On the hyperplane conjecture for random convex sets

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1 O the hyperplae cojecture for radom covex sets arxiv:math.mg/06257v 8 Dec 2006 Bo az Klartag ad Gady Kozma Abstract Let N +, ad deote by K the covex hull of N idepedet stadard gaussia radom vectors i R. We prove that with high probability, the isotropic costat of K is bouded by a uiversal costat. Thus we verify the hyperplae cojecture for the class of gaussia radom polytopes. Itroductio The hyperplae cojecture suggests a positive aswer to the followig questio: Is there a uiversal costat c > 0, such that for ay dimesio ad for ay covex set K R of volume oe, there exists at least oe hyperplae H R with Vol (K H > c? Here, of course, Vol deotes ( -dimesioal volume. This seemigly iocuous questio, cosidered two decades ago by Bourgai [3, 4], has ot bee aswered yet. We refer the reader to, e.g., [2], [8] or [4] for partial results, history ad for additioal literature regardig this hyperplae cojecture. I particular, there are large classes of covex bodies for which a affirmative aswer to the above questio is kow. These iclude ucoditioal covex bodies [3, 8], zooids, duals to zooids [2], bodies with a bouded outer volume ratio [8], uit balls of Schatte orms [5] ad others (e.g., [, 7]. A potetial couter-example to the hyperplae cojecture could have stemmed from radom covex bodies, that typically belog to oe of these classes. Recall that, startig with Gluski s work [8], radom polytopes are a major source of couter examples i highdimesioal covex geometry (i additio to the distace problem [8], oe has, e.g., the basis problem [23] or the gaussia perimeter problem [9]. The goal of this short ote is to show that gaussia radom polytopes, ad related models of radom covex sets, do ot costitute a couter-example to the hyperplae cojecture. Suppose K R is a covex body. The isotropic costat of K, deoted here by L K, is defied by L 2 K = if Tx 2 dx, ( T:R R Vol (K + 2 K Supported by the Clay Mathematics Istitute ad by NSF grat #DMS

2 where the ifimum rus over all volume-preservig affie maps T : R R, ad stads for the stadard Euclidea orm i R. Directly from the defiitio, the isotropic costat is ivariat uder affie trasformatios. It is well-kow (see [0] or [8] that whe Vol (K =, c L K if sup Vol (T K H c 2, (2 T:R R H R L K where the ifimum rus over all volume-preservig affie maps T : R R, the supremum rus over all hyperplaes H R, ad c, c 2 > 0 are some uiversal costats. Throughout the text, the symbols c, C, c, C, c, c 2 etc. deote various positive uiversal costats, whose value may chage from oe lie to the ext. Thus, accordig to (2, the hyperplae cojecture is equivalet to the existece of a uiversal upper boud for the isotropic costat of a arbitrary covex body i a arbitrary dimesio. It is well-kow that L K > c for ay fiite-dimesioal covex body K (see, e.g., [8]. The best kow geeral upper boud is L K < C /4 for a covex body K R (see [4] ad also [5], [6] ad [7]. There are two atural models for radom covex bodies (a body is a compact with oempty iterior. I the cetrally-symmetric cotext, the first model is the symmetric covex hull of N radom idepedet poits, while the secod (its dual, is the itersectio of N radom strips. For the secod model, however, it is quite easy to demostrate the hyperplae cojecture. Ideed, if N 2, the a simple calculatio shows that it has a bouded outer volume ratio, ad hece has a bouded isotropic costat (see [8]. If N < 2, the the hyperplae cojecture holds determiistically wheever the resultig set is a body, accordig to [2]. Thus we will focus o the first model. We say that a radom vector X = (X,..., X R is a stadard gaussia vector if its coordiates X,..., X are idepedet, stadard ormal variables. Theorem. Let, N ad let G 0,..., G N be idepedet stadard gaussia vectors i R. Deote K = covg 0,..., G N ad T = cov±g,..., ±G N where cov deotes covex hull. The, with probability greater tha Ce c, Here, C, c > 0 are uiversal costats. L K < C ad L T < C. Let us sketch the proof i the case that N > 2 (for smaller N the argumet is oly slightly more opaque. It is easy to see that i this case the radius of the iscribed ball is, with high probability, c log N just calculate the probability that i a give directio all poits are iside a strip of width ǫ log N ad do a uio-boud over a dese et i S. O the other had, with probability all faces of K are ( -dimesioal simplices. Further, 2

3 with high probability all ceters of gravity of all simplices are with distace C log N from 0 the ceter of gravity of each simplex is a gaussia vector whose coordiates have variace /, ad agai all that is eeded is to do a uio-boud over all -tuples of vertices. The cocetratio of the volume of a simplex aroud its ceter of gravity shows that almost all the mass is withi distace of log N which implies the required estimate for L K. O first glace it seems that some miracle is at work why should we get the same log N i the lower ad upper boud? However, this is some maifestatio of the pheomeo that the maximum of may idepedet variables is strogly cocetrated. Of course, the differet faces of our body are ot idepedet, but it turs out that they are sufficietly idepedet to display a similar cocetratio pheomeo. Thus our proof is robust ad would admit direct geeralizatios to other types of distributios, i place of the stadard gaussia distributio. We will prove it for some other distributios that iclude the uiform distributio o the cube ad o its corers ±, see Theorem 3.2 below. The techique should also work for poits uiform o S. Ackowledgemet. We would like to thak Jea Bourgai for motivatig us to work o this problem. BK would also like to thak Alai Pajor for discussios o the subject. 2 Simplices I this sectio, we assume that N are itegers, ad that G 0,..., G N are idepedet radom vectors i R which eed ot be idetically distributed. We write G i = (G i,,..., G i, R, ad we make the followig assumptios regardig G 0,..., G N : ( a The radom variables G i,j (i = 0,..., N, j =,..., are idepedet. ( b For ay i = 0,..., N, j =,...,, EG i,j = 0, EG 2 i,j = ad E exp ( G 2 i,j /0 0. ( c The G i (i = 0,..., N are absolutely cotiuous. The costat 0 plays o special rôle. Note that ( a, ( b ad ( c hold whe G 0,..., G N are idepedet stadard gaussia vectors. Our mai techical tool is the followig Berstei s iequality for variables with expoetial tail ( ψ, see, e.g. [26, Sectio 2.2.2]. Theorem 2. Let L > 0, let m be a iteger, ad let X,..., X m be idepedet radom variables with zero mea. Assume that The, for ay t > 0, P m E exp( X i /L 20 for i m. m X i > t i= where c > 0 is a uiversal costat. ( t 2 exp cm mi L, t2, L 2 3

4 The followig lemma is a cosequece of Theorem 2.. Lemma 2.2 Fix 0 k < k 2 < < k N. The, (i P G ki > C log 2N ( <, 0N i= N (ii P G i > C log 2N ( <, N + 0N i=0 ( 2 (iii P G 2 ki,j + G 2 k i,j > C log 2N i= j= j= Here, C > 0 is a uiversal costat. Proof. Fix i, j. We have, for ay t, ( <. 0N E exp(tg ki,j exp( 5 2 t2 E exp(g 2 ( b k i,j/0 0 exp( 5 2 t2. By idepedece, for ay j =,.., ad t, ( E exp t so ( E exp t i= G ki,j ( E exp t i= i= G ki,j G ki,j 0 exp ( 5 2 t2, + exp Deote Y j = i= G k i,j mea zero ad variace oe. Moreover, by (3, for j =,...,, ( Y 2 E exp j 00 ( t i= G ki,j 20 exp( 5 2 t2. (3 (j =,...,. The the Y j are idepedet radom variables of 2E exp ( Y 2 j P Y 2 j 00 > log t t 0 Ee log t Y j dt ( dt t 5/2 dt < 9. (4 Hece, we may apply Theorem 2. for the idepedet radom variables Yj 2, with L = 00. We coclude that P G ki > C log 2N P i= P j= j= [Y 2 j ] > (C/22 log 2N Yj 2 > C 2 log 2N 2 exp ( c C2 400 log 2N < (, 0N 4

5 for a appropriate choice of a large uiversal costat C > 50. This completes the proof of (i. The argumet that leads to (ii is similar. We defie Ỹj = N N+ i=0 G i,j (j =,...,. Arguig exactly as above, we fid that for j =,...,, ( Ỹj 2 E exp < We may ivoke Theorem 2. for the idepedet radom variables Ỹ 2 j, with L = 00. This yields N P G i > C log 2N = P N + N + i=0 j= P [Ỹ j 2 ] > (C/2 2 log 2N 2 exp j= Ỹ 2 j > C 2 log 2N ( c C2 400 log 2N < (, 0N provided that C is a sufficietly large uiversal costat. This proves (ii. It remais to prove (iii. The radom variables G 2 i,j (i = 0,..., N, j =,..., are idepedet, have mea zero ad they satisfy that E exp [ G 2 i,j /0] 2E exp(g 2 i,j /0 20, accordig to ( b. Hece, we may apply Theorem 2. for the idepedet radom variables G 2 k i,j, with L = 0. We coclude that for ay j, P [G 2 k i,j ] > C log 2N ( 2 exp c C 0 log 2N ( <, (5 20N i= for a large uiversal costat C. We sum (5 over j =,..., ad coclude that P G 2 2 k i,j > (C + 2 log 2N i,j= P [G 2 k i,j ] > C log 2N (5 ( ( < < 20N 2. 0N j= i= Deote Z i = j= G k i,j (i =,...,. The the Z i are idepedet ad EZi 2 = for i =,...,. Repeatig the argumet from (3 ad (4 (the oly differece is that here we sum over j ad before we summed over i, we obtai that E exp ( Zi 2 /00 < 9. Thus, we may use Theorem 2. for the radom variables Zi 2 with L = 00, ad deduce that P [Zi 2 ] > C log 2N ( 2 exp c C 00 log 2N < ( 2. (7 0N i= 5 (6

6 The desired coclusio (iii follows at oce from (6 ad (7. The ext lemma is a simple, cocrete calculatio for the regular ( -simplex. We write e,..., e for the stadard basis i R, ad deote the ( -dimesioal regular simplex. = cove,..., e R, Lemma 2.3 Let X = (X,..., X be a radom vector that is distributed uiformly i. The, where δ i,j is Kroecker s delta. EX i X j = + δ i,j ( + Proof. Examie X without its last coordiate, (X,..., X. This is distributed uiformly i the simplex x R ; x i, i, x i 0. i= Cosequetly, the desity of the radom variable Y = X + + X is proportioal to t t 2 i the iterval (0,, ad is zero elsewhere. Hece, E(X + + X 2 = EY 2 = Note that i= X i. Therefore ( 2 = E X i = i= 0 t 2 ( t 2 dt = +. (8 i= EX 2 i + i j EX i X j. (9 From (8 ad (9 we get that, whe 2, ( EX 2 + ( ( 2EX X 2 = +, EX2 + ( EX X 2 =, ad the lemma follows. Corollary 2.4 Fix 0 k < k 2 <... < k N, ad set F = covg k,..., G k. Deote Z N N+ i=0 G i. The with probability greater tha 4 (, 0N the set F is a ( - dimesioal simplex that satisfies (i x 2 dx < C log 2N Vol (F F, (ii x Z 2 dx < C log 2N Vol (F. F 6

7 Here, C > 0 is a uiversal costat. Proof. The radom vectors G ki are idepedet ad absolutely cotiuous accordig to ( c. Hece, with probability oe, the liear spa of G k,..., G k equals R, ad the set F is a ( -dimesioal simplex i R whose vertices are the poits G k,..., G k. Deote T(G ki,j i,j=,...,, a matrix. The, FT ( ad hece, x 2 dx Tx 2 dx. (0 Vol (F F Vol ( Accordig to Lemma 2.3, Tx 2 dx = Vol ( = ( + i= j,j 2 = G ki,j G ki,j 2 x Vol i= j,j 2 = ( j x j2 dx ( ( 2 G ki,j G ki,j 2 ( + δ j,j 2 G ki,j + G 2 k ( + i,j. i= j= j= We coclude from (0, ( ad Lemma 2.2(iii that P x 2 dx > C log 2N Vol (F F < (. (2 0N As for the secod part of the corollary, accordig to Lemma 2.2(i ad Lemma 2.2(ii we kow that N P G ki + G j < 2C log 2N ( 2. (3 N + 0N Additioally, Vol (F F = N + i= j=0 x Z 2 dx = Z 2 2 Vol (F N 2 G j 2 j=0 G ki, N + i= F xdx, Z + x 2 dx Vol (F F N G j + By combiig (4 with (2 ad (3, we obtai P x Z 2 dx > C log 2N Vol (F F j=0 x 2 dx. (4 Vol (F F ( < 3. (5 0N From (2 ad (5 the corollary follows. For a poit x R ad a set A R, we write d(x, A = if y A x y. 7

8 Lemma 2.5 Set K = covg 0,..., G N, T = cov±g,..., ±G N, ad deote Z N N+ i=0 G i. The with probability greater tha Ce c, (i x 2 dx < C log 2N Vol (T T, (ii x Z 2 dx < C log 2N Vol (K. K Here, c, C > 0 are uiversal costats. Proof. The radom vectors G i,j are absolutely cotiuous, by assumptio ( c. Hece, with probability oe, the poits G 0,..., G N are i geeral positio i R ; that is, with probability oe, o + distict poits from G 0,..., G N lie i the same affie hyperplae i R. Cosequetly, all the ( -dimesioal facets of the polytopes K ad T are simplices, with probability oe. Note that, i the case of T we use the fact that G i ad G i could ever belog to the same face. Let F,..., F l be a complete list of the ( -dimesioal facets of T. Sice a facet is determied by poits from ±G,..., ±G N, the ( ( 2N 2eN l. Accordig to Corollary 2.4(i, with probability greater tha 4 ( 2e F i x 2 dx < C ( log 2N 0, Vol (F i, for i =,..., l. (6 Each poit x T (except for the origi may be uiquely represeted as x = ty with 0 < t ad y T. We itegrate with respect to these stadard polar coordiates, ad obtai that Vol (T T x 2 dx = Vol (T T 0 ty 2 t y, ν y dt dy (7 where ν y is the uit outward ormal to T at y (ν y is uiquely defied almost everywhere as T is covex. Whe y F i for some i =,..., l, we have that y, ν y = d(0, asp F i where asp F i is the affie subspace spaed by F i. Hece, from (7, Vol (T T x 2 dx = Vol (T l i= d(0, asp F i + 2 F i y 2 dy. (8 8

9 Recall that l i= d(0, asp F i Vol (F i = Vol (T. We combie (8 with (6, ad coclude that with probability greater tha 4 ( 2e Vol (T T x 2 dx < Vol (T l i= 0, d(0, asp F i Vol (F i + 2 C log 2N < C log 2N. This completes the proof of (i. The proof of (ii is very similar; we supply some details. Let G,..., G k deote the ( -dimesioal facets of K. Observe that Z K, ad that ay x K (except for the poit Z is uiquely represeted as x = Z + t(y Z with 0 < t ad y K. As before, itegratio i polar coordiates yields Vol (K K x Z 2 dx Vol (K K 0 t t(y Z 2 y Z, ν y dt dy = = + 2 k i= d(z, asp G i Vol (K G i y Z 2 dy. Agai, k i= d(z, asp G i Vol (G i = Vol (K. Thus, i order to prove (ii, we may simply reproduce the argumet from the proof of (i, with Corollary 2.4(ii replacig the role of Corollary 2.4(i. This completes the proof. 3 Radom Polytopes We summarize the results of Sectio 2 i the followig corollary. Note that the covex bodies discussed i this corollary have diameter that is larger tha c with high probability. Nevertheless, it is still possible to prove a much better estimate regardig the secod momet of the Euclidea orm. Corollary 3. Let N ad suppose that G 0,..., G N are idepedet radom vectors i R that satisfy coditios ( a ad ( b above. Set K = covg 0,..., G N, T = cov±g,..., ±G N, ad deote Z = N N+ i=0 G i. The with probability greater tha Ce c, Vol (T T x 2 dx < C log 2N where C, c > 0 are uiversal costats. ad x Z 2 dx < C log 2N Vol (K K, Proof. Suppose first that G 0,..., G N satisfy also ( c; that is, assume that they are absolutely cotiuous radom variables. The the desired coclusio follows from Lemma 2.5. For the geeral case, ote that the quatity P Vol (T T x 2 dx < C log 2N, Vol (K 9 K x Z 2 dx < C log 2N (9

10 depeds cotiuously o the distributio of G 0,..., G N i the weak topology at measures where P(Vol (K = 0 = P(Vol (T = 0 = 0. At other measures (9 may have a discotiuity of o more tha P(Vol (K = 0 + P(Vol (T = 0. The corollary follows by approximatig G 0,..., G N with absolutely cotiuous radom vectors that satisfy ( a, ( b ad ( c ad otig that by ( a ad ( b we have P(Vol (K = 0+P(Vol (T = 0 < Ce c, accordig to [22]. The ext theorem is cocered with o-gaussia aalogs of Theorem.. The mai ew case covered by that theorem is that of radom sig vectors, i.e., idepedet radom vectors whose coordiates are idepedet, symmetric Beroulli variables. We remark i passig that i the Beroulli case the probability of K or T to be degeerate was kow before [22]. See [3, 25]. Theorem 3.2 Let ad 2 N 2. Suppose that G,..., G N are idepedet radom vectors i R that satisfy coditios ( a ad ( b above. Set T = cov±g,..., ±G N. The with probability greater tha Ce c, where C, c > 0 are uiversal costats. L T < C Proof. We may clearly assume that exceeds a certai uiversal costat. It was proved i [6, Theorem 4.8] that, uder the assumptios of the preset theorem, log(2n/ (Vol (T / > C (20 with probability greater tha Ce c. From Corollary 3. we kow that with probability larger tha Ce c, x 2 dx < C log 2N (2 Vol (T T The theorem follows by substitutig the estimates (20 ad (2 ito the defiitio (. Geerally speakig, the restrictios o N i Theorem 3.2 are typically quite easy to work aroud. For example, i the case of Beroulli variables, if 2 N < 3 the (20 ad hece the coclusio of Theorem 3.2 hold with a differet costat, while if N 3 the with very high probability T is a hypercube. As for N < 2, oe ca show (20 by otig that eve a sigle simplex has eough volume see e.g., [24]. Our ext lemma shows the same volume estimates i the Gaussia case. It is stadard ad well-kow. Lemma 3.3 Let N ad suppose that G 0,..., G N are idepedet stadard gaussia vectors i R. Deote K covg 0,..., G N ad T = cov±g,..., ±G N. 0

11 The, with probability greater tha Ce c, Vol (K > c log(2n/ where c, C > 0 are uiversal costats. ad Vol (T log(2n/ > c, Proof sketch. We start with the lower boud for Vol (T. For the rage N 2, it is wellkow (see, e.g., [9, 6] ad refereces therei that with probability greater tha Ce c, c log 2N D T (22 where D = x R ; x is the uit Euclidea ball i R. Sice Vol / (D > c/, the desired lower boud for Vol (T follows from (22 i this case. It remais to deal with the rage N < 2. It turs out that i this rage a sigle simplex supplies eough volume for our eeds. We thus assume that N =. Elemetary Euclidea geometry shows that Vol (T = 2! d (G i, spg,..., G i i= where sp stads for liear spa (we defie sp( = 0. Deote ξ i = d (G i, spg,..., G i for i =,...,. The ξ,..., ξ are idepedet radom variables, with ξi 2 beig distributed chi-square with i degrees of freedom. Stadard estimates (oe may use, e.g., Berstei s iequality above show that P Vol (T < (2c P! i= [ log ξ ] i > log < Ce c i c for a suitable choice of uiversal costats c, C, c > 0. This completes the proof of the desired lower boud for Vol (T. Regardig Vol (K, deote G i = G i G 0 for i =,...,, ad set K = cov±g,..., ±G. The, Vol (K = Vol (cov0, G G 0, G 2 G 0,..., G N G 0 4 Vol (K (23 by the Rogers-Shephard iequality [2]. With probability oe, the vectors G,..., G are liearly idepedet. Let S : R R be the uique liear map that satisfies S(G i = G i G 0 for i =,...,. The K = S(T. Hece, Vol (K = det(s Vol (T. (24 Let v R be such that v, G i = for i =,...,. The vector v is idepedet of G 0. Moreover, v / G, ad with probability greater tha Ce c we have G C. Cosequetly, P v < c/ Ce c.

12 Clearly, Sx = x x, v G 0 for all x R. Therefore det(s = v, G 0. Coditioig o v, we see that det(s is a gaussia radom variable with mea ad variace v 2. Hece, P det(s < 2 = E v 2 ( exp (t 2 dt Ce c + C 2. (25 2π v 2 2 v 2 2 The desired lower boud for Vol (K follows from (23, (24, (25 ad from the lower boud for Vol (T, that was already prove. Proof of Theorem.. From Corollary 3. ad Lemma 3.3, we kow that with probability greater tha Ce c, x 2 dx < C ad x Z 2 dx < C Vol (T + 2 T Vol (K + 2 K for some poit Z R depedig o K. The theorem follows from the defiitio (. Refereces [] K. Ball, Logarithmically cocave fuctios ad sectios of covex sets i R. Studia Math. 88: (988, [2] K. Ball, Normed spaces with a weak-gordo-lewis property. Fuctioal aalysis (Austi, Texas, 987/989, Lecture Notes i Math., Vol. 470, Spriger, Berli, (99, [3] J. Bourgai, O high-dimesioal maximal fuctios associated to covex bodies. Amer. J. Math. 08:6 (986, [4] J. Bourgai, Geometry of Baach spaces ad harmoic aalysis. Proceedigs of the Iteratioal Cogress of Mathematicias, (Berkeley, Calif., 986, Amer. Math. Soc., Providece, RI, (987, [5] J. Bourgai, O the distributio of polyomials o high-dimesioal covex sets. Geometric aspects of fuctioal aalysis (989 90, Lecture Notes i Math., Vol. 469, Spriger, Berli, (99, [6] J. Bourgai, O the isotropy-costat problem for PSI-2 -bodies. Geometric aspects of fuctioal aalysis ( , Lecture Notes i Math., Vol. 807, Spriger, Berli, (2003, 4 2. [7] S. Dar, Remarks o Bourgai s problem o slicig of covex bodies. Geometric aspects of fuctioal aalysis (Israel, , Oper. Theory Adv. Appl., 77, Birkhäuser, Basel, (995, [8] E. D. Gluski, The diameter of Mikowski compactum roughly equals to. Fuktsioal. Aal. i Prilozhe., 5: (98, 72 73; Eglish traslatio i Fuct. Aal. Appl., 5 (98,

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14 [25] T. Tao, V. Vu, O the sigularity probability of radom Beroulli matrices. See [26] A. W. Va der Vaart, J. A. Weller, Weak covergece ad Empirical Processes. Spriger- Verlag, 996. Departmet of Mathematics, Priceto uiversity, Priceto, NJ 08544, USA address: bklartag@priceto.edu Departmet of Mathematics, Weizma Istitute of Sciece, Rehovot 7600, Israel address: gady.kozma@weizma.ac.il 4