ISOMORPHIC PROPERTIES OF INTERSECTION BODIES

Transcription

1 ISOMORPHIC PROPERTIES OF INTERSECTION BODIES A KOLDOBSKY, G PAOURIS, AND M ZYMONOPOULOU Abstract We study isomorphic properties of two geeralizatios of itersectio bodies - the class I of -itersectio bodies i R ad the class BP of geeralized -itersectio bodies i R I particular, we show that all covex bodies ca be i a certai sese approximated by itersectio bodies, amely, if K is ay symmetric covex body i R ad 1 1 the the outer volume ratio distace from K to the class BP ca be estimated by ( ) 1 C ovr(k, BP ) := if{ : C BP, K C} c K log e, where c > 0 is a absolute costat Next we prove that if K is a symmetric covex body i R, 1 1 ad its -itersectio body I (K) exists ad is covex, the d BM (I (K), B 2 ) c(), where c() is a costat depedig oly o, d BM is the Baach-Mazur distace, ad B2 is the uit Euclidea ball i R This geeralizes a well-ow result of Hesley ad Borell We coclude the paper with volumetric estimates for -itersectio bodies 1 Itroductio Let K be a symmetric star body i R Followig Lutwa ([35]), we say that a body I(K) is the itersectio body of K if the radius of I(K) i every directio is equal to the volume of the cetral hyperplae sectio of K perpedicular to this directio, ie for every ξ S 1, ρ I(K) (ξ) = K ξ, where ξ is the cetral hyperplae perpedicular to ξ ad stads for the volume A more geeral class of itersectio bodies ca be defied as the closure i the radial metric of the class of itersectio bodies of star bodies Itersectio bodies play a importat role i the solutio of the Busema-Petty problem posed i [9] i 1956: suppose that K ad L are origi symmetric covex bodies i R so that, for every ξ S 1, K ξ L ξ Does it follow that K L? The problem was completely solved i the ed of the 90 s, ad the aswer is affirmative if 4 ad egative if 5 The solutio has appeared as a result of wor of may mathematicias (see [12, Ch8] or [25, Ch5] for details) A coectio betwee itersectio bodies ad the Busema- Petty problem was established by Lutwa ([35]): the aswer to the Busema-Petty 1

2 2 A KOLDOBSKY, G PAOURIS, AND M ZYMONOPOULOU problem i R is affirmative if ad oly if every symmetric covex body i R is a itersectio body A more geeral cocept of a -itersectio body was itroduced i [28], [29] For a iteger, 1 < ad star bodies K, D i R, we say that D is the -itersectio body of K if for every ( )-dimesioal subspace H of R, D H = K H The -itersectio body of K is uique, but for > 1 it does ot always exist If the -itersectio body of K exists, we deote it by I (K) Taig the closure i the radial metric of the class of all D s that appear as -itersectio bodies of star bodies, we defie the class of -itersectio bodies The class of -itersectio bodies is related to a certai geeralizatio of the Busema-Petty problem i the same way as itersectio bodies are related to the origial problem (see [28] for details; this geeralizatio offers a coditio o the volume of sectios that allows to compare the volumes of two bodies i arbitrary dimesios) We deote the class of -itersectio bodies i R by I I [29] the first amed author also gave a Fourier characterizatio of I : K I if ad oly if K is a positive defiite distributio i R Aother geeralizatio of itersectio bodies was itroduced by Zhag ([50]) Let 1 1 The spherical Rado trasform R is a operator actig from C(S 1 ) to the space C(G, ) of cotiuous fuctios o the Grassmaia: R f(h) = f(x)dx, f C(S 1 ), H Gr, S 1 H We say that a star body K is a geeralized -itersectio body if there exists a liear positive fuctioal µ o R (C(S 1 )) such that for every f C(S 1 ), x K f(x)dx = µ(r (f)) S 1 Followig the otatio of [38] we deote the class of all geeralized -itersectio bodies i R by BP A characterizatio of the class BP was obtaied by Griberg ad Zhag ([17]) as a geeralizatio of the correspodig result of Goodey ad Weil for the origial itersectio bodies ([18]): a star body K belogs to the class BP if ad oly if it is the limit (i the radial metric) of -radial sums of ellipsoids The class of geeralized itersectio bodies is related to the so-called lower dimesioal Busema-Petty problem (LDBP-problem) Suppose that 1 < ad symmetric covex bodies K, L i R satisfy K H L H, for every ( )-dimesioal subspace H of R Does it follow that K L? It was proved i [50] that the aswer to this questio is affirmative if ad oly if every symmetric covex body i R is a geeralized -itersectio body Usig this, Bourgai ad Zhag ([6]) (see also [46]) proved that for the dimesios of sectios > 3 the aswer to the LDBP-problem is egative Aother proof of this result was give later i [29] The problem is still ope i the cases where the dimesio of sectios = 2, 3 I the case = 1 the classes I1 = BP 1 coicide with the class of origial itersectio bodies Also i the case = 1 the two classes cotai all star bodies i R It was proved i [31] (see also [25, Corollary 49]) that the class I 3 cotais all symmetric covex bodies i R, ad this is o loger true for the classes

3 ISOMORPHIC PROPERTIES OF INTERSECTION BODIES 3 I with < 3 ([28] or [25, Theorem 413]) It is ow that BP I ([29]; see also [38] or [25, p92] for simpler proofs) The latter two results immediately imply the egative aswer to the LDBP-problem with the dimesio of sectios greater tha 3 The aswer to the problem with two- or three-dimesioal sectios would be positive, if the classes BP ad I with = 2 or = 3 were equal However, Milma ([39]) proved that BP is a proper subclass of I for 2 2 The example of Milma is ot covex, so LDBP-problem is still ope for two- ad three-dimesioal sectios Aother ope problem is whether the classes I icrease with It was proved by Yasi ([49]) that for 1 < 2 there exists a symmetric covex body that belogs to I 2 but ot to I 1 However, the iclusio I 1 I 2 is ow oly i the case where 1 divides 2 (see [38]) For more results o these classes of bodies see [32] ad refereces there I spite of all these results, the isomorphic properties of itersectio bodies are ot very well uderstood The first result of this id was established by Hesley ad Borell (see [20], [4]): if K is symmetric ad covex, the the Baach- Mazur distace d BM (I(K), B2 ) c, where c > 0 is a absolute costat, which meas that itersectio bodies of covex bodies are isomorphic to ellipsoids (ote that if K is a symmetric covex body, the the classical result of Busema ([7]) guaraties that I(K) is also symmetric ad covex) Busema ([8]) also showed a isoperimetric type iequality: if K is symmetric covex ad K = 1, the I(K) I(D ), where D is the Euclidea ball with volume 1 This result ca be exteded to a class more geeral tha covex bodies (eve for Borel measurable sets) (see [16], [13]) However, itersectio bodies of covex bodies form oly a small part of the class of covex itersectio bodies As it was proved by the first amed author ([30])(usig the Fourier characterizatio of the itersectio bodies), the uit ball of ay fiite dimesioal subspace of L p, p (0, 2] is a -itersectio body for every, ad i particular all polar projectio bodies (uit balls of subspaces of L 1 ) are itersectio bodies The class of itersectio bodies is strictly larger tha the class of polar of projectio bodies; see [26] for examples A log stadig questio is if the two classes are isomorphic, ie whether for every itersectio body I there exists a projectio body Π, such that d BM (I, Π ) c, where c > 0 is a absolute costat (this questio is related to the 1970 problem of Kwapie ([33]) from the Baach space theory through the coectio betwee itersectio bodies ad L p -spaces; see [22]) A closely related result was proved i [22]: for ay q (0, 1) ad ay <, there exists a costat c(q, ) such that for ay covex -itersectio body K i R there exists a subspace of L q whose uit ball D satisfies d BM (K, D) < c(q, ) Note that the costat c(q, ) goes to ifiity whe q teds to 1, ad, if ot for that, the case q = 1 would imply the desired result for polar projectio bodies I this article we prove several isomorphic results for itersectio bodies ad their geeralizatios We have already metioed the fact that the class I 3 cotais all symmetric covex bodies, but this is o loger the case for the classes I with < 3 We start with a result showig that -itersectio bodies with < 3 are still i some sese dese i the class of all covex bodies Let K be a symmetric covex body i R ad let A be a class of star bodies We defie the outer volume ratio distace by ovr(k, A) := if{ovr(k, C), C A},

4 4 A KOLDOBSKY, G PAOURIS, AND M ZYMONOPOULOU where ( ) 1 T C ovr(k, C) := if{ : K T C, T GL } K Theorem 11 Let K be a symmetric covex body i R ad 1 1 The log (11) ovr(k, BP e ) c, where c > 0 is a absolute costat Recall that BP I, so the result also applies to -itersectio bodies Our secod result exteds to the class of -itersectio bodies the classical result of Hesley ad Borell that a itersectio body of a covex body is isomorphic to a ellipsoid Here oe faces two additioal difficulties First the -itersectio body of a covex body does ot ecessarily exist ad, secodly, eve if it exists it may ot be covex So ay result must tae ito accout these two coditios as additioal assumptios We prove Theorem 12 Let K be a symmetric covex body i R, 1 1 ad assume that I (K) exists ad it is covex The (12) d BM (I (K), B 2 ) c(), where c() depeds oly o Fially, we get some estimates for the volume radius of -itersectio bodies: Theorem 13 Let K be a symmetric star-shaped body i R with K = B2 Assume that the -itersectio body of K exists The ( ) 1 I (K) L B (13) 2 I (B2 ), L K with equality if ad oly if K is a symmetric ellipsoid Here L K stads for the isotropic costat of K Moreover, if I (K) is a covex body, we have that ( ) 1 I (K) (14) c log (1 + I (B2 ) dbm (I (K), B2 )) c mi{log, log }, where c > 0 is a uiversal costat The paper is orgaized as follows: I sectio 2 we itroduce basic defiitios ad otatio I sectios 3, 4 ad 5 we give the proof of Theorems 11, 12 ad 13 respectively We provide some fial remars i sectio 6 2 Notatio ad Defiitios We wor i R, which is equipped with a Euclidea structure, We deote by 2 the correspodig Euclidea orm, ad write B2 for the Euclidea uit ball, ad S 1 for the uit sphere Volume is deoted by We write ω for the volume of B2 ad σ for the rotatioally ivariat probability measure o S 1 We will write D for the euclidea ball of volume 1 (D := B2 1 B2 ) The Grassmaia maifold G, of -dimesioal subspaces of R is equipped with the Haar probability measure µ, We deote GL the set of liear ivertible trasformatios ad SL for the measure preservig liear trasformatios

5 ISOMORPHIC PROPERTIES OF INTERSECTION BODIES 5 A compact set K i R is called a star body if the origi is a iterior poit of K, every straight lie passig through the origi crosses the boudary of K at exactly two poits ad the Miowsi fuctioal of K defied by x K = mi{a 0 : x ak}, x R is a cotiuous fuctio o R The radial fuctio of K is give by ρ K (x) = max{a > 0 : ax K Throughout this article we say that K is symmetric if K = K The radial metric o the class of star bodies is defied by d r (K, L) = max ρ K(ξ) ρ L (ξ) ξ S 1 If K, L are star bodies ad N, the the -radial sum of K ad L is a star body D whose radial fuctio is equal to ρ D = ( ρ K + ρ L) 1/ If K is a covex body, we write h K for the support fuctio of K: h K (x) = max ξ K x, ξ x R We defie the geometric distace betwee two bodies K 1 ad K 2, d G (K 1, K 2 ) as the ifimum of positive umbers r such that there exists some a > 0 so that K 1 ak 2 rak 1 The Baach-Mazur distace is defied as d BM (K 1, K 2 ) := if d G (K 1, T K 2 ) T GL Let K be a star body i R of volume 1 We defie the isotropic costat of K, L K as L 2 K := 1 if{ x 2 2dx : T SL } T K For two covex bodies K ad L i R, the coverig umber of K by L, deoted by N(K, L), is defied as the miimal umber of traslates of L with their ceters i K, eeded to cover K The otatio a b meas that there exist uiversal costats c 1, c 2 > 0 such that c 1 a b c 2 b We refer to the boos [12], [25], [41], [45], [47] for basic facts from the Bru-Miowsi theory, the asymptotic theory of fiite dimesioal ormed spaces ad itersectio bodies 3 O the outer volume ratio of covex bodies with respect to the class BP The mai idea of the proof of Theorem 11 is to first cover a give covex body with Euclidea balls ad the show that, if the umber of balls is ot too big, oe ca approximate the uio of the balls by a body i BP, where will be related to the coverig umber of K We will use the followig theorem for coverig umbers of Pisier (see [45]) Theorem 31 Let K a symmetric covex body i R ad α (0, 2) The there exist a T SL such that if K 1 = T K the (1) K 1 = B 2,

6 6 A KOLDOBSKY, G PAOURIS, AND M ZYMONOPOULOU (2) log N(K 1, tb2 ) c t a (2 α), t 1, where c > 0 is a uiversal costat It is ow that the positio of the body K, costructed i the previous theorem, satisfies the reverse Bru-Miowsi iequality of V Milma ([40]) For completeess we provide a proof Corollary 32 Let K be a symmetric covex body with the properties of the body K 1 from the previous theorem The, for t 1 (31) K + tb 2 1 ct K 1, where c > 0 is a absolute costat Proof Choose α = 1 The usig Theorem 31, we have K + tb 2 1 t K 1 = K + tb 2 1 t B 2 1 N (K + tb 2, 2tB 2 ) 1 We will eed the followig elemetary N (K, tb 2 ) 1 c Lemma 33 Let z R ad t > 0 The there exists a cetered ellipsoid E such that (32) z + tb 2 E co{2z + 2 2tB 2, 2z + 2 2tB 2 } Proof We may assume that z := z 1 e 1 Let C be the cylider defied as The oe ca chec that C := {(s, y) R : s z 1 + t, y 2 t} (33) z + tb 2 C co{z + 2tB 2, z + 2tB 2 } =: K Let Q := {(s, y) R : s 1, y 2 1} The C := T Q, where T GL Defie E 1 := T B 2 The (34) B 2 Q 2B 2, or E 1 C 2E 1 The, by (33) we get that z + tb 2 C 2E 1 2C 2K We set E := 2E 1 ad the proof is complete Note that the class BP is closed uder -radial sums ad that ellipsoids belog to this class for all 1 1 Let V 1, V 2 two star bodies i R We defie the distace betwee V 1 ad V 2 as {ρ V1 (x) (35) d(v 1, V 2 ) := sup x R,x 0 ρ V2 (x), ρ V 2 (x)} ρ V1 (x) Observe that the defiitio implies that (36) where d := d(v 1, V 2 ) 1 d V 1 V 2 dv 1,

7 ISOMORPHIC PROPERTIES OF INTERSECTION BODIES 7 Lemma 34 Let 1 1 ad log N Let V i, i N, be symmetric star bodies i BP ad let V = N i=1 V i The there exists C BP such that (37) d(v, C) e Proof Let ρ Vi be the correspodig radial fuctios of V i The (38) ρ V := max i N ρ V i (x), x R \ {0} For ay a R N we have that (39) a a p N 1 a e a Let C be the star body defied by (310) ρ C(x) := N ρ V i (x), 0 x R Note that C BP ad by (39) we also have that d(v, C) e i=1 We ca ow give a proof of Theorem 11: Let α := 2 1 log e Note that ovr(t K, BP ) = ovr(k, BP ) for ay T GL So we may assume that K is i the positio described i Theorem 31 We have that for every t 1, there exists N, such that log N c t a (2 a), ad z 1,, z N K such that K N i=1z i + tb 2 Give ay z i, let E i as i Lemma 33 The we have that ( K N i=1e i 2 co{z 1,, z } + ) 2tB2 2(K + 2tB2 ) Choose t such that = c t a (2 a) The log N So, by Lemma 34, there exist a C BP such that d(c, V ) e, where V = N i=1 E i So, 1 (311) e K 1 ( e V C ev 2e K + ) 2tB2 ( Recall that t := (312) (2 a) ) 1 a ( 1 e K C 2e K + c Moreover, by Corollary 32, we have that log e The (311) becomes ) log e B 2 C 1 2e K + tb 2 1 ct K 1 Let C 1 := ec We have that C 1 BP, K C 1 ad C 1 1 (313) c t c log e K 1 This fiishes the proof

8 8 A KOLDOBSKY, G PAOURIS, AND M ZYMONOPOULOU 4 Distaces for -itersectio bodies Let K a star body i R Recall that the -itersectio body of K (if it exists) is a star body that satisfies (41) I (K) F = K F, F G, Note that I (K) (if it exists) is uique Also if t > 0 ad T GL, oe has that (42) I (tk) = t I (K) ad I (T K) = dett T (I (K)) The last equality follows from the Fourier characterizatio of the -itersectio bodies (see [29]) Let K be a symmetric covex body of volume 1 ad p 1 We defie the L p - cetroid body of K ([36],[37],[43]), as the symmetric covex body that has support fuctio ( ) 1 (43) h Zp(K)(θ) := x, θ p p dx We will use the followig L q -versio of Rogers-Shephard iequality ([43],[44]): Propositio 41 Let K be a symmetric covex body of volume 1 i R ad 1 1 The for every F G,, (44) K F 1 PF Z (K) 1 1 We will also use the Sataló ad reverse Sataló iequality ([5]): If K is a symmetric covex body i R, the (45) ( K K ) 1 1 K Propositio 42 Let K be a symmetric covex body i R of volume 1 ad 1 1 Assume that I (K) exists The for F G,, ( ) (46) I (K) F 1 Z (K) F 1 Proof Usig the defiitio (41) ad equatios (44), (45), we get that I (K) F 1 = K F 1 PF Z (K) 1 ) Z(K) F 1 F 1 = ( Z (K) The followig lemma is a well ow applicatio of the Bru-Miowsi iequality: Lemma 43 Let K be a symmetric covex body of volume 1 i R ad let 2 The (47) Z 2 (K) Z (K) cz 2 (K), where c > 0 is a absolute costat Note that Z 2 (K) is a ellipsoid Moreover, sice Z p (T K) = T Z p (K) for T SL, there exists T SL such that Z 2 (T K) := L K B 2 I this case we say that K is isotropic (see [42], [14] for more iformatio o isotropicity) Note that i the case

9 ISOMORPHIC PROPERTIES OF INTERSECTION BODIES 9 where K is covex the defiitio of the isotropic costat L K that we give here is equivalet to the defiitio give i 2 (see eg [14]) We have the followig Corollary 44 Let K be a isotropic covex body i R, 1 1 ad assume that I (K) exists The for all F 1, F 2 G,, (48) where c 1 > 1 is a absolute costat 1 (c 1 ) I (K) F 1 I (K) F 2 (c 1), Proof Usig equatios (46) ad (47) we have that ( ) I (K) F 1 Z (K) F1 c ) I (K) F 2 F2 ( Z (K) ( ) Z2(K) c F1 c 0 Z 2 (K) F c Z2 (K) F 1 2 c 0 Z2 (K) F 2 (c 1), sice K is isotropic We wor similary for the left had side iequality We will also eed the followig Lemma 45 Let K be a symmetric covex body i R m+1 Let r := r K (e m+1 ) ad R := h K (e m+1 ) The (49) 2r m + 1 K Rm K 2R K R m Proof To estimate the left-had side observe that re m+1 K, so K co{k R m r, re m+1, re m+1 } = 2 m + 1 K Rm For the right-had side, observe that the fuctio f(s) := K (se m+1 + e m+1) is eve ad log-cocave by the Bru-Miowsi iequality, therefore attais its maximum at 0 The by Fubii s theorem, K = 2 This fiishes the proof R 0 f(s)ds 2Rf(0) = 2R K R m Propositio 46 Let K be a symmetric covex body i R, 3 ad 2 1 Assume that there exists δ 1 such that for every F 1, F 2 G,, The K F 1 K F 2 δ (410) d G (K, B 2 ) δ Proof Let θ 1 S 1 be such that R(K) := h K (θ 1 ) = ρ K (θ 1 ) Let θ 2 S 1 be such that r(k) := h K (θ 2 ) = ρ K (θ 2 ) It is eough to show that R r δ

10 10 A KOLDOBSKY, G PAOURIS, AND M ZYMONOPOULOU We may assume that θ 1 θ 2, or else we have othig to prove Let F 0 := spa{θ 1, θ 2 } Let F G, 1, F F 0, F 1 := spa{θ 1, F } ad F 2 := spa{θ 2, F } Let S := K F The by Lemma 45 we have that (411) The R r δ 2RS K F 1 2RS ad 2rS R r K F 1 K F 2 R r K F 2 2rS or Proof of Theorem 12: The case = 1 is covered by the result of Hesley as oted i the Itroductio So we assume that 3 ad 2 1 Usig (42) we may assume that K is isotropic The we wat to show that d G (I (K), B 2 ) (c), where c > 0 absolute costat This follows from Corollary 44 ad Propositio 46 The depedece o i Theorem 12 is very bad, c() (c) This becomes meaigless for log log log We ca give a (slightly) better boud (but still expoetial) usig certai tools that were developed i order to attac the Hyperplae cojecture For a proof of the followig lemma see [42] Lemma 47 Let K be a isotropic covex body i R ad let 1 1 The for ay F G, there exists a symmetric covex body B i F such that, B := B(F ) is also isotropic ad (412) K F 1 L B L K Secod proof of Theorem 12: Agai we may assume that K is isotropic We will use the best ow boud for the isotropic costat due to B Klartag [23] (see also [24]): for every B i R, (413) L B c 1 4, where c > 0 is a uiversal costat Moreover, it is ow (eg([42]) that L K L B 2 1 So usig (112) we get that for every F 1, F 2 G,, I (K) F 1 (414) I (K) F 2 = K F 1 ( K F2 c L ) B(F 1) (c ) 4 L B(F2) So, usig Propositio 46 agai we get that d G (K, B2 ) (c) 4 This fiishes the proof 5 Volumetric estimates for -itersectio bodies Let p 0 ad C be a symmetric star body i R We defie ( ) 1 (51) M p (C) := θ p C dσ(θ) p S 1 Moreover, if C is covex we write ( ) 1 (52) W p (C) := h p C (θ)dσ(θ) p S 1

11 ISOMORPHIC PROPERTIES OF INTERSECTION BODIES 11 Let K be a compact set i R with K = 1 Let p >, p 0 We defie ( ) 1 (53) E p (K) := x p 2 dx p K We have the followig idetity (see [44] for a geeralizatio to the case of measures): Lemma 51 Let K be a symmetric star body i R of volume 1 ad 1 1 The (54) E (K) ( G, K F dµ(f ) ) 1 = E (D ) ( G, D F dµ(f ) Proof Ideed, writig i polar coordiates, we get that E (K) = ω dσ(θ) G, S E θ dµ(e) = K ω dσ(θ) ω ω ( )ω G, S E θ dµ(e) = K E dµ(e) = ( )ω K G, ω K F dµ(f ) ( )ω G, Sice the same holds also for D, equatio (54) follows We have the followig applicatio of the previous lemma ad a defiitio of - itersectio bodies Lemma 52 Let K be a symmetric star body i R of volume 1 ad 1 1 Assume that I (K) exists The ( ) 1 (55) M (I (K)) K F dµ(f ) = ω 1 G, Moreover, (56) M (I (K)) M (I (D )) = E (K) E (D ) Proof Itegratig (41) over G, we have that (57) I (K) F dµ(f ) = K F dµ(f ) G, G, Moreover, I (K) F dµ(f ) = G, = ω dσ(θ) S 1 θ I (K) G, ω S F dσ F (θ) dµ(f ) θ I (K) = ω M (I (K)) So, by (57) we get (55) (56) follows from (55) ad (54) The followig two propositios deal with the behavior of the ratio Ep(K) E p(d ) It is ot difficult to obtai a lower boud for the quatity Ep(K) E p(d ) The mai tool is a argumet of Milma ad Pajor (see [42]) ) 1

12 12 A KOLDOBSKY, G PAOURIS, AND M ZYMONOPOULOU Propositio 53 Let K a compact set i R of volume 1 ad let p >, p 0 The (58) E p (K) E p (D ), with equality if ad oly if K D = 1 I particular, if K is a star body, the we have equality if ad oly if K = D Proof Note that K \ D = D \ K Also if x K \ D the x 2 B2 1 while if x D \ K, x 2 B2 1 So, if p > 0, Ep(K) p = x p 2 dx = x K K\D p 2 dx + x p 2dx K D x p 2 K D dx + x p 2 dx = Ep 2 (D ) D \K It is clear that we have equality if ad oly if K D = 1 We wor similarly if p < 0 Propositio 54 Let K a star body i R of volume 1, < p < q, p, q 0 The E p (K) (59) E q (K) E p(d ) E q (D ), with equality if ad oly if K = D Proof We follow a argumet from [3] A simple computatio shows that E q (D ) (510) E p (D ) = ( +q )1/q ( +p )1/p For every q >, q 0, by itegratio i polar coordiates we have (511) E q q (K) = ω 0 ( ) 1 r +q 1 σ r K dr := 0 r +q 1 g(r), dr The fuctio g(r) := ω σ ( 1 r K) is o-icreasig o (0, ) ad it is supported i some [r(k), R(K)] If we assume that K is ot the Euclidea ball, the [r(k), R(K)] is a iterval ad g(r) ca be assumed absolutely cotiuous I this case we ca write ρ(s) (512) g(r) = ds, (r > 0) s r for some o-egative fuctio ρ o (0, ) The, agai by itegratio i polar coordiates, 1 = K = r 1 1 ρ(s) g(r) dr = r dr ds = ρ(s) ds s 0 0 0<r<s Hece, ρ represets a probability desity of a positive radom variable, say, ξ The (511) becomes Eq q (K) = r q+ 1 g(r) dr = s q ρ(s) ds = 0 + q 0 + q E(ξq ) Applyig Hölder s iequality for < p < q, we see that (513) (E(ξ q )) 1/q > (E(ξ p )) 1/p 0

13 ISOMORPHIC PROPERTIES OF INTERSECTION BODIES 13 Note that, sice ξ is a o-zero radom variable with a absolute cotiuous desity, there is o equality case i (513) (see [19, Th188]) So, ( ) 1/q ( ) 1/q E q (K) E p (K) = +q E(ξq ) +q ( ) 1/p > ( ) 1/p = E q(d ) +p E(ξp E ) p (D ), +p as claimed We will use the followig immediate applicatio of Hölder s iequality Lemma 55 Let C be a star symmetric body i R ad p q, p, q 0 The (514) M p (C) M q (C), with equality if ad oly if C = ab 2 for some a > 0 Moreover writig the volume of C i polar coordiates we get that ( ) 1 B (515) M (C) = 2 C Results of Lewis [34], Figiel ad Tomcza-Jaegerma [11], Pisier [45] establish the followig reverse Uryso iequality: Propositio 56 Let C be a symmetric covex body i R T SL such that (516) W (C 1 ) c C 1 1 log (1 + dbm (C, B 2 )), where c > 0 is a absolute costat ad C 1 = T C Proof of Theorem 13: Usig Lemma 55 ad (515) we have that ( ) 1 M (I (K)) (517) M (I (D )) I (D ) I (K) The there exists Usig (42), we may assume that K is isotropic The E 2 (K) = L K The, by (56), ad Propositio 54, (518) M (I (K)) M (I (D )) = E (K) E (D ) E 2(K) E 2 (D ) = L K L D So, (517) ad (518) imply equatio (13) We ow assume that I (K) is covex We cosider I (K) to be i the positio described i Propositio 56 (Agai by usig (42)) The by Lemmas 55 ad 56, M (I (K)) M 1 (I (K)) = W (I (K)) c I(K) 1 log (1 + dbm (I (K), B2 )) c log (1 + d BM (I (K), B2 )), I (K) 1 usig also Sataló iequality This implies that ( ) 1 M (I (K)) (519) M (I (D )) c I (D ) log (1 + dbm (I (K), B2 )) I (K) So agai by (56) ad Propositio 53, we have that (520) M (I (K)) M (I (D )) = E (K) 1 E (D )

14 14 A KOLDOBSKY, G PAOURIS, AND M ZYMONOPOULOU So, equatios (519) ad (520) imply (14) d BM (K, B2 ) ad Theorem 12 To coclude we use the fact that 6 Cocludig remars Theorem 11 idicates that the classes BP icrease i a caoical way It is ot clear to us if the reverse iequality (up to the logarithmic term) holds true eve for the Baach-Mazur distace We pose this as a Questio 1: Is it true that for every ad every, 1 1, there exists a covex symmetric body K i R such that d BM (K, BP ) c, where c > 0 is a uiversal costat? We ca show that this is true i the case = 1: Propositio 61 There exists c > 0 such that for every 1, d BM (B, I 1 ) c The proof of the latter Propositio depeds o the followig fact: all covex itersectio bodies have bouded volume ratio Let VR (a) be the class of symmetric ( ) 1 covex bodies i R with vr(k) := if{ T K B : T GL 2 } a I this otatio we have the followig Propositio 62 There exists c > 0 such that for every 1 every covex itersectio body i R belogs to the class VR (c) Proof The proof is simply a combiatio of certai ow results Let K I 1 ad covex, ad let X the -dimesioal Baach space with orm := K The it has bee show i [22] that X embeds isomorphically to L 1/2 It is ow (eg [21]) that every Baach subspace of L 1/2 has (Rademacher) cotype 2 Next, by a result of Bourgai ad Milma ([5]), every fiite dimesioal subspace with bouded cotype 2 costat, has the property that its uit ball has bouded volume ratio This fiishes the proof Proof of Propositio 61: We will show that d BM (B, VR (a)) c(a) The by Propositio 62 the proof would be complete For simplicity we assume that = 4 for some N The geeral case follows easily Let C VR (a) such that d BM (B, VR (a)) = d G (B, C) =: d The we have that C B dc By a well ow geeralizatio of Kashi s theorem ([48]) there exists F G, ad 2 a ellipsoid E such that c 1 (a)e C F c 2 (a)e Moreover it is ow (see eg [51]) that for ay ellipsoid i R m there exists E G m, m ad some r > 0 such that 2 E E = rb E, where B E is the Euclidea ball of E So we get that there exists E G, ad r 4 1 := r 1 (a) > 0, r 2 := r 2 (a), such that r 1 B E B E dr 2 B E Hece, it is eough to show that d G (B E, B E ) c, for every E G, 4 Cosiderig the polar body of B E it is eough to show that the covex hull of at most 8 poits i R has geometric distace from the Euclidea ball at least c Let N +1, v 1,, v N B 2 ad v 1 2 = 1 Let K := co{v 1,, v N } Note that

15 ISOMORPHIC PROPERTIES OF INTERSECTION BODIES 15 R(K) = 1 Now it is eough to show that if N = 8 the r K := mi θs 1 ρ K (θ) c But (see eg [1], [2], [10], [15]) oe has that K 1 c log en Writig the volume of K i polar coordiates we get that there exists at least oe θ S 1 such that ρ K (θ) c log en We complete the proof by choosig N = 8 The estimate i Theorem 12 is expoetial with respect to Eve if we assume that the Hyperplae cojecture has a positive aswer the existig proof would still give a estimate expoetial with respect to We believe that a better estimate (polyomial) must be true Havig i mid equatio (46), we pose the followig questio: Questio 2: Is it true that if K is symmetric ad covex i R, 1 ad I (K) exists ad it is covex the I (K) is isomorphic to Z (K)? Note that a positive aswer to the previous questio would easily imply a liear i estimate i Theorem 12 The secod coclusio of Theorem 13 ca be viewed as a geeralizatio of the classical Busema iequality (see Itroductio) We do t ow if the assumptio that I (K) is covex is ecessary i Theorem 13 ad whether the estimate ca be replaced by 1: Questio 3: Is it true that if K is a symmetric star body of volume 1, 1 1 ad if I (K) exists, the I (K) I (D )? Acowledgmets: The first ame author wish to tha the US Natioal Sciece Foudatio for support through grats DMS ad DMS , ad the Max Plac Istitute for Mathematics for support ad hospitality durig his stay i Sprig 2011 The secod ame author was partially supported by NSF Part of this wor was carried out whe the third ame author was visitig the Mathematics Departmet of Texas A& M which she thas for its hospitality Refereces 1 I Báráy, Z Füredi, Approximatio of the sphere by polytopes havig few vertices, Proc Amer Math Soc 102 (1988), o 3, I Báráy, Z Füredi, Computig the volume is difficult, Discrete Comput Geom 2 (1987), S G Bobov ad A Koldobsy, O the cetral limit property of covex bodies, Geom Aspects of Fuct Aalysis (Milma-Schechtma eds), Lecture Notes i Math 1807 (2003), C Borell, Complemets of Lyapuov s iequality, Math A 205 (1973), J Bourgai ad V D Milma, New volume ratio properties for covex symmetric bodies i R, Ivet Math 88, o 2, (1987), J Bourgai ad G Zhag, O a geeralizatio of the Busema-Petty problem, Covex Geometric Aalysis (Bereley, CA, 1996), 65 76, Math Sci Res Ist Publ, 34, Cambridge Uiv Press, Cambridge, H Busema, A theorem o covex bodies of the Bru-Miowsi type, Proc Nat Acad Sci USA 35 (1949), H Busema, Volume i terms of cocurret cross-sectios, Pacific J Math 3, (1953), 1 12

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