ON THE COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES

Transcription

1 ON THE COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES MARK RUDELSON Abstrat We show that for ay 1 t 1/2 log 5/2, the set of uoditioal ovex bodies i R otais a t-separated subset of ardiality at least exp exp t 2 log t This implies the existee of a uoditioal ovex body i R whih aot be approximated withi the distae d by a projetio of a polytope with N faes uless N expd We also show that for t 2, the ardiality of a t-separated set of ompletely symmetri bodies i R does ot exeed exp exp C log2 log t 1 itrodutio I [1] Barviok ad Veomett posed a questio whether ay -dimesioal ovex symmetri body a be approximated by a projetio of a setio of a simplex whose dimesio is subexpoetial i The importae of this questio stems from the fat that the ovex bodies geerated this way allow a effiiet ostrutio of the membership orale The questio of Barviok ad Veomett has bee aswered i [4], where it was show that for all 1 N, there exists a -dimesioal symmetri ovex body B suh that for every - dimesioal ovex body K obtaied as a projetio of a setio of a N-dimesioal simplex oe has db, K l 2N l2n where d, deotes the Baah-Mazur distae ad is a absolute positive ostat Moreover, this result is sharp up to a logarithmi fator Departmet of Mathematis, Uiversity of Mihiga Partially supported by NSF grats DMS , DMS , ad USAF Grat FA ,

2 2 MARK RUDELSON Oe of the mai steps i the proof of this result was a estimate of the omplexity of the set of all ovex symmetri bodies i R, i e, the Mikowski or Baah Mazur ompatum The omplexity is measured i terms of the maximal size of a t-separated set with respet to the Baah Mazur distae dk, D = if{λ 1 D T K λd}, where the ifimum is take over all liear operators T : R R A set A i a metri spae X, d is alled t-separated if the distae betwee ay two distit poits of A is at least t It follows from [4] that for ay 1 t, the set of all -dimesioal ovex bodies otais a t-separated subset of ardiality at least 11 expexp/t More preisely, Theorem 23 [4] asserts that for ay 2 M e, there exists a probability measure P M o the set of ovex symmetri polytopes suh that } 12 P M P M {K, K dk, K 2e M lm/ This probabilisti estimate together with the uio boud implies the required lower boud o the maximal size of a t-et Note that for t = O1, the estimate above shows that the omplexity of the Mikowski ompatum is doubly expoetial i terms of the dimesio This fat has bee idepedetly established by Pisier [6], who asked whether a similar statemet holds for the set of all uoditioal ovex bodies ad for the set of all ompletely symmetri bodies We show below that the aswer to the first questio is affirmative, ad to the seod oe egative Cosider uoditioal ovex bodies first A ovex symmetri body K R is alled uoditioal if it symmetri with respet to all oordiate hyperplaes This property a be oveietly reformulated i terms of the orm geerated by K For x R, set x K = mi{a 0 x ak} The body K is uoditioal if for ay x = x je j, ad for ay J [], x j e j x j e j, K K j J j [] ie, wheever all oordiate projetios are otratios

3 COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES 3 Our mai result shows that the omplexity of the set K u of uoditioal ovex bodies at the sale t is doubly expoetial as log as t = O1 More preisely, we prove the followig theorem Theorem 11 Let 1 t 1/2 log 5/2 The set of -dimesioal uoditioal ovex bodies otais a t-separated set of ardiality at least exp exp t 2 log t Here, ad are positive absolute ostats Note that ulike the estimate 11, whih is valid for 1 t, the estimate above holds oly i the rage 1 t 1/2 log 5/2 By a theorem of Lidestrauss ad Szakowski [3], the maximal Baah Mazur distae betwee two -dimesioal uoditioal bodies does ot exeed C 1 ε 0 estimate of the ardiality of a t-separated set i K u for some ε 0 1/3 This meas that a o-trivial is impossible wheever t > 1 ε 0 Followig the derivatio of Theorem 11 [1], oe a show that Theorem 11 implies a result o the hardess of approximatio of a uoditioal ovex body by a projetio of a setio of a simplex refiig the solutio of the problem posed by Barviok ad Veomett Corollary 12 Let N There exists a -dimesioal uoditioal ovex body B, suh that for every -dimesioal ovex body K obtaied as a projetio of a setio of a N-dimesioal simplex oe has 1/4 db, K log 1 log N log N where is a absolute positive ostat I partiular, Corollary 12 meas that to be able to approximate all uoditioal ovex bodies i R by projetios of setios of a N-dimesioal simplex withi the distae O1, oe has to take N exp Cosider ow the set of ompletely symmetri bodies We will all a -dimesioal ovex body ompletely symmetri if it is uoditioal ad ivariat uder all permutatios of the oordiates This term is ot ommoly used I the laguage of ormed spaes, ompletely symmetri ovex bodies orrespod to the spaes with 1-symmetri basis However, sie the term symmetri ovex bodies has a differet meaig, we will use ompletely symmetri for this lass of bodies,

4 4 MARK RUDELSON The set of ompletely symmetri ovex bodies is muh smaller tha the set of all uoditioal oes This maifests quatitatively i the fat that the ardiality of a t-separated set of ompletely symmetri bodies is sigifiatly lower Namely, we prove the followig propositio i Setio 5 Propositio 13 Let t 2 The ardiality of ay t-separated set i K s does ot exeed exp exp C log2 log t This propositio meas, i partiular, that the omplexity of the set of ompletely symmetri ovex bodies is ot doubly expoetial i the dimesio, whih aswers the seod questio of Pisier Akowledgemet The author is grateful to Olivier Guédo for several suggestios whih allowed to larify the presetatio 2 Notatio ad a outlie of the ostrutio Let us list some basi otatio used i the proofs below By P ad E we deote the probability ad the expetatio If N is a atural umber, the [N] deotes the set of all itegers from 1 to N Let 1 p By x p we deote the stadard l p -orm of a vetor x = x 1,, x R : 1/p x p = x j p, ad B p deotes the uit ball of l p If A : R R m is a liear operator, ad K 1, K 2 are ovex symmetri bodies, the A : K 1 K 2 stads for the operator orm of A osidered as a operator betwee ormed spaes with uit balls K 1 ad K 2 : A : K 1 K 2 = max x K 1 Ax K2 The orm A : B2 B2 m is deoted simply by A The Hilbert Shmidt orm of A is m 1/2 A HS = a ij 2 i=1

5 COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES 5 For K 1,, K L R, we deote by absovk 1,, K L their absolute ovex hull L absovk 1,, K L = { λ l x l l=1 L λ l 1, x l K l for l [L]} l=1 Also, we defie the uoditioal ovex hull of the poits x 1,, x L R with oordiates x j = x j 1,, x j by uovx 1,, x L = ov ε 1 1x 1 1,, ε 1 x 1,, ε L 1 x L 1,, ε L x L, where the ovex hull is take over all hoies of ε l i { 1, 1} Obviously, uovx 1,, x L is the smallest uoditioal ovex set otaiig x 1,, x L Fially, C,, 0 et deote absolute ostats whose value may hage from lie to lie The radom ovex bodies K, K appearig i 12 are geeralized Gluski polytopes Suh polytopes were itrodued by Gluski [2] to prove that the diameter of Mikowski ompatum is of the order Ω These polytopes are ostruted as the absolute ovex hull of N idepedet vetors uiformly distributed over S 1 ad a few determiisti uit vetors Suh ostrutio, however, aot be adopted to prove Theorem 11 Ideed, a argumet based o measure oetratio shows that if x 1,, x N, x 1,, x N are idepedet radom vetors uiformly distributed over S 1 ad N, the with high probability duovx 1,, x N, uovx 1,, x N C, makig it impossible to ahieve distaes greater tha O1 Moreover, if N/, the this distae teds to 1, whih does ot allow to prove the doubly expoetial omplexity boud for distaes of order O1 either To avoid the problems arisig i attempts to use the stadard ostrutio of the Gluski polytopes, we give up o the assumptio that the radom vetors are uiformly distributed o the sphere Istead, we fix umber δ > 0 ad N N depedig o the desired distae ad osider idepedet radom sets I 1,, I N [] uiformly hose amog the sets of ardiality δ Here ad below, we assume for simpliity that the umbers δ, /2 et are iteger Alteratively, oe a take the iteger part of these umbers For eah l [N], set

6 6 MARK RUDELSON x l = i I l e i The radom ovex body K will be defied as K = KI 1,, I N = absov uovx 1,, x N, δb1, δ B2, Here the saled opies of B1 ad B2 appear oly for tehial reasos, ad the mai role is played by the uoditioal ovex hull of x 1,, x N The mai advatage of this ostrutio is that the distae betwee two idepedet opies of suh bodies a be large ad a be otrolled i terms of δ ad N Oe of the importat features of this ostrutio is that the radom poits x l, l [N] are defied via radom sets I l of a fixed ardiality This meas that the oordiates of x l are ot idepedet A alterative defiitio of radom verties y l = i=1 ν i,le i, where ν i,l, i [], l [N] are idepedet Beroulli radom variables takig value 1 with probability δ would have bee muh easier to work with Yet, with suh defiitio, P ν 1,1 = = ν,1 = 1 = δ, whih is oly expoetially small i This would have made the doubly expoetial boud for probability uattaiable We will show i Setio 4 that the distae betwee two idepedet opies of the polytope K is large with probability lose to 1 This will allow us to derive Theorem 11 by a appliatio of the uio boud The large deviatio ad small ball probability estimates istrumetal for the proof of the mai result of Setio 4 are obtaied i Setio 3 3 Small ball probability ad large deviatio bouds for the liear image of a radom vetor We start with establishig a oetratio estimate for radom quadrati forms similar to the Haso Wright iequality The followig Lemma is based o Theorem 11 [7] Lemma 31 Let J be a radom subset of [] of size m < uiformly hose amog all suh subsets Deote by R J = j j e je T j the oordiate projetio o the set J Let Y = ε 1,, ε be the vetor whose oordiates are idepedet symmetri ±1 Beroulli radom variables The for ay matrix A ad ay t > 0, P Y T R J AR J Y EY T R J AR J Y t [ ] 2 exp t 2 m A 2 t A

7 COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES 7 Proof Let us separate the diagoal ad off-diagoal terms We have P Y T R J AR J Y EY T R J AR J Y t P a jj m a jj t + P ε 2 j ε k a jj t 2 j J =: p 1 + p 2 j J,j k We estimate p 1 ad p 2 separately To estimate p 1, osider a futio F o the permutatio group Π defied by m F π = a πj,πj For k <, deote by A l the algebra of subsets of Π, whose elemets are the sets of permutatios π for whih π1,, πl is the same Let X 0 π = EF π, ad for l m, set X l = E[F π A l ] The sequee X 0,, X m defied this way is a martigale with martigale differees Hee, by Azuma s iequality X l+1 X l max j [] a jj A p 1 = P X m X 0 t/2 2 exp t2 m A 2 The estimate for p 2 follows diretly from Theorem 11 [7] Ideed, let Y J be the oordiate restritio of the vetor Y to the set J Similarly, let A J be the square submatrix of the matrix A whose rows ad olums belog to J Deote by J the diagoal of A J The A J J 2 A ad A J J HS J A J J 2 m A Therefore, by Theorem 11 [7] we have p 2 = P YJ T A J J Y J t/2 = E J P [ YJ T A J J Y J t/2 J ] [ ] 2 exp t 2 m A 2 t A The small ball probability boud follows immediately from Lemma 31 Corollary 32 Let B be a matrix I the otatio of Lemma 31, m P BR J Y 2 2 B HS 2 exp m B 4 HS 2 B 4

8 8 MARK RUDELSON Proof We apply Lemma 31 with A = B T B, ad t = m 2 B 2 HS I this ase, EY T R J AR J Y = m tra = 2t The the left had side of the iequality above a be bouded by [ ] m exp B 4 HS m 2 B 4 B 2 HS B 2 To derive the orollary, ote that B 2 HS B 2, so the first term i the miimum is always smaller tha the seod oe Lemma 31 a be also applied to derive the large deviatio iequality for BR J Y 2 However, the boud obtaied this way will ot be strog eough for our purposes To prove the large deviatio estimate we employ a differet tehique Lemma 33 Let J be a radom subset of [] of size m < uiformly hose amog all suh subsets Deote by R J = j j e je T j the oordiate projetio o the set J Let Y = ε 1,, ε be vetor whose oordiates are idepedet symmetri ±1 Beroulli radom variables The for ay matrix B ad ay t > 4m/ B HS, P BR J Y 2 t 2 exp t2 B 2 Proof Coditio o the set J first Note that BR J B Applyig Talagrad s ovex distae iequality [8], we obtai P BY 2 M s J 2 exp s2 2 B 2, where M is the media of BR J Y 2 Sie M E BR J Y 2 2 1/2 = BR J HS, the previous iequality implies 31 P BR J Y 2 t + BR J HS J 2 exp t2 B 2 Set A = B T B To fiish the proof, we have to obtai a large deviatio boud for the radom variable U := BR J 2 HS = trrt J AR J = j J depedig o J The set J is hose uiformly from the sets of ardiality m, so the elemets of J are ot idepedet To take advatage of idepedee, let us itrodue auxiliary radom variables Let δ 1,, δ be idepedet Beroulli {0, 1} radom variables takig value 1 with probability 2m/ The P δ j < a jj

9 COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES 9 m < 1/2 Cheroff s iequality provides more preise boud for this probability, but this estimate would suffie for our purposes Set Z = F δ 1,, δ = δ j a jj Sie max j [] a jj A, we derive from Berstei s iequality that τ P Z > τ exp A for ay τ 4m tra = 2EZ Compare the radom variables U ad Z Notie that the radom variable Z oditioed o the evet m δ j = m has the same distributio as U Also, for ay m < m, m m P Z > τ δ j = m P Z > τ δ j = m This observatio allows to olude that for ay τ 4m m P Z > τ δ j = m P δ j = m Thus, m m P P U > τ = P Z > τ m m m δ j = m Z > τ 1 P m δ j = m tra, τ P δ j = m exp A 1 m τ τ δ j < m exp 2 exp A A Combiig this with iequality 31, we obtai that for ay t > 4m/ B HS, P BR J Y 2 2t E J P BR J Y 2 t + BR J HS BR J HS t + P BR J 2 HS > t2 t 2 3 exp B 2

10 10 MARK RUDELSON 4 Distae betwee uoditioal radom polytopes We follow the lassial sheme of estimatig the distaes developed for Gluski s polytopes, see eg [5] Fix a matrix V GL Deote the sigular values of V by V = s 1 V s 2 V s V > 0 For the Baah Mazur distae estimate, we a ormalize V by assumig s /2 V 1 ad s /2 V 1 1 Let K ad K be idepedet radom uoditioal ovex bodies, ad let Y be a vertex of K We start with estimatig the probability that V Y dk for some d > 1 For the stadard Gluski polytopes, suh estimate is obtaied by volumetri osideratios I our settig, this argumet is uavailable, ad we use the results of Setio 3 istead Propositio 41 Let δ > C 1/2, ad let N = expδ 2 For l [N], let I l [] be a set of ardiality I l = m = δ Defie a ovex body K by K = KI 1,, I l = absov δb I 1 2,, δb I N 2, δ B 2 Let J [] be a radom subset of ardiality δ uiformly distributed i [], ad let ε 1,, ε be idepedet symmetri ±1 Beroulli radom variables Set Y = ε j e j j J The for ay liear operator V : R R with s /2 V 1 ad log V 1/ δ, P V Y K exp δ 2 δ log V Proof Let V = s j u i v i i=1 be the sigular value deompositio of V The there exists a iterval I = [i 1, i 2 ] [1, /2] of ardiality i 2 i 1 = I =: r 0 log V suh that s i1 /s i2 2 Ideed, otherwise we would have V /2r k=1 s k 1r+1 s kr 2 /2r > V

11 COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES 11 provided that the ostat 0 is hose small eough Set Q = i I s i u i v i ad P = i I u i u i Sie the ratio of the maximal ad the miimal sigular values of Q does ot exeed 2, the operator Q satisfies r Q 2 Q 2 HS r/4 Q 2 Note that for ay α > 0, V Y α K implies QY = P QY = P V Y αp K Corollary 32 applied to B = Q yields P QY 2 s i2 δr exp m 2 r2 δ 2, where the last iequality follows from the defiitio of r ad the assumptio log V 1/ δ For l [N], set E l = spa{p e j : j I l }, ad let P El be the orthogoal projetio oto E l Sie Q = s i1 2s i2, P El Q 2s i2 ad P El Q HS 2s i2 P El HS 2s i2 δ Applyig Lemma 33 with t = Cs i2 δ > 2 δ P El Q HS, we get Let Ω be the evet that P P El QY 2 Cs i2 δ e δ2 1 QY 2 s i2 δr ad 2 P El QY 2 Cs i2 δ for ay l [N] The previous estimates show that 41 P Ω e δ2 + Ne δ2 e δ 2, where we used the assumptio o N with a suffiietly small ostat Sie P B I l 2 B 2 E l, for ay y R, we have max Qy, x = max Qy, P x x B I l 2 x B I l 2 max Qy, u = P u B2 E El Qy 2 l Assume that Ω ours Coditios 1 ad 2 imply 42 P El QY 2 /r δ QY 2

12 12 MARK RUDELSON for all l [N] If QY αp K, the QY 2 2 α max x P K α max l [N] α δ QY, x = α max QY, x x K max x δb I l 2 max l [N] C r αδ QY 2, QY, x + max x δ QY, x B2 P El QY 2 + δ QY 2 where the last iequality holds beause of 42 Combiig this with 1 ad reallig that s i2 1, we obtai α r 1 δ > δ log V if is hose small eough This meas that the evet V Y implies Ω, ad so, the propositio follows from estimate 41 δ log V K Propositio 41 pertais to oe radom vetor Y The body K = KI 1,, I N otais may idepedet opies of Y, ad we a use this idepedee to derive the desired doubly expoetial boud for probability If K ad K are idepedet ovex bodies, deote by P K the probability with respet to K oditioed o K beig fixed Corollary 42 Let δ > C 1/2, ad let N = expδ 2 Let I l, l [N] be idepedet radom subsets of [] uiformly hose amog the sets of ardiality δ For l [N] deote x l = j I l e j Cosider a radom ovex body K = KI 1,, I N = ov uovx 1,, x N, δb1, δ B2, ad let K be a idepedet opy of K Assume that K = KI 1,, I N satisfies N l=1 I l = [] Let V : R R be a liear operator suh that s /2 V 1 The 0 P K V K K exp expδ 2 δ log1/δ Proof Deote for shortess α = δ log1/δ

13 COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES 13 Assume that V K αk Sie K δ B2 have V α/ δ, whih implies log V C log1/δ ad K δb 2, we The oditio o K implies that K KI 1,, I N, where the last set is defied i Propositio 41 Let ε j,l, j [], l [N] be idepedet symmetri ±1 radom variables The the vetors Y l = j I l ε j e j are otaied i K Hee, by Propositio 41, N N P V K αk P V Y l αk exp δ 2 as required l=1 exp expδ 2 Our mai tehial result, Theorem 43 below, will imply Theorem 11 almost immediately log Theorem 43 Let δ > C, ad let N = expδ2 For l [N], let I l [] be a set of ardiality I l = δ For l [N] deote x l = j I l e j Cosider a radom ovex body K = KI 1,, I N = absov uovx 1,, x N, δb1, δ B2, ad let K be a idepedet opy of K The P K,K dk, K 1 δ log 2 exp 1/δ exp 2 δ 2 Proof The proof follows the geeral sheme developed by Gluski Fix K = KI 1,, I suh that l=1 I l = [] Let be a ostat to be hose later Deote by W K the evet W K, K = { V : R R s /2 V 1 ad V K K } δ log1/δ We start with provig the followig Claim P K W K, K exp expδ 2 Set ε = 0, where 2 log1/δ 0 is the ostat from Corollary 42 By Corollary 8 [5], the set of all operators V : R R suh that

14 14 MARK RUDELSON s /2 V δ/ε ad V δb1 K possesses a δ-et N of ardiality 2 N volk / δ C 2, δ volb2 where we used K δb 1 to obtai the last iequality Hee, W K, K possesses a ε-et N = ε/ δn of ardiality C 2 Assume ow that there exists a operator V : R R suh that s /2 V 1 ad V K ε/ δk Let V 0 N be suh that V V 0 < ε The V 0 : K K V : K K + V V 0 : K K ε + V V 0 : δb2 δ B2 δ This meas that 2ε δ = 0 δ log1/δ P K W K, K P K V 0 N V 0 K N max V 0 N P KV 0 K 0 δ log1/δ K 0 δ log1/δ K Combiig the boud for N appearig above with Corollary 42, we show that this probability does ot exeed exp expδ 2 provided log that δ > C for a suffiietly large C This ompletes the proof of the Claim To derive the Theorem from the Claim, ote that the iequality dk, K guaratees the existee of a liear operator V : δ log 2 1/δ R R suh that V : K K V 1 : K K δ log 2 1/δ Without loss of geerality, we may assume that s /2 V 1 ad s /2 V 1 1 This meas that the evet W K, K ours with operator V, or W K, K ours with V 1 Also, P K N I l [] l=1 1 δ N exp expδ 2

15 COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES 15 Therefore, P K,K dk, K δ log 2 1/δ N N E K P K W K, K I l = [] + P K I l [] + E K P K W K, K 4 exp expδ 2 Theorem 43 is proved Proof of Theorem 11 Let δ = l=1 l=1 N N I l = [] + P K I l [] l=1 1 t log t, where 1 is the ostat from Theorem 43 Choosig the ostat i the formulatio of Theorem 11 suffiietly small, we esure that the oditio δ > C log M = exp exp 2 4 δ2 l=1 holds, ad Theorem 43 applies Set = exp exp t 2 log t, with some ostat > 0 Cosider M idepedet uoditioal radom ovex bodies K 1,, K M whih are ostruted as i Theorem 43 By this theorem ad the uio boud, P m, m [M] m m, dk m, K m δ log 2 1/δ M 2 exp exp 2 δ 2 2 exp exp 2 δ2 This iequality implies that K u otais a t-separated set of ardiality at least M We ow pass to the proof of Corollary 12 Sie this orollary follows from Theorem 11 ad [4], we will provide oly a sketh of the proof istead of a omplete argumet Proof of Corollary 12 sketh Fix m, m N Followig the proof of Theorem 11 [4], we estimate of the ardiality of a speial 2- et i the set of all -dimesioal projetios of m-dimesioal setios

16 16 MARK RUDELSON of the simplex N R N By Lemmas 31, 32 [4], the setios of the N- dimesioal simplex a be eoded by the poits of the Grassmaia G N+1,m so that a ε-et A 1 o the Grassmaia orrespods to a 1 + εm N et N 1 i the set of the setios of the simplex i the Baah-Mazur distae Similarly, by Lemma 33 [4], for ay K N 1 we a eode all -dimesioal projetios of K by poits of the Grassmai G m, so that a ε-et A 2 o this Grassmaia orrespods to 1+εm N et i the set of the projetios of K i the Baah- Mazur distae Combiig these two results, we see that the poits of A 1 A 2 orrespod to some 1 + εm N et i the set of the projetios of of the setios of the simplex The ets A 1, A 2 a be hose so that A 1 A 2 C[N+1m+m] C exp C N 2 log C ε ε Choosig ε suh that 1+εm N = 2, we derive that there exists a 2-et M m i the set of all -dimesioal projetios of m-dimesioal setios of N of ardiality M m expn 2 log N Settig M = N m=m m, we obtai a 2-et i the set of all -dimesioal projetios of setios of N satisfyig a similar estimate If ay -dimesioal uoditioal ovex body a be approximated by a projetio of a setio of N withi the distae d, the M is a 2d-et i the set K u This meas that the ardiality of ay 2d 2 -separated set i K u does ot exeed expn 2 log N Let us show that this implies the desired lower boud o d Assume that 1/4 d log 1, log N log N where the ostat > 0 will be hose later If is suffiietly small, the the iequality 2d 2 1/2 log 5/2 holds, ad so Theorem 11 applies By this theorem, there exists a 2d 2 -separated set of ardiality at least exp exp 2d 4 log 4 2d 2 + 1

17 COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES 17 Combiig this with the upper estimate for this ardiality proved above, we obtai 2d 4 log 4 3 log N 2d This otradits our assumptio o d if the ostat is hose suffiietly small 5 Complexity of the set of ompletely symmetri bodies I this setio, we prove Propositio 13 establishig the upper boud o the ardiality of a t-separated set i the set of all ompletely symmetri bodies Deote the set of all ompletely symmetri bodies i R by K s Proof Fix 1 < τ < Let L N be the smallest umber suh that 51 τ L < 1 τ 1 Deote by Y the set of all o-dereasig futios ψ : [L] [] suh that ψ1 < ψ Note that + L Y + L L L Defie a futio Φ : K s R Y by L Φ ψ K = τ l l=1 ψl 1<j ψl e j, K where we use the ovetio ψ0 = 0 Assume that K, D K s We will prove that if for all ψ Y, the τ 1 Φ ψ D Φ ψ K τφ ψ D 52 dk, D τ 6 ψ Y, K K s To this ed, take ay vetor x R suh that x K = 1 ad x 1 x 0 Defie the vetor y = y j with oordiates y j takig values i the set {0} {τ l, l [L]} so that y j x j < τy j, if x j τ L ; y j = 0, if x j < τ L The x K τ y K + x j e j τ y K + τ L τ y K + 1 τ 1, x j <τ L K

18 18 MARK RUDELSON where we used 51 i the last iequality Hee, 1 = x K τ 2 y K Also, the iequalities Φ ψ K τφ ψ D, ψ Y imply y K τ y D Combiig this with y D x D, we obtai x K τ 3 x D, ad reversig the roles of K ad D, we derive τ 3 x D x K τ 3 x D, whih implies 52 Defie ow a ew futio Θ : K s R Y by settig Θ ψ K = log Φ ψ K The ΘK s [ logτ 2, log ] Y Hee, there exists a log τ-et N ΘK s i the -orm of ardiality Y log N log τ + 2 For ay x N, hoose a body K x K s suh that ΘK x = x ad osider the set M = {K x : x N } The for ay K K s, there exists K x M with ΘK ΘK x log τ This meas that for ay ψ Y, τ 1 Φ ψ K x Φ ψ K τφ ψ K x for all ψ Y, ad by 52, dk, K x τ 6 Thus, we ostruted a τ 6 -et M i K s of ardiality Y +L L log log M log τ + 2 log τ + 2 Assume ow that τ 2 1/12 previous iequality implies M exp The, by 51, L < log, ad the log τ exp C log2 log τ By the multipliative triagle iequality, the same iequality holds for the ardiality of ay τ 12 -separated set i K s To derive the statemet of the Propositio, set τ = t 1/12 Remark 51 The same proof works for all values t > 1 However, i the ase 1 < t 2 the estimate of L i 51 i terms of ad t is differet, whih leads to a differet estimate of the ardiality of a t-separated set Referees [1] A Barviok, E Veomett, The omputatioal omplexity of ovex bodies Surveys o disrete ad omputatioal geometry, , Cotemp Math, 453, Amer Math So, Providee, RI, 2008 [2] E D Gluski, The diameter of the Mikowski ompatum is roughly equal to, Fuktsioal Aal i Prilozhe , o 1, 72 73

19 COMPLEXITY OF THE SET OF UNCONDITIONAL CONVEX BODIES 19 [3] J Lidestrauss, A Szakowski, O the Baah-Mazur distae betwee spaes havig a uoditioal basis Aspets of positivity i futioal aalysis Tbige, 1985, , North-Hollad Math Stud, 122, North- Hollad, Amsterdam, 1986 [4] A Litvak, M Rudelso, N Tomzak-Jaegerma, O approximatio by projetios of polytopes with few faets, Israel Joural of Math, to appear [5] P Makiewiz, N Tomzak-Jaegerma, Quotiets of fiite-dimesioal Baah spaes; radom pheomea, Hadbook of the geometry of Baah spaes, Vol 2, , North-Hollad, Amsterdam, 2003 [6] G Pisier, O the metri etropy of the Baah-Mazur ompatum, Mathematika , o 1, [7] MRudelso, RVershyi, Haso-Wright iequality ad sub-gaussia oetratio, Eletro Commu Probab , o 82, 9 pp [8] M Talagrad, A ew look at idepedee, A Probab , o 1, 1 34