Symmetrization and isotropic constants of convex bodies

Transcription

1 Symmetrizatio ad isotropic costats of covex bodies J. Bourgai B. lartag V. Milma IAS, Priceto el Aviv el Aviv November 18, 23 Abstract We ivestigate the effect of a Steier type symmetrizatio o the isotropic costat of a covex body. We reduce the problem of boudig the isotropic costat of a arbitrary covex body, to the problem of boudig the isotropic costat of a fiite volume ratio body. We also add two observatios cocerig the slicig problem. he first is the equivalece of the problem to a reverse Bru-Mikowski iequality i isotropic positio. he secod is the essetial mootoicity i of L = sup R L where the supremum is take over all covex bodies i R, ad L is the isotropic costat of. 1 Itroductio Let R be a covex body whose baryceter is at the origi (i.e. b() = x dx = ). he iertia matrix of is the matrix M whose etries are M i,j = x ix j dx. he isotropic costat of, deoted by L, is defied as L 2 = det(m ) 1. V ol() 1+ 2 he isotropic costat is ivariat uder liear trasformatios of the body. If M is a scalar matrix ad V ol() = 1, we say that is isotropic, or that is i isotropic positio. I this case, for ay θ R, x, θ 2 dx = L 2 θ 2 where is the stadard Euclidea orm i R. Ay covex body has a uique affie image of volume oe which is i isotropic positio. We refer the reader to [MP] for more iformatio cocerig the isotropic positio ad the isotropic costat. 1

2 A major usolved problem asks whether there exists a umerical costat C such that L < C for every covex body i ay fiite dimesio. his problem is called the slicig problem or the hyperplae cojecture. A positive aswer to this questio has may iterestig cosequeces, see [MP]. Oe of these is that every covex body of volume oe, has a 1 dimesioal sectio whose 1 dimesioal volume is greater tha some costat c >. he curret best estimate is L < c 1/4 log, for a arbitrary covex body R (see [Bou], or the presetatio i [D]. See [Pa] for the o-symmetric case). For certai classes of covex bodies the questio is affirmatively aswered, such as for ucoditioal bodies (as observed by Bourgai, see [MP]), zooids, duals of zooids (see [Ba2], also for the coectio with the Gordo-Lewis costat), duals to bodies with fiite volume ratio (see [MP]), ad more (e.g. [J]). Here, we preset a reductio of the geeral problem to the boudess of the isotropic costat of a certai class of covex bodies: those which have a fiite volume ratio. For R, the volume ratio of is defied as, v.r.() = sup E ( ) 1 V ol() V ol(e) where the supremum is over all ellipsoids cotaied i. prove the followig coditioal propositio: Here we Propositio 1.1 here exists v > 1 such that the followig holds: If there exists c 1 > such that for ay ad for ay R, the iequality v.r.() < v implies that L < c 1, the there exists c 2 > such that for ay ad for ay R we have L < c 2. Next, we shall state a qualitative versio of Propositio 1.1. Deote L = sup R L where the supremum is over all covex sets i R, ad set L (a) = sup{l ; R, v.r.() a}. he we ca boud L by a fuctio of L (a) for a suitable a > 1. As a matter of fact, this fuctio is almost liear: Propositio 1.2 For ay δ >, there exist umbers v(δ) > 1, c(δ) > such that for ay, L < c(δ) L (v(δ)) 1+δ. A proof of these propositios, usig a symmetrizatio techique, is preseted i Sectio 4. he techique itself is preseted i Sectio 2. We prove the followig propositio i Sectio 3. 2

3 Propositio 1.3 If m <, the L m < cl where c is a umerical costat. As observed by. Ball (see [MP]), the hyperplae cojecture implies that a reverse Bru-Mikowski iequality holds i the isotropic positio. Aswerig a questio posed by. Ball to oe of the authors, we show that the slicig problem is actually equivalet to a reverse Bru-Mikowski iequality i the isotropic positio. he followig coditioal statemet is proved i Sectio 5: Propositio 1.4 Assume that there exists a costat C >, such that for ay, ad for ay two isotropic covex bodies, R, ( V ol( + ) 1/ C V ol() 1/ + V ol( ) 1/). (1) he it follows that for ay covex body R, L < C (C) where C (C) is a umber that depeds solely o C. Actually, Propositio 1.4 is correct eve if we restrict to be a Euclidea ball, as is evidet from the proof. Note that as proved i [M1], iequality (1) which is a reverse Bru-Mikowski iequality, holds whe ad are i a special positio called M-positio (see defiitio i Sectio 3). However, the coectio of a M-ellipsoid with the isotropic positio is ot yet clear. hroughout the paper we deote by c, c, c, C etc. some positive uiversal costats whose value is ot ecessarily the same o differet appearaces. Wheever we write A B, we mea that there exist uiversal costats c, c > such that ca < B < c A. Also, V ol( ) deotes the volume of a set R, relative to its affie hull. he paper [BM] serves as a exteded itroductio to this paper. 2 Symmetrizatio 2.1 Defiitio Let R be a covex body, let E R be a subspace of dimesio k, ad let E be a k-dimesioal covex body, whose baryceter is at the origi. We defie the (, E)-symmetrizatio of as the uique body such that: (i) for ay x E, V ol( (x + E)) = V ol( (x + E)). (ii) for ay x E the body (x + E) is homothetic to, ad its baryceter lies i E. 3

4 I other words, we replace ay parallel sectio of, with a homothetic copy of of the appropriate volume. his procedure of symmetrizatio is kow i covexity, see [BF], page 79. For completeess, we shall ext prove that this symmetrizatio preserves covexity, as follows from Bru-Mikowski iequality. Lemma 2.1 is a covex body. Proof: For ay z E, the sectio (z + E) is covex, as a homothetic copy of. Let x, y P roj E ( ) be ay poits, where P roj E is the orthogoal projectio oto E i R. We will show that cov((x + E), (y + E) ) = λ [(x + E) ] + (1 λ) [(y + E) ]. λ 1 For z E, deote v(z) = V ol((z + E) ) = V ol((z + E) ). Sice is covex, by Bru-Mikowski, v(λx + (1 λ)y) 1/k λv(x) 1/k + (1 λ)v(y) 1/k (2) where k = dim(e). Sice (z + E) = z + poit z E, iequality (2) etails that ( v(z) V ol( )) 1/k for ay (λx + (1 λ)y + E) λ [(x + E) ] + (1 λ) [(y + E) ] ad the lemma is proved. 2.2 he effect of a symmetrizatio o the isotropic costat Let us determie the eigevectors of the iertia matrix M. hese eigevectors are also called axes of iertia of the body. If is a arbitrary body of volume oe with its baryceter at zero, ad {e 1,.., e } are its axes of iertia, the sice L 2 = det(m ) 1/, ( L 2 = i=1 x, e i 2 dx ) 1 Lemma 2.2 Assume that is isotropic. Let e 1,.., e k be axes of iertia of the body E, ad let e k+1,.., e be ay orthoormal basis of E. he the orthoormal basis {e 1,.., e } is a basis of iertia axes of.. 4

5 Proof: By property (i) from the symmetrizatio defiitio, for ay v E x, v 2 dx = x, v 2 dx = L 2 v 2 (3) sice is isotropic. By property (ii), for ay v E, u E, x, v x, u dx = y, v z, u dzdy = P roj E ( ) [y+e] sice the baryceter of is at zero. Hece, E ad E are ivariat subspaces of M. Accordig to (3), the operator M restricted to E is simply a multiple of the idetity. herefore ay orthogoal basis e k+1,.., e of E is a basis of eigevectors of M. All that remais is to select k axes of iertia i E. Let e 1,.., e k be axes of iertia of the k-dimesioal body. It is straightforward to verify that for ay u 1, u 2 E, x, u 1 x, u 2 dx = c(, E, ) x, u 1 x, u 2 dx P roj E () V ol( (x+e))1+2/k dx where c(, E, ) = depeds oly o V ol( ) 1+2/k, E,. herefore e 1,.., e k are also axes of iertia of. We postpoe the proof of the followig lemma to Sectio 6. Lemma 2.3 Let f be a compactly supported o-egative fuctio o R, such that f 1/k is cocave o its support, ad f(x)dx = 1. R Deote M = max x R f(x). he, (k + 1)(k + 2) ( + k + 1)( + k + 2) M 2/k R f(x) 1+ 2 k dx M 2/k. Now we ca estimate L 2 orms of some liear fuctioals over. Lemma 2.4 Let R be a covex body of volume oe whose baryceter is at the origi. Let E R be a subspace with dim(e) = k, ad let E be a k-dimesioal covex body of volume oe with zero as a baryceter. Deote by the (, E)-symmetrizatio of. he for ay v E, ad, x, v 2 dx x, v 2 dx ( ) 2 k + 1 V ol( E) 2/k x, v 2 dx + 1 ( ) V ol( E) 2/k x, v 2 dx. k + 1 5

6 Proof: x, v 2 dx = = P roj E ( ) P roj E ( ) (E+x) V ol( (E + x)) 1+ 2 k dx y, v 2 dydx y, v 2 dy. Deote g(x) = V ol( (x + E)) = V ol( (x + E)). he by Bru-Mikowski iequality, g 1/k is cocave o its support i E ad g = vol() = 1. By Lemma 2.3, (k + 1)(k + 2) ( + 1)( + 2) M 2/k y, v 2 dy x, v 2 dx M 2/k y, v 2 dy where M = max x E g(x). Sice the baryceter of is at the origi, by heorem 1 i [F], g() M ad sice g() = V ol( E), we get ( ) 2 k + 1 (k + 1)(k + 2) + 1 ( + 1)( + 2) ( ) k + 1 g() k + 1 x, v 2 dx V ol( E) 2 k x, v 2 dx ( ) k + 1 he followig theorem coects the isotropic costat of the symmetrized body with the isotropic costats of,. heorem 2.5 Let be a isotropic body of volume oe, E a subspace of dimesio k, a k-dimesioal covex body with its baryceter at the origi, ad the (, E)-symmetrizatio of. he L L 1 k L k/ V ol( E) 1/. I fact, the ratio of these two quatities is always betwee ( ) k/. +1 ad k+1 ( ) k/ k+1 +1 Proof: We may assume that V ol( ) = 1. Let {e 1,.., e } be selected accordig to Lemma 2.2. he, L = ( i=1 x, e i 2 dx ) 1/ = L 1 k ( k i=1 x, e i 2 dx ) 1/ 6

7 where the right-most equality follows from (3). By Lemma 2.4, k ( ) 2 L L 1 k k + 1 V ol( E) 2 k x, e i dx = L 1 k i=1 ( ) k k V ol( E) + 1 k L sice the vectors e 1,.., e k are iertia axes of. herefore, L > cl 1 k Lk/ V ol( E)1/. Regardig the iverse iequality, accordig to the opposite iequality i Lemma 2.4 we get, ( ) k + 1 L 1 L k k + 1 V ol( E) 1 k L 1/ < cl 1 k for a differet costat c. Lk/ V ol( E)1/ 3 Use of a M-ellipsoid We will eed to use a special ellipsoid associated with a arbitrary covex body, called a M-ellipsoid. A M-ellipsoid is defied by the followig theorem (see [M1], or chapter 7 i the book [P]): heorem 3.1 Let R be a covex body. he there exists a ellipsoid E with V ol(e) = V ol() such that N(, E) = mi{ A; A + E} < e c where A is the umber of elemets i the set A, ad c is a umerical costat. We say that E is a M-ellipsoid of (with costat c). A M-ellipsoid may replace i various volume computatios. For example, assume that E is a M-ellipsoid of. If E R is a subspace, ad P roj E is the orthogoal projectio oto E i R, the by heorem 3.1, V ol(p roj E ()) 1/ (e c V ol(p roj E (E))) 1/ = c V ol(p roj E (E)) 1/. Lemma 3.2 Let R be a covex body of volume oe whose baryceter is at the origi. Let E R be a subspace of ay dimesio. he, V ol( E) 1/ 1 > c V ol(p roj E ()). 1/ 7

8 Proof: Deote m = dim(e) ad M = max x E V ol( (E + x)). By Fubii ad by heorem 1 i [F], ( ) m + 1 V ol() MV ol(p roj E ) V ol(p roj m + 1 E )V ol( E). Sice V ol() = 1, we obtai V ol( E) 1/ > ( ) m m V ol(p roj E ()) 1/ ( ) m m+1 ad sice +1 > c, the lemma follows. Proof of Propositio 1.3: First assume that m 2. Recall that L = sup C R L C where the supremum is take over all isotropic covex bodies i R. his supremum is attaied by a compactess argumet (the collectio of all covex sets modulu affie trasformatios is compact). Defie to be oe of the bodies where the supremum is attaied; i.e. L = L ad is isotropic ad of volume oe. Let E be a M-ellipsoid of. Sice E is a ellipsoid of volume oe, it has at least oe projectio oto a subspace E of dimesio m, such that By Lemma 3.2, V ol(p roj E ) 1/ < cv ol(p roj E E) 1/ < C. V ol( E) 1/ > c. Let be a m-dimesioal body such that L = L m, ad is of volume oe ad isotropic. Deote by the (, E)-symmetrizatio of. he L = L L, ad by heorem 2.5, L L or equivaletly, > cl 1 m L m V ol( E) 1/ > cl 1 m L = L > c m L = c m Lm. Sice we assumed that m 2, we get L m < c L. Regardig the case i which m < 2 : Note that L m L 2m, sice the 2m dimesioal body which is the cartesia product of with itself, has the same isotropic costat as. If s is the maximal iteger such that 2 s m, the clearly 2 s m > 2, ad therefore L m L 2 s m < c L. L m 8

9 Remark 3.3: I the proof of Propositio 1.3 we showed that for every covex body R of volume oe, ad for ay 1 k, there exists a k-dimesioal subspace E such that V ol( E) 1/ > c. his fact is a direct cosequece of the existece of a M-ellipsoid, but may ot be very trivial to obtai directly. We would like to metio a additioal property attributed to a body R, which has the largest possible isotropic costat. For this purpose, we will quote a useful result which appears i [Ba1] ad i [MP]. Our formulatio is closer to the oe i [MP] (Lemma 3.1, ad Propositio 3.11 there). Although results i that paper are stated oly for cetrally-symmetric bodies, the symmetry assumptio is rarely used. he geeralizatio to o-symmetric bodies is straightforward, ad reads as follows: Lemma 3.4 Let R be a isotropic covex body of volume oe. Let 1 k ad let E be a k-codimesioal subspace. Defie C as the uit ball of the (o-symmetric) orm defied o E as θ = θ 1+ p p+1 /( x, θ p dx E(θ) ) 1 p+1 for p = k + 1, where E(θ) = {x + tθ; x E, t > } is a half of a k 1-codimesioal subspace. he ideed C is covex, ad L C L V ol( E) 1/k. Corollary 3.5 Let R be a covex isotropic body of volume oe, such that L = L. he for ay subspace E R of codimesio k, where c is a umerical costat. Proof: By Lemma 3.4, V ol( E) 1/k < c V ol( E) 1 k L C L = L C L L k L < c where the last iequality follows from Propositio Proof of the reductio to bodies with fiite volume ratio I this sectio, assume that R is a covex isotropic body of volume oe, such that L = L. Apriori, a M-ellipsoid of may be 9

10 very differet from a Euclidea ball. We shall see that Corollary 3.5 imposes striget coditios o the axes of a M-ellipsoid. 4.1 Cotrollig the axes of a M-ellipsoid Deote by κ m the volume of a uit{ Euclidea ball i R m. It is well kow that κ 1/m m 1 m. Let E = x R ; } x 2 i i 1 be a M- λ 2 i ellipsoid of, whose existece is guarateed i heorem 3.1. he axes of this ellipsoid are of legths λ 1,.., λ, ad ( i=1 λ i) 1/ 1, sice the volume of a M-ellipsoid is oe. Assume that the λ i s are ordered, i.e. λ 1... λ. For coveiece, ad without loss of geerality, we assume that is divisible by four. Claim 4.1 λ /2 < c, for some umerical costat c. Proof: Let E R be ay subspace of ay dimesio. By Lemma 3.2 ad Corollary 3.5, V ol(p roj E ()) 1/ > c V ol( E ) 1/ > c. Let E = sp{e 1,.., e /2 }, the liear space spaed by e 1,.., e /2. he, c < V ol(p roj E ()) 1/ N(, E) 1/ /2 κ /2 i=1 1/ λi /2 /2 because V ol(p roj E (E)) = κ /2 1 λi. Sice (κ /2 ) 1/ 1, we get that 2/ /2 > c. Hece we obtai, λ /2 i= 2 +1 λ i 2/ i=1 4.2 Fiite volume ratio λ i ( ) 2/ /2 = λ i i=1 i=1 λ i 2/ < c. (4) he followig lemma, whose proof ivolves the otio of a M-ellipsoid, origially appears i [M2]. It ca also be deduced from the proof of Corollary 7.9 i [P]. 1

11 Lemma 4.2 Let R be a covex body. Let < λ < 1. he there exists a subspace G of dimesio λ such that if P : R R is a projectio (i.e. P is liear ad P 2 = P ) such that ker(p ) = G, the P () has a volume ratio smaller tha c(λ), where c(λ) is some fuctio which depeds solely o λ. he cetral theme uderlyig the proof which follows, is the coectio betwee a M-ellipsoid ad the isotropy ellipsoid of a body with the largest possible isotropic costat. his coectio arises whe we project oto the subspace E = sp{e 1,.., e /2 }, together with its coverig ellipsoid. Accordig to (4) we get that P roj E (E) c D, so i fact the ormalized Euclidea ball is a M-ellipsoid for P roj E (). I other words, the isotropy ellipsoid ad the selected M-ellipsoid of are equivalet i a large projectio. herefore, we may combie the properties of a M-ellipsoid with the properties of the isotropy ellipsoid, to create a fiite volume ratio body. Apply Lemma 4.2 to the body P roj E (). here exists a subspace F E such that dim(f ) = /4 ad v.r.(p roj F ()) = v.r.(p roj F (P roj E ())) < C. Ideed, F is the orthogoal complemet i E, to the subspace G from Lemma 4.2. Deote as the (D F, F )-symmetrizatio of, where D F is the stadard Euclidea ball i F. he, F = P roj F ( ) = P roj F () is a fiite volume ratio body, i.e. there exists a ellipsoid F F ( ) such that V ol( 4/ F ) V ol(f) < C. We claim that has a bouded volume ratio. Ideed, the ellipsoid E = { λx + µy; λ 2 + µ 2 1, x F, y F } satisfies 1 2 E cov{f, F }, V ol(e ) 1/ 1 2C V ol(p roj F ( )) 1/ V ol( F ) 1/ 1 2C, by Lemma 3.2. Hece E is evidece of the fiite volume ratio property of. Note also that accordig to Claim 4.1, V ol(p roj F ()) 1/ N(, E) 1/ V ol(p roj F ( λ /2 D)) 1/ < c. Hece by Lemma 3.2 ad heorem 2.5 L L 1/4 L 1/4 V ol( F ) 1/ > c V ol(p roj F ()) > 1/ c L 1/4 11

12 ad therefore, L < c(l ) 4 where L = L ( c) is the largest possible L amog all covex bodies i R, havig volume ratio ot larger tha c, ad Propositio 1.1 is proved. Remark 4.3: Regardig the coectio betwee v, L ad L (v); Formally, we have proved for some v > 1 that L (L (v)) 4 for all. However, by adjustig the dimesios of the subspaces E ad F, we ca reduce the power of L (v), at the expese of icreasig the volume ratio costat, v. he depedece obtaied usig this method is quite poor: For ay < θ < 1, L e c 1 θ L(e c 1 θ ) 1 θ. 5 he isotropic positio ad a M-ellipsoid Proof of Propositio 1.4: Deote D m = {x R m ; x κ 1/m m }, a Euclidea ball of volume oe. Let R be a covex isotropic body of volume oe. Deote, = { (x 1, x 2 ); x 1 LD L, x 2 L L D D } R 2. Let E R 2 be the subspace spaed by the first stadard uit vectors, ad let F = E. We claim that is a isotropic body. By a reasoig similar to that i Lemma 2.2, the subspaces E ad F are ivariat uder the actio of the matrix M. I additio, M acts as a multiple of the idetity i both subspaces. Let us show that it is the same multiple of the idetity i both subspaces, ad hece M is a scalar matrix. For ay v E, Also, for ay v F, x, v 2 dx = L D L x, v 2 dx = L L D x, v 2 dx = L D L. D x, v 2 dx = L L D. herefore is isotropic. Accordig to our assumptio, a reverse Bru-Mikowski iequality holds. Hece by (1), ( V ol( + D 2 ) 1/2 < C V ol( ) 1/2 + V ol(d 2 ) 1/2) = 2C. (5) 12

13 But L L D D + D 2 + D 2. Hece, V ol( + D 2 ) 1/2 > V ol ( ) 1/2 ( L L D + D 2 > c L D L D Combiig (5) ad (6), ad usig the fact that L D < c we get L < ( cc) 4 ) 1/4. ad sice is arbitrary, the isotropic costat of a arbitrary covex body i R is uiversally bouded. Remark: he proof of Propositio 1.1 uses the close relatio betwee a M-ellipsoid ad the isotropy ellipsoid of the body whose isotropic costat is as large as possible. As follows from Propositio 1.4, if we could deduce such a relatio betwee a M-ellipsoid ad the isotropy ellipsoid of a arbitrary covex body R, the a uiversal boud for the isotropic costat will follow. 6 Appedix: Cocave Fuctios his sectio proves Lemma 2.3 i a way similar to the proofs preseted i [Ba1], [F]. he followig lemma reflects the fact that amog all cocave fuctios o the lie, the liear fuctio is extremal. Lemma 6.1 Let f : [, ) [, ) be a compactly supported fuctio such that f 1/k is cocave o its support ad a = f() >. Let > ad choose b such that ( ) k f(x)x dx = a 1/k bx + x dx where x + = max{x, }. he for ay p > 1 f(x) p x dx (6) ( ) pk a 1/k bx + x dx. (7) Proof: Sice f has a compact support, f(x)x dx <, so b >. Deote h(x) = a 1/k f(x) 1/k. he h is a covex fuctio ad h() =. herefore h(x) = h(x) x is icreasig. Sice (a 1/k x h) k +x dx = (a 1/k bx) k +x dx it is impossible that h is always smaller or always larger tha b. he fuctio h is icreasig, so there exists x [, ) such that h b o 13

14 [, x ] ad h b o [x, ). Deote g(x) = ( a 1/k bx ) k. I order to + obtai (7) we eed to prove that p f(x) y p 1 dyx dx p g(x) y p 1 dyx dx. Sice (g(x) f(x))(x x ), ad g p 1 is a decreasig fuctio, x f(x) g(x) g(x) x f(x) y p 1 dyx dx y p 1 dyx dx Subtractig (9) from (8), we obtai f(x) g(x) ad the lemma is prove. x f(x) g(x) g(x) x f(x) g(x ) p 1 dyx dx, (8) g(x ) p 1 dyx dx. (9) y p 1 dyx dx g(x ) p 1 (f(x) g(x))x dx = Proof of Lemma 2.3: he iequality o the right has othig to do with log-cocavity: Sice R f = 1, R f 1+ 2 k = R f f 2 k R f M 2 k = M 2 k. Let us prove the left-most iequality. By traslatig f if ecessary, we may assume that f() = M. We shall begi by itegratig i polar coordiates: R f(x) 1+ 2 k dx = S 1 f(rθ) 1+ 2 k r 1 drdθ. Fix θ S 1, ad deote g(r) = f(rθ). he g 1/k is cocave, as a restrictio of a cocave fuctio to a straight lie. Now, by Lemma 6.1 for p = k, g(x) 1+ 2 k x 1 dx ( ) k+2 a 1/k bx + x 1 dx (1) where a = g() ad b is chose as i Lemma 6.1, i.e. g(x)x 1 dx = ( a 1/k bx ) k + x 1 dx. A elemetary calculatio yields that ( a 1/k bx ) k+2 + x 1 dx ( a 1/k bx ) k + x 1 dx = (k + 1)(k + 2) a2/k ( + k + 1)( + k + 2) (11) 14

15 where we used the fact that 1 xa (1 x) b dx = a!b! (a+b+1)!. Deote c,k =. Combiig (1) ad (11) we obtai (k+1)(k+2) (+k+1)(+k+2) g(x) 1+ 2 k x 1 dx a 2/k c,k = c,k g() 2/k g(x)x 1 dx or i other words, for every θ S 1, ( ) k a 1/k bx x 1 dx + f(rθ) 1+ 2 k r 1 dr c,k f() 2/k f(rθ)r 1 dr. By itegratig this iequality over the sphere S 1, R f(x) 1+ 2 k dx c,k f() 2/k R f(x)dx = c,k f() 2/k. Refereces [Ba1]. M. Ball, Logarithmically cocave fuctios ad sectios of covex sets i R. Studia Math. 88 (1988) [Ba2]. M. Ball, Normed spaces with a weak-gordo-lewis property, Proc. of Fuct. Aal., Uiversity of exas ad Austi ( ), Lecture Notes i Math., vol. 147, Spriger (1991) [BF]. Boese, W. Fechel, heory of covex bodies. raslated from Germa ad edited by L. Boro, C. Christeso ad B. Smith. BCS Associates, Moscow, Idaho USA (1987). [Bou] J. Bourgai, O the distributio of polyomials o high dimesioal covex sets, Geometric aspects of fuctioal aalysis (1989 9), Lecture Notes i Math., vol. 1469, Spriger Berli (1991) [BM] J. Bourgai, B. lartag, V.D. Milma, A reductio of the slicig problem to fiite volume ratio bodies, C. R. Acad. Sci. Paris, Ser. I 336 (23) [D] [F] S. Dar, Remarks o Bourgai s problem o slicig of covex bodies, Geometric aspects of fuctioal aalysis, Operator heory: Advaces ad Applicatios, vol. 77 (1995) M. Fradelizi, Sectios of covex bodies through their cetroid, Arch. Math 69 (1997)

16 [J] M. Juge, Hyperplae cojecture for quotiet spaces of L p. Forum Math. 6, o. 5, (1994) [M1] V.D. Milma, Iégalité de Bru-Mikowski iverse et applicatios à le théorie locale des espaces ormés. C.R. Acad. Sci. Paris, Ser. I 32 (1986) [M2] V.D. Milma, Geometrical iequalities ad mixed volumes i the local theory of Baach spaces. Colloquium i hoor of Lauret Schwartz, Vol. 1 (Palaiseau, 1983). Astérisque No. 131 (1985), [MP] V.D. Milma, A. Pajor, Isotropic positio ad iertia ellipsoids ad zooids of the uit ball of a ormed -dimesioal space. Geometric aspects of fuctioal aalysis ( ), Lecture Notes i Math., vol. 1376, Spriger Berli, (1989) [Pa] G. Paouris, O the isotropic costat of o-symmetric covex bodies. Geometric aspects of fuctioal aalysis, Lecture Notes i Math., 1745, Spriger, Berli (2) [P] G. Pisier, he volume of covex bodies ad Baach space geometry, Cambridge racts i Mathematics, Cambridge uiv. Press, vol. 94 (1997). 16