Concentration on the l n p ball

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1 Cocetratio o the l p ball Gideo chechtma Joel Zi Abstract We prove a cocetratio iequality for fuctios, Lipschitz with respect to the Euclidea metric, o the ball of l p,1 p<equipped with the ormalized Lebesgue measure. 1 Itroductio I [Z] the authors proved a iequality which ca be iterpreted as givig the right order of the tail distributio of the l q orm o the l p ball equipped with the ormalized Lebesgue measure. More precisely ad specializig to the case of q =adp= 1, we proved there that µ({x; x >t} is bouded above by C exp( ct for t>t/ (where C, c ad T are absolute costats ad bouded from below by a similar quatity (with differet absolute costats. The measure µ ca be either the ormalized Lebesgue measure o B1 - the ball of l 1 or the ormalized Lebesgue measure o B 1 - the sphere of l 1.(Forotherp-s the relevat measure o the sphere is a differet oe tha the usual surface measure. Followig a questio of M. Gromov, we geeralize here this iequality so as to give the right deviatio iequality for a geeral Lipschitz fuctio with respect to the Euclidea metric ad for ay deviatio, i.e., ot oly for large eough t. More precisely we prove i Theorem 3.1 that µ({x : f(x fdµ >t} Cexp( ct For some absolute positive costats C, c ad for all t>. We also prove a similar result for µ replaced with that ormalized Lebesgue measure o the l p ball, 1 <p<. Here the right had side of the iequality above takes the form C exp( ctp. This is treated i ectio 4. omewhat surprisigly, for the fuctio f(x = x we get better cosetratio results tha for a geeral Lipschitz fuctio. This is doe i sectio 5. upported i part by BF upported i part by BF ad NF 1

2 elimiaries Let λ deote the ormalized Lebesgue measure o B 1 - the uit sphere of l 1. We recall the followig kow lemma (a equivalet versio of which was also used ad proved i [Z]. Lemma.1 Let X 1,X,...,X be idepedet radom variables each with desity fuctio 1 e t ad put = ( i=1 X i. The ( X 1, X X,..., iduces the measure λ o B 1. Moreover, ( X 1, X X,..., is idepedet of. We also recall the followig fact which is essetially Theorem 3.1 i [Z] (combie the statemet of (1 of Theorem 3 with the last lie o page. Theorem. ([Z] There are absolute positive costats T,c such that for all t>t/, puttig X =(X 1,X,...,X, ( X / > t exp( ct. We shall also make a essetial use of a theorem of Talagrad [Tal] givig a fie deviatio iequality, with respect to the probability measure iduced by X o R, for fuctios which have cotrolled Lipschitz costats with respect to both the l ad the l 1 orms. Bobkov ad Ledoux [BL] discovered a relatively simple proof ad a far reachig geeralizatio of this iequality. We also refer to the lecture otes [Led] by Ledoux which gives a very ice treatmet of this ad related iequalities. Theorem.3 ([Tal],[BL] Let F : R R be a fuctio satisfyig The F (x F (y α x y ad F (x F (y β x y 1 ( F (x EF >r Cexp( c mi(r/β, r /α for some absolute positive costats C, c ad all r>. I particular, ( 1 Cexp( c >r mi(r, r. We refer to [Led], (4.3 o p. 53 from which a similar iequality, with EF replaced by the media of F follows immediately. Replacig the media with the mea is stadard (see e.g. [M] op V.4, page 14. The I particular part follows from the fact that the fuctio F (x 1,...,x = 1 i=1 x i satisfies F (x F (y 1 x y ad F (x F (y 1 x y 1. We shall also use the followig simple corollary to Theorem.3. Corollary.4 For some absolute positive costats C, c ad all r>, ( 1 Cexp( c >r r

3 3 The mai result Theorem 3.1 There exist positive costats C, c such that if f : B1 R satisfies f(x f(y x y for all x, y B1 the, for all t>, λ({x: f(x E(f >t} Cexp( ct. Remark: Cosiderig fuctios that deped oly o the first 1 variables, we easily get a similar statemet for the full ball B1 1. The mai techical part of the proof of theorem 3.1 is cotaied i the followig lemma. The probability distributio i its statemet is the oe itroduced i the previous sectio. Lemma 3. Put X =(X 1,X,...,X. The for some absolute positive costats C, c ad for every Lipschitz fuctio with Lipschitz costat 1 ( ( ( X X f f >t Ce ct, for all <t.. oof: ice f has Lipschitz costat 1, ( ( ( ( X X X f f >t 1 >t. We distiguish betwee two cases. Case 1. t T,whereT is from Theorem.. Usig first Corollary.4 ad the Theorem. ( ( X 1 >t = E X 1 > t X CE X exp( ct X / T/ ct C exp( ct du (1 u u ct + C exp( ct exp( cudu. ( u u T/ Here ad elsewhere i this ote C ad c deote absolute costats, ot ecessarily the same i each istace. The first summad, (1, is equal to C exp( ct T/ u = Cexp( c t. For the secod summad, (, observe that the maximum of exp( ct cu occurs whe u = t/ ad at u this poit the maximum is: exp( c t 3/4. Hece ( is domiated by C exp( c t 3/4 ct C t exp( c t 3/4. T/ u 3

4 ice, for t T, t 1/ 3/4 T 1/ t, the last quatity is bouded by Ctexp( ct. ice we may assume that t 1, this quatity is bouded by C exp( ct. Case. We write T t. ( X 1 >t ( X = 1 >tad X ( X + 1 t >tad X >t. (3 Usig Lemma.1 ad Theorem.3, ( X 1 >tad X t ( ( ct E X exp I X X t t ( t C u exp( ct u u X t t C u exp( ct u du. t du Now, / While if T is large eough, t u exp( ct/u du = C exp( ct3/. t 1/ t exp( ct/u du u t t exp( c du 1/ u = te c [ 1/t] 3/ te c Ce c. This takes care of the first summad i (3. For the secod summad we use a similar lie of iequalities to that of case 1, usig Corollary.4 ad Theorem.. ( X 1 >tad X >t C t t u C exp( ct C exp( ct. exp( ct cu du u t exp( ct u u t du oof of Theorem 3.1: Give a fuctio o B1 which is Lipschitz with costat oe, exted it to a fuctio o all of R satisfyig the same Lipschitz coditio. (There are may ways to do it, for example f(x =if y B 1 (f(y+ x y. We shall cotiue to call the exteded fuctio f. 4

5 Theorem.3 implies easily that it Y is a idepedet copy of X =(X 1,...,X the for all t> ( ( ( X Y f f >t Ce ct. Combiig this with Lemma 3. ad Lemma.1 we get that λ λ ({(x, y : f(x f(y >t Ce ct from which the statemet of the theorem follows by stadard argumets (see e.g. [M] op V.4, page 14. Next we show i opositio 3.4 below that, for each fixed t 1/4, the result of Theorem 3.1 is best possible, except for the choice of the uiversal costats c, C. Note that the fuctio f(x = x =max{ x 1,..., x } is Lipschitz with costat oe. We first prove a somewhat weaker result whose proof is much simpler. opositio 3.3 For each log/ t 1/4, ({ } λ x : x dλ >t e 4t. oof: Note first that by Lemma.1, dλ = E(max{ X 1,..., X }/E. Itisoweasy to deduce that dλ < log/. Thus, ( ( λ {x: x dλ } >t max X i ( log / + t i ( ( X 1 t X 1 t X i 1 t = Ee t 1 t P i= X i =(1 t 1 e t( 1 1 t e 4t. i= opositio 3.4 For each t 1/4, ({ λ x : x } dλ >t ce 4t for some absolute costat c>. oof: By opositio 3.3 it is eough to assume t<log/. LetM= X,α=t+ EM < 1,β = α ad let 1 α k = k i=1 X i. ( ( M P >α P (X >α P(X 1,X >α=i II. 5

6 But, while, Factorig, II = = = I = P (X > α 1 α 1 =Eexp( β 1 = (Eexp( βx 1 1 = ( 1+β 1 ( P(X 1 βx >β,x βx 1 >β ( P((1 β X 1, (1 β X > (β + β ( ( ( (E exp( ( β + β 1 1 β X = 1+ (β +β 1 β ( ( ( ( 1 β 1. 1+β 1+β I II (1 + β 1 [ 1 (1 α 3]. (4 Notice that = (1 α 1 α exp( ( 1 (1 + β 1 1 α α = exp( exp( (t/(1 α exp( EM/(1 α 1 α α α log 1 α exp( exp( (t/(1 α exp( (EM. 1 α 1 α But, EM = = P (M >tdt = 1 u 1 u du = 1 [ 1 (1 exp( t ] dt u j du = j= 1 j= 1 j +1, so EM log coverges to a positive limit (Euler s costat. ice both α 1 α ted to 1, we get from (4 that I II c (1 (1 α 3 e 4t c (1 exp( α( 3 e 4t. ad exp( α 1 α Hece, it suffices to have ( 3α log or t log EM +3α. This last quatity is asymptotically egative, sice log EM, is asymptotically egative ad α. 6

7 4 cocetratio o the l p ball, 1 <p< Theorem 3.1 ad the well kow cocetratio estimate for Lipschitz fuctio o the Euclidea sphere or ball (which is of the form C exp( ct suggest that a similar result with estimate C exp( ct p holds o the l p ball, 1 <p<. This is ideed the case but the proof seems to require a very ew result due to Lata la ad Oleszkiewicz [LO]. Actually [LO] was motivated i part by a questio of the authors whose motivatio was Theorem 4.1 below. ice the proof is very similar to that of the case p = 1 we oly sketch it. Theorem 4.1 There exist positive costats C, c such that if 1 <p<ad f : Bp satisfies f(x f(y x y for all x, y Bp the, for all t>, R λ({x: f(x E(f >t} Cexp( ct p. Here λ deotes the measure o the l p sphere which asig to each set A the ormalized Lebegue measure, o the l p ball, of the set {ta : t 1}. It is the easy to deduce the same result (with differet costats for the ormalized Lebegue measure o the l p ball. As we idicated above, the role of Theorem.3 will be replaced here with the followig result which is a combiatio of Theorems 1 ad of [LO]. Theorem 4. ([LO] Let F : R R be a fuctio satisfyig F (x F (y α x y, let deotes the probability distributio o R with desity c p exp( x 1 p x p ad deote =( X 1 p + + X p 1/p. The ( F (X EF >r Cexp( c(r/α p for some absolute positive costats C, c ad all r>. I particular, ( E 1/p 1/p Cexp( c >r p/ r p. The aalogues of Lemma.1 ad Theorem. also hold, i.e. with the ew iterpretatio of, λ ad, ( X 1, X X,..., iduces the measure λ o B p, ad ( X 1, X X,..., is idepedet of. Ad Theorem 4.3 ([Z] There are absolute positive costats T,c such that for all t>t/ 1/ 1/p, puttig X =(X 1,X,...,X, ( X / > t exp( ct p. Deote α = α(, p = E ad ote that α(, p is bouded away from zero ad by 1/p uiversal costats, for 1 < p <. As for the case p = 1, Theorem 4 will follow (usig Theorem 4. oce we establish Lemma 4.4 For every Lipschitz fuctio f with costat 1 ( ( ( X αx f f >t 1/p Ce ctp For some absolute positive costats C, c ad for all <t. 7

8 ketch of oof: As i the proof of Lemma 3., the proof reduces to estimatig ( ( X α =E X 1/p >t α > t X which by Theorem 4. is domiated by CE X exp( ctp p/ ( X / =C pct p p/ p Case 1. t T 1/ 1/p. The right had side of (5 is domiated by exp( ctp p/ u p+1 u p ( X >u du. (5 T 1/ 1/p pct p p/ exp( ctp p/ pct p p/ du + exp( c( tp p/ + u p du u p+1 u p T u 1/ 1/p p+1 u p = I + II. I is equal to exp( c t p while i II the itegrad is domiated by ctp p/ exp( c t p/ p/4+1/ u p+1 ad thus II is domoated by t p exp( ct p/ p/4+1/. The boud t T 1/ 1/p implies ow that II is also domiated by exp( c t p. Case. t>t 1/ 1/p. write the right had side of (5 as C ( T 1/ 1/p t + + T 1/ 1/p t ( pct p p/ exp( ctp p/ X u p+1 u p >u du ( Estimatig X >u by 1, I is bouded by exp( c t p. II is bouded by = I + II + III. (6 t exp( c p/ pct p p/ du ct p exp( c p/ exp( c p/. T u 1/ 1/p p+1 Fially III of (6 is domiated by pct p p/ exp( c( tp p/ + u p du exp( ct p pct p p/ exp( c tp p/ du t u p+1 u p t u p+1 u p < exp( ct p. 5 The fuctio f(x = x As we remarked i the itroductio, for t larger tha a absolute costat divided by,the coclusio of Theorem 3.1 is best possible for the fuctio f(x = x. It turs out that for smaller values of t a stroger iequality holds. 8

9 opositio 5.1 Let t, for t> 1/ α(, t = 3/4 t 1/, for 5/6 <t 1/ t, for <t 5/6. The, for some absolute costats <c,c<, ({ } λ x : x dλ >t Ce cα(,t. For t>t/ the propositio was proved i [Z]. For the lower values of t, as i the proof of the mai result, it is eough to prove ( X 1 Ce >t cα(,t (7 ad, for Y idepedet of X, ( X Y >t Ce cα(,t. (8 oof of (7: AsiCase1itheproofofLemma3.(usigTheorem.3isteadofCorollary.4, ( ( X 1 >t = E X 1 > t ( ( X ( ( t t CE X exp c. X X Now, E X [ ( ( ( ( t t exp c X X ] 1 X t e c while by Theorem., E X [ ( ( ( ( t t exp c X X ( ct ct /u X = e >u t u 3 T/ ct ct /u e + u 3 T/ 1 X >t ] ct ct /u e e cu du. (9 u 3 The first summad i (9 is equal to e c t /T. Noticig that the maximum of e ct /u e cu occurs at u = 1/3 t /3 ad is equal to e c t /3, we get that the secod summad i (9 is domiated by e c t /3 T/ ct du = c t u 3 T t/3 e c C e c t /3 9

10 for the relevat rage of t (i.e., t larger tha a absolute costat times 1/. ummarizig, we get that (with differet absolute costats c, C ( X 1 Ce >t c t + Ce ct/3. (1 ad it is easy to see that this is domiated by Ce cα(,t. oofof(8:first ote that by Theorem.3, ( Y <c <Ce c for a appropriate C ad that is larger tha a costat times α(, t i the rage i questio. Thus, ( X Y >t = ( X > Y + Y t + t Ce cα(,t + ( X > Y +c3/ t (11 ad it is eough to prove that ( (Xi Y i >c3/ t i=1 Ce cα(,t. (1 To prove P (1 ote that for each p the left had side of (1 is domiated by a absolute costat times E ɛ i X i p. Apply ow the result of Hitczeko, Motgomery-mith ad Oleszkiewicz ( 3/ t p [HMO, Th. 4.] to the variables Xi ad get that, for each p, the left had side of (1 is domiated by ( C( 1/p (p p 1/ 1/ p. (13 3/ t For t larger tha 5/6 take p = c 3/4 t 1/ for c small with respect to C. The (13 is domiated by C e c 3/4 t 1/.Fortsmaller tha 5/6 but larger tha a costat times 1/ take p = c t ad get that (13 is domiated by C e c t. Remarks: 1. The use of [HMO] was suggested to us by. Kwapie. The result of [HMO] was cosiderably geeralized by Lata la [Lat]. Origially we had aother (more direct but also more special to the variables i questio proof of (1: After appropriately trucatig the X i s evaluate exp(t ɛ i Xi 1 Xi <a by carefully evaluatig the terms i the Taylor expasio. This follows closely [Bou, Lemma 1].. Is the statemet of Lemma 5.1 best possible? for t>c 1/ this was proved i [Z]. For 5/6 <t<c 1/ this is still the case. This easily follows from the followig three facts: a. The proof of (7 i this case gives a estimate, (1 which is of better order of magitude tha e cα(,t. b. The iequalities i (11 ca be iverted (with differet costats of course. c. ( ɛ i Xi >c3/ t ( X1 >c3/ t =e 3/4 t 1/. i=1 For 1 <t< 5/6 oe ca show the right lower boud for the quatity i the left ed sides of (11 but it does t seem to combie icely with upper boud for (7. 1

11 Refereces [BL] [Bou] [GM]. Bobkov ad M. Ledoux, Poicaré s iequalities ad Talagrad s cocetratio pheomeo for the expoetial distributio, obab. Theory Related Fields 17 ( J. Bourgai, Radom poits i isotropic covex sets, Covex geometric aalysis (Berkeley, CA, 1996 Math. ci. Res. Ist. Publ. 34, Cambridge Uiv. ess, Cambridge (1999 M. Gromov ad V. D. Milma, A topological applicatio of the isoperimetric iequality, Amer. J. Math. 15 ( [HMO] P. Hitczeko,.J. Motgomery-mith ad K. Oleszkiewicz, Momet iequalities for sums of certai idepedet symmetric radom variables tudia Math. 13 ( [Lat] [LO] [Led] [M] R. Lata la, Estimatio of momets of sums of idepedet real radom variables A. obab. 5 ( R. Lata la ad K. Oleszkiewicz, Betwee obolev ad Poicare, This volume. M. Ledoux, Cocetratio of measure ad logarithmic obolev iequalities, eprit (1997. V. D. Milma ad G. chechtma, Asymptotic theory of fiite dimesioal ormed spaces (Lecture Notes i Mathematics, 1. priger-verlag, Berli-New York, [Z] G. chechtma ad J. Zi, O the volume of the itersectio of two L p A.M.. 11 ( balls, oc. [cc] G. chechtma ad M. chmuckeschläger, Aother remark o the volume of the itersectio of two L p balls, Geometric aspects of fuctioal aalysis (1989 9, Lecture Notes i Math., priger 1469 ( [Tal] M. Talagrad, A ew isoperimetric iequality ad the cocetratio of measure pheomeo, Geometric aspects of fuctioal aalysis (1989 9, Lecture Notes i Math., priger 1469 ( Departmet of Mathematics Weizma Istitute of ciece Rehovot, Israel gideo@wisdom.weizma.ac.il 11

12 Departmet of Mathematics Texas A&M uiversity College tatio, Texas 1