n

Prop osition Let kk k k b e norms on IR Then

T

q

T

max kxk kxk kxk c minfT ng max kxk max kxk

T T

n n n

S S S

n

Pro of Let x S b e such that

i

kx k b max kxk

i i i i

n

S

Let k b e the dimension of the span of the x s which we assume without loss

i

k

of generalityis IR Then

T T

Y Y

sup jhy x ij exp sup ln jhy x ij

i i

n

k

y S

y S

i i

Z

T

X

lnjhy x ij d y exp

i

k

y S

i

Z

T

Y

exp lnjhy x ij d y

i

k

y S

i

Z

T

Y

jhy x ij d y c

i

k

y S

i

p

T

c k

where the last inequality follows from and c is the same constantasin

Lemma The fact already used ab ove that b jhy x ijky k concludes

i i i

the pro of

Remarks i It should b e clear from the discussion ab ove that the extreme

case in Corollary and Prop osition is attained for the seminorms

kk jhx ij The optimal constantintheright hand side of the inequality

i i

T

of Prop osition is thus C where

T

T

T

Y

C min max jhx xij

T i

n n

x x S xS

T

i

Taking x x to b e orthogonal if T n and an appropriate repitition of

T

eed the geometric an orthogonal basis if Tn and using the inequalitybetw

and arithmetic means one gets easily that C minfT ng ie the

T

constant in Prop osition is optimal except for the choice of the absolute

constant c As one of the referees p ointed out it maybeofinterest to

determine the actual value of C

T

ii A related question is the following Given T and b nd the conguration

n n

of T pro jective caps A fx S bjhx xij gwithcenters x S

i i i

for which the measure of their intersection is minimal We remark that it is

not hard to see that even for Tnthe extremal situation is not when the

T

x are orthogonal Wemayeven have pro jective caps fA g with orthogonal

i i

i

T

u A for some centers with non emptyintersection but such that

i i

i

orthogonal transformations u i T The analogous question for

i

nonpro jective caps was solved byGromovGfor T nFor the b est of

our knowledge the case Tnis still op en

n

iii Let kk b e a norm on IR satisfying then sp ecializing to the two

norms kk and kk wehave the following curious inequalities

ab ab max kxkkxk

n

S

The formal use of Corollary gives in the right side However the case

of two norms ie T is simpler and stronger as we noted in the Remark

p

after Claim Wemayuse instead of T twice in the right

side of the displayed inequality of Corollary which gives a factor of

Simple examples show that can not b e improved even in twodimensional

n case

q averages of norms

In this section we consider averages of norms under unitary rotations

The expression in Theorem is a typical one

We will study a normed spaces equipp ed in addition with an Euclidean

n

norm ie the spaces X IR kk jj where kk is a norm and jj is some

n

xed Euclidean norm on IR which without loss of generalitywe assume is

the canonical one The parameter M M X belowwas intro duced at the

q q

b eginning of the Intro duction Throughout this section as b efore we denote

n n

kxk b kId IR jj IR kkk max

n

S

n

ie the b est p ossible constant in the inequality kxk bjxj x IR

n

Statement For q n and any normed space X IR k k

p p

b b q q

p p

max M c M max M c

q

n n

where c c are some absolute constants Moreover

p

q

b M

q

p

C

M M n

Pro of By the usual concentration inequalities MS

n o

n

x S jkxkM j t exp ct nb

So

Z Z

q

q

jkxk M j d x q t exp ct nb dt

n

S

p

p

q q

Z

q b

cn

q q

p

q s exp s ds C

b n

where C is an absolute constant Thus

p

b q

p

M M kkxkM k C

q

L

q

n

whichgives the right hand side inequality

Toprove the left hand side inequality notice that the unit ball K of X

n

is contained in a symmetric strip of width b Indeed let x S b e such

that kx k b then K fy jjhy x ij bg It follows that for every t

tK fy jhy x ij tbg

and

n o n o

n n

x S kxk t S y S jhy x ij tb

So

o n

n

S kxkt S x

p

t

We shall show b elow that S c n exp cnt b for t b and

b

p

some absolute constant cThus for every t b n b

n o

q

cnt

n

ctexp M t x S kxkt

q

qb

p

b q

p

Cho osing t we get the result

n

p

t

It remains to provethat S c exp cnt b for t b Let n

b

Z

n

I cos d

n

q

p

n see eg ch of MS and then I

n

Z

n

S cos d

I

n

for arcsin tb Hence

s s

Z

n n

n n

S cos d cos

for some

So if t b wecanchose arcsin tb and obtain

p

nt

S c exp cnt b

b

for some absolute constant c and n

Remarks Obviously M b It follows that if b is of order of magnitude

q

larger than M then M is of the same order as b if and only if q is larger

q

than a constanttimesn

It follows from a recent results of La that for every q andevery

normed space X wehave cM M M

q

Let q and let T b e a p ositiveinteger Denote

q

T

X

q

kx k E E

i qT

T

i

T

n

where E is the exp ectation with resp ect to the pro duct measure on S

Lemma Let q n and max fq g There is an absolute

n

constant C such that for any normed space X IR kk j j

p

b q

p

M E C

q qT

nT

In particular there exists some absolute constant C such that if T C

then

E M E

qT q qT

Pro of Since

q

T T

X X

q

kx k kx k

i i

T T

i i

T

n

we get from concentration inequalities for S cf MS that

q

T

X

q

A

Prob t E exp ct nT b kx k

qT i

T

i

for some absolute constant cFollowing the rst part of the previous pro of

weget

q

p

T

X

qb

q

p

E C kx k

qT i

T nT

i

L

q

Thus for q n

q q

p

T T

X X

qb

q q

p

M E C kx k kx k E

q qT i i qT

T T nT

i i

q

L L

By

p

b q

p

M c

q

n

So if

p p

b q qb

p p

c C

n nT

then

p

qb

p

E C

qT

nT

ie if T C then E M E

qT q qT

n

Prop osition Let X IR k k jj and let q For every

b

there exists a constant C dep ending on only such that if T C

M

q

with max fq g then there exist orthogonal op erators u u such

T

that

q

T

X

q

E jxj ku xk E jxj

qT i qT

T

i

n

for all x IR and E M E Moreover one can take

qT q qT

q

ln

C c

n

Pro of Let andletN be a net in S By Lemma of MS N

n

Using again the concentration inequalities can b e chosen with jN j

T

n

on S MS we get for the normalized Haar measure Pr on

T

O n that

q

T

X

q

A

c exp c E nT b E E Pr kU xk

qT qT i

qT

T

i

n

for all x S Hence if

b ln

T c

E

qT

then with p ositive probability fu u g satisfy

T

q

T

X

q

E E ku xk

qT qT i

T

i

for all x in N A standard successive approximation argument gives for

n

all x S as long as

q

ln

b

T c

E

qT

Take

q

ln

C max c C

where C is the constant from Lemma By this lemma if T C bM

q

then M E and hence

q qT

q

ln

b b

c C

M E

q qT

Thus if T C bM we get the result

q

Remark The case q is implicitly contained in BLM The proba

bilistic estimates there are obtained in a dierent form using the fact that

one can estimate the norm of sums of indep endent random variables in

term of the norms of the individual variables A similar pro of can also b e

used here One needs to use a generalization due to Schmuckenschlager S

of the fact concerning norm ab ove to the setting of general norms

We turn now to the study of k X and t X whichweintro duced in the

q

Intro duction

It was proved in MS that for every ndimensional normed space X the

pro duct k X t X is of order n or in other words t X nk X

bM We will study now the relation b etween t X andt X for any

q

q This will provide a similar asymptotic formula in nfor t X

q

Theorem There are absolute constants c c such that for every

n

normed space X IR kk

i if q then

M M

q

c t X X c t X t

q

M M

q q

ii if q then

c t X t X c t X

q

Pro of By Corollary wehaveforevery qthat

s

b

t X

q

AM

q

where

for q

A and s max fqg

c for q

Since t X bM we get the left side inequality

b

Prop osition and Lemma imply that if T C with

M

q

max fq g and C C an absolute constant then there exist orthogonal

op erators u u such that

T

q

T

X

q

E jxj M jxj M jxj E jxj ku xk

qT q q qT i

T

i

n

for all x IR Hence we get that for q

b

t X C

q

M

q

Thus

b M

t X t X

q

M M

q q

averages intersection of rotated b o dies

Let k k b e a norm and K b e the unit ball of this norm Let q and

given orthogonal op erators u u denote

T

q

T

X

q

ku xk and kxk maxku xk kxk

i T i qT

iT

T

i

Toavoid cumb ersome notation we ignore in this notation the concrete

choice of the op erators fu g which of course inuence the resulted norms

i

We shall b e mostly interested in the dep endence of the quantities ab oveon

T Let K denote the unit ball of kk Of course

qT qT

T

K K u

T

i

i

and for q ln T K K eK

T qT T

The question wewould like to study is Given r with M rb

how many orthogonal op erators we need in order to have

K rD

T

More precisely what is a correct order of the minimal number T r such

that there exist T T r orthogonal op erators such that

K rD

T

In the following theorem M denotes the median of the function kk on

n

the sphere S A similar statement with almost identical pro of holds when

M denotes the average of kk in whichcaseM M

Theorem There are absolute constants c C c C such that if

br M then

n

rM n

T r C n log n C exp

rb rb

q

log n

b M

In particular if r M say and then

b n

c n c n

exp T r exp

rb rb

Moreover for

n

T C n log n

rb

q

log n

b M c n



in the case r M and a random or T exp



rb b n

choice u u satises with high probability K rD

T T

n

Pro of Let x S b e suchthat kx k b and let besuchthat

n

For x S and denote by S x the cap cos

rb

n o n o

n n

S x y S y x y S hy xicos

n

where is the geo desic distance on S

n

For to b e chosen later cho ose net N on the sphere S with

n

resp ect to the geo desic distance satisfying jN j cf Lemma

T

of p oints on the sphere the union of the of MS If for some set fx g

i

i

T

caps S x covers N then the union of the caps fS x g covers

i i

i

n

S In this case for any orthogonal op erators u with x u x wehave

i i i

max ku xkb cos

i

iT

r

n

for any x S ie K rD

T

T

Let Pr b e the normalized Haar measure on O n Then

T T

X

Pr x N x S u x Pr x S u x

i i

i i

xN

T

jN j Pr x S u x

T

jN j S x

exp n ln TS x

So if

n ln

TB

S x

then there are u suchthat

i

xk max ku

i

iT

r

n

for all x S To estimate B note that as wesaw in the pro of of State

ment

s

n

n

S x sin

for any Therefore

A S x

s

n

n

sin

then since sin sin for as long as Set

n

and

s

n

n

n

sin A

n n

Hence

n ln n ln lnn

cn ln n

B

n

A A sin

Since cos for wehave

cn ln n

B

n

cos

and we get that for

n

C n log n T

rb

with an absolute constant C there are orthogonal op erators u u such

T

that K rD

T

Toprovethelower b ound let us p oint out that if

T

u K rD

i

i

n

then for S fx S rx int u K g where int K is the interior of K

i i

S

T

n

S covers S So if A S S for

i i

i

n

S x S kxk

r

then T A ie T

A

ie r Recall the concentration inequality Denote

rM M

see for example ch of MS See also Prop osition V in the same b o ok

for a similar inequality when M denotes the average

r

n

fx kxk M bg exp

M

for any Take then since b M

b r

r r

M n rM n

exp exp A

b

rb

Hence

s

n

exp T rM

e

rb

whichproves the theorem

Remarks Let K beastripfx jx j g or a b ounded approximation

p

n and we need at least n rotations to get a of the strip Then Mb

b ounded K This shows that a condition ensuring that Mb is of order of

p

magnitude larger than n is necessary In fact a more careful examination

q

ln n

is the right condition shows that Mbc

n

It may b e instructive to notice that for any xed C the in

equality in the In particular part of the theorem for r in the interval

CM M can b e written as

c k X c k X



rM T r rM

The left hand side inequalitycontinues to hold also for r close but smaller

than M More precisely for r M one has

C exp c k X T r

for C b eing the constant from Theorem and some absolute constant c

Note that in contrast to the exp onen tial b ehavior of T r above if r

M then

b

T r C

M

This easily follows from the main result of BLM see also S together with

the fact that

T

X

max ku xk ku xk

i i

iT

T

i

Averages of quasinorms

We explained the notion of pnorms in section The results of section

can b e applied to the case of pnorms in a similar manner as they were applied

for norms However the use of the ptriangle inequality leads sometimes to

gaps b etween the upp er and the lower estimates We still get some interesting

versions of the convex case As the pro ofs are quite similar to the resp ective

ones in the convex case we do not rep eat them here The real dierence

between the pro ofs here and those in the convex case is the use of the non

linear separation result Lemma in the pro of of Theorem and

through it in Corollary b elow

n

Claim Let q n and let X IR k k be a pnormed space

Then

p

p

q b

q

p

M max M c max M c b

q

n n

and

p

cb q

p

E M max E

qT q qT

nT

where max fq g and c c c dep end on p only

Let us p oint out that using ideas of La one can get that for every

q and every pnormed space X

cM M M

q

where c dep ends on p but not on q L chapter In La this is proved

in the normed case

Note also that if kk is pnorm then for every integer T

q

T

X

q

kxk ku xk

qT i

T

i

is snorm for s minfp q gFor snorm it is more convenient to use concen

s

tration inequalities not for the function kxk but for the function kxk

qT

qT

Thus if we dene

s

s

L E kxk

qT

qT

T

where E is the exp ectation with resp ect to the pro duct measure on O n

whichisequivalentto E by Latalas theorem then rep eating the pro of of

qT

Prop osition we obtain

n

Prop osition Let q p and X IR kk be a pnormed

space For denote

s

s

c

log C

s

b

where s minfp q gIfT C with max fq g then there are

M

q

orthogonal op erators u u such that

T

q

T

X

q

L jxj ku xk L jxj

qT i qT

T

i

n

for all x IR

These statements allow us to extend the results of MS and of the pre

vious section concerning the relation b etween k X t X and t X inthe

q

following way

n

Corollary Let q Let X IR kk be a pnormed space for

some p Denote s minfp q g then there exists an absolute constant

csuch that for q

qp q

c b

t X

q

p M

q

and for q

s

c b

t X

q

s M

q

As we noted after Claim in the second case q the term bM can

q

b e estimated by c bM where c is a constant dep ending on p only

p p

The pro of of this fact is analogous to the pro of of Theorem

The following fact is a corollary of Theorem Recall that for the

pconvex part of Theorem a crucial role was played by the nonlinear

form of HahnBanach theorem for pconvex sets Lemma

n

Corollary Let X IR kk be a pnormed space for some p

Let qThen

b

t X

q

AM

q

where

p

p

q for q

p

p

and A C pC q

p p

for q

p p

for C p and C q dened in Theorem

n

Claim Let X IR kk be a pnormed space for some p

Then there exists a constant C p dep ending on p only such that

p

p

M M

k X n C p n

b b

Pro of The standard concentrationphenomena metho ds MS on the

sphere implies the lower b ound This fact was already used bySJDilworth

in D

Using the same scheme as in the pro of of Theorem b of MS and

pconvexityof kk we get that

p

n

kxk c jxj

k X

n

for all x IR Thatproves the upp er b ound

We conclude this section with a variant of Theorem for pnorms

n

Prop osition Let X IR kk be a pnormed space for some p

There are absolute constants C C such that if br M then

p

n

p

rM n

p

T r C n log n c rb C exp

p

p

rb

M

p

where c p In particular if c b r M say and

p p

b

p

p

log n

then there are constants c c dep ending on p onlysuch that

p p

n

p

p

p

n rb T r exp c n rb exp c

p p

Pro of The pro of of this prop osition essentially rep eats the pro of of Theo

rem To prove the upp er b ound we only need to substitute the inclusion

n n

x S kxk x S jhx x ij

r rb

with the inclusion

p

p

n n

x S jhx x ij x S kxk

r c rb

p

whichfollows from Lemma In other words in the pro of of Theorem

p

p

such that cos one should chose

c rb

p

Toprovethelower b ound wehave to apply concentration inequalities to

p

the function kk

Averages of quasiconvex b o dies in M p osition

n

Recall that for any subsets K K of IR the covering number N K K

n

is the smallest number N such that there are N p oints y y in IR such

N

that

N

K y K

i

i

Dene the volume radius r of a starb o dy K by the formula jK j jrDj

ndimensional volume of K where jK j denotes the

Let C for p b e the constant from J Bastero J Bernues and

p

A Penas extension BBP of the second named authors reverse Brunn

Minkowski inequality for pconvex b o dies ie

cp

C

p

p

see L for the dep endence of the constanton p Let us recall this result

Theorem Let p For all n and all symmetric pconvex

n n n

bodies K K IR there exists a linear op erator U IR IR with

jdetU j and

n n n

jUK K j C jK j jK j

p

In terms of covering numb ers this theorem can b e formulated in the fol

lowing way

n

olume Theorem For every symmetric pconvex body K in IR with v

n n

radius r there exists a linear op erator U IR IR with jdetU j such

that

p

for t C exp n C t

p p

p

max fN UKtrDNrD tU K g

exp n ln C t for tC

p p

n

for t C

p

If the op erator U in Theorem can b e taken to b e the identity op erator

then wesay that the b o dy K is in M p osition

K K K are in M p osition then as in the Remark If the b o dies

l

convex case P pp

n p n n n

jK K K j C l jK j jK j jK j

l p l

The following theorem has b een recently proved in MS

n

Theorem Let kk b e a norm on IR and assume that its unit ball

K is in M p osition Assume further that for some T orthogonal op erators

u u and for some constant C

T

T

X

jxj ku xkC jxj

i

T

i

n

for all x IR Then there are an orthogonal op erator u and a constant C

dep ending on T and C only such that for some R

Rjxjkxk kuxkC Rjxj

n

for all x IR

By duality this theorem is equivalent to the following statement

Theorem Let a symmetric convex body K be in M p osition Assume

that for some T orthogonal op erators u u and for some constant C

T

T

X

D u K CD

i

T

i

Then there exist an orthogonal op erator u and a constant C dep ending on

T and C onlysuch that for some R

RD K uK C RD

In this section we shall extend b oth theorems to the quasiconvex case

Because duality arguments can not b e applied in the nonconvex case these

two theorems b ecome dierent statements

Lemma Let q and B Let K b e a star b o dy such that for any

t B

q

B

N K tD exp n

t

Then there exists an orthogonal op erator u suchthat

q

K uK C BD

Remark An immediate corollary is that if a pconvex body K is in M

n

p osition then there exists an orthogonal op erator u such that for all x IR

kxk kuxk jxj

K K

rCp

where r is the volume radius of K Here and everywhere in this section we

cp

denote a function of the typ e p by C p The absolute constant c may

b e dierent in dierent places Therefore the pro duct of two functions of

that typ e is again a function denoted by C p

Pro of of Lemma Let the constant C satisfy

q

B

C

n N

By the denition of covering numb ers for N e there exist fx g such

i

that

N

K x C D

i

i

n

Consider the normalized rotation invariant measure on the sphere RC S

where R will b e sp ecied later Since the measure of the intersection

n

x C D RC S

i

do es not exceed

r

R n

A exp

for any x MS ch we obtain that if N A then there exists an

i

orthogonal op erator u suchthat

RC ux jx j x C D

i i j

for any i and j But the union of ux C D covers

i

n n

u K RC S uK RC S

Therefore

n

K uK RC S

Take

p

q

p

q B and R C

q

q

B This completes the pro of then N A andRC e

q

Lemma Let B Let K i T b e symmetric pconvex b o dies

i

such that jK j jD j and for any t B

i

q

B

N D tK exp n

i

t

for all i T Then

n

n

f T jD j K

i

i

where

Tp q

f T c B A

for

A min T max

q p

In particular if all the K s are in M p osition then

i

p

Tp

A

f T C p min T

p

This lemma easily follows from the following claim

Claim Under the assumptions of Lemma

q

B

Tp

A N D t K exp n

i

t

i

Pro of Let two starb o dies B and B satisfy

N

B x B

i

i

Then it is not hard to see that there are p oints y in B such that

i

N

B y B B

i

i

Indeed

N

B x B B

i

i

and if

z B x B

i i

then

B x B z B B

i i

Analogously

B x B x B B B

for any x B Hence using pconvexity and the assumptions of the claim

we get from that for N N D t K andforx D

i

N N

p p

D x t K D D x t K

i i

i i

T

p

we get by and For N N D t K and for y D t K

i

that

N N



p p

A A

t K y D t K x D

j i

j i

N N



p p

z D t K t K

i

i

e get for z x y Continuing in this wayw

i j k

N N

T

T Tp T p T p

D v D t K t K t K

i T T

i

for some v s where

i

q

B

N N D t K exp n

i i i

t

i

Setting

ip

t t

i

we obtain that

T

Y

Tp

N D t K N D t K

i i i

i

qT q

X

B B

q ip

exp n exp n A

t t

i

Lemma Let K i T b e symmetric pconvex b o dies in M

i

n

p osition Let kk denote the gauge of K Assume that for any x IR

i i

jxj max kxk

i

iT

Then there exist a number k fTg and an orthogonal op erator u such

that

f T

jxjkxk kuxk

k k

C p

where f T was dened in Lemma

Pro of Since jxjmax kxk

iT i

K D

i

Let r r denote the volume radii of K K and

T T

r minr

k i

iT

Then by Lemma

n

T

T

n

K

i

j K j

i

r

i

f T

n n

jD j r jD j r

k k

Hence by Lemma there exists an orthogonal op erator such that

jxj jxj

kxk kuxk

K K

r C p f T C p

k

Nowwe are ready to extend Theorem to the quasiconvex case

n

Theorem Let kk be a pnorm on IR and assume that its unit ball

K is in M p osition Assume further that for some T orthogonal op erators

u u and for some constant C

T

p

X

p

jxj jjjxjjj ku xk C jxj

i

T

i

n

for all x IR Then there exists an orthogonal op erator u suchthat

f T

p

jxj kxk kuxk T C jxj

C p

This theorem is a direct consequence of the previous lemma To extend

Theorem we need the following two lemmas

Lemma Let B Let K b e a symmetric pconvex body suchthat

for any t B

q

B

N rD tK exp n

t

where r is the volume radius of K Then there exists an orthogonal op erator

u suchthat

q

C p B c

K uK D

r

The pro of of this lemma is almost identical to the pro of of Theorem of

q

C p LMP Note also that if the b o dy K is in M p osition then B c

q

and C p B c can b e replaced by C p

Lemma Let K i T b e symmetric pconvex b o dies in M

i

p osition Assume that for some constant C

T

X

K D D

i

C T

i

Then there exist a number k fTg and an orthogonal op erator u such

that

p

D CCp T K uK

k k

Pro of Let r r denote volume radii of K K and

T T

r max r

k i

iT

By the assumption of the lemma and the remark after Theorem wehave

n

P

P

T

T

K T

i

r

i

i

i

p p

C p T C p T r

k

n

C T

jD j

Therefore by Lemma weget

C p

p

K uK CCp T K uK D

k k k k

r

k

for some orthogonal op erator u

This lemmagives us the following extension of Theorem

Theorem Let a symmetricconvex body K be in M p osition Assume

that for some T orthogonal op erators u u and for some constant C

T

T

X

D u K CD

i

T

i

Then there exists an orthogonal op erator u suchthat

D K uK TD

p

CCp T

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AE Litvak and VD Milman G Schechtman

Department of Mathematics Department of Theoretical Mathematics

Tel Aviv University The Weizmann Institute of Science

Ramat Aviv Israel Rehovot Israel

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