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Averages of Norms and Quasi-Norms

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Averages of Norms and Quasi-Norms 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
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