ESTIMATING VOLUME AND SURFACE AREA OF A CONVEX BODY VIA ITS PROJECTIONS OR SECTIONS

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1 ESTIMATING VOLUME AND SURFACE AREA OF A CONVEX BODY VIA ITS PROJECTIONS OR SECTIONS ALEXANDER KOLDOBSKY, CHRISTOS SAROGLOU, AND ARTEM ZVAVITCH arxiv:submit/73588 [math.mg] 27 Nov 206 Abstract. The mai goal of this paper is to preset a series of iequalities coectig the surface area measureofacovex body adsurfaceareameasure ofitsprojectios adsectios. We asweraquestio from [GKV] regardig the compariso of the surface over volume fuctioal of a zooid ad of its projectios. I additio, we preset a solutio of a questio from [CGG] regardig the asymptotic behavior of the best costat i a recetly proposed reverse Loomis-Whitey iequality. Next we give a ew sufficiet coditio for the slicig problem to have a affirmative aswer, i terms of the least outer volume ratio distace from the class of itersectio bodies of projectios of at least proportioal dimesio of covex bodies. Fially, we show that certai geometric quatities such as the volume ratio ad miimal surface area (after a suitable ormalizatio) are ot ecessarily close to each other.. Itroductio I the past decades a lot of effort has bee put i the study of problems of estimatig volumetric quatities of a covex body (i.e. a covex compact set with o-empty iterior) i terms of the correspodig fuctioals of its sectios or projectios. We refer to the books [Ga2, Ko, KoY, RZ, S] for more iformatio, examples ad the history of those problems. Our aim is to cotiue this ivestigatio by cosiderig a umber of problems of this type. Let K be a covex body i R, 3 ad u be a uit vector. Defie the followig quatity: R u (K) := K S(K) S(K u ) K u where u = {x R : x,u = 0} deotes the subspace which is orthogoal to the uit vector u, K u deotes the orthogoal projectio of K oto u ad S( ), deote the surface area ad the volume fuctioals respectively i the give space. I [GHP] it was show that R u (K) 2( )/ ad i [FGM] it was show that this iequality is sharp i the sese that there exist covex bodies K ad a uit vector u such that R u (K) is arbitrarily close to 2( )/. May iequalities, ivolvig projectios, ca be improved or simplified if cosidered for the class of zooids. The followig questio was posed i [GKV]: Questio.. Are there better bouds for R u (K) if we restrict ourselves i the class of zooids? I Sectio 3 (Theorem 3.) we show that the aswer to this questio is egative; we ca fid a zooid Z R such that R u (Z) is arbitrarily close to 2( )/. I Sectio 4, we cotiue the study of iequalities ivolvig the size of projectios of -dimesioal covex body ad study the followig problem proposed i [CGG]:, Questio.2. Fid the smallest costat Λ, such that the followig iequality holds for all covex bodies i R : () mi K e i Λ K, {e,...,e } F where F deotes the set of all orthoormal basis i R. 200 Mathematics Subject Classificatio. 52A20, 52A2, 52A38, 52A40. Key words ad phrases. Covex bodies; Hyperplae sectios; Orthogoal Projectios; Zooids; Itersectio bodies. The third amed author is supported i part by the U.S. Natioal Sciece Foudatio Grat DMS-0636.

2 Iequality () ca be viewed as a reverse to the classical Loomis-Whitey [LW] iequality for compact subset A R : (2) A e i A. The authors i [CGG] asked for the correct asymptotical behavior of Λ /. I sectio 4 (Theorem 4.) we show that Λ / c, for some absolute costat c > 0. Moreover, we prove that this estimate is the best possible up to a absolute costat c. We ote that other variats of reverse Loomis-Whitey type iequalities where cosidered i [CGaG]. I Sectio 5, we study a umber of iequalities which arise i the study of compariso problems. For example, what iformatio o covex bodies we ca have from compariso iequalities o its curvature fuctios. We also study the relatioship of volume ratio, curvature measure ad surface area of covex bodies. For istace we prove that a covex body (eve highly symmetric) ca have large volume ratio (close to the volume ratio of the cube of the same volume) but small miimal surface area (close to the surface area of the ball of the same volume). Notice that it is well kow that the opposite caot happe (see (7) below). Bourgai s slicig problem [Bou] asks whether ay covex body of volume has a hyperplae sectio of volume greater tha c, where c is a absolute costat. It follows from the work of the first amed author [Ko2, Theorem ] (Theorem C i Sectio 6) that if a cetrally symmetric covex body K has bouded outer volume ratio with respect to the class of itersectio bodies (see Sectio 6 for defiitio), the the slicig problem has a affirmative aswer for K (actually this result exteds to geeral measures i the place of volume). I Sectio 5, we exted this result (however, ot bodywise) as follows (Theorem 6.): If every covex body i R has a projectio of dimesio at least proportioal to which is close to a itersectio body (i the sese of outer volume ratio), the the slicig problem will have a affirmative aswer. 2. Backgroud ad otatio We use the otatio a b to declare that the ratio a/b is bouded from above ad from below by absolute costats. We deote by D the stadard -dimesioal Euclidia ball of volume ad by B 2 the Euclidia ball of radius. Deote, also, by ω the volume of B 2. A body L is called a star body if it is star-shaped at the origi ad its radial fuctio ρ L is positive ad cotiuous. We remid that the radial fuctio ρ L of L is defied as: ρ L (u) = max{λ > 0 : λu L}, u S, where S = {x R : x = } is the uit sphere i R. I this sectio, K will always deote a covex body i R. The support fuctio of K is defied as: h K (x) = max y K x,y, x R. The surface area measure (or curvature measure) S K of K is a measure o S, defied by: S K (Ω) = H ( { x bdk : u Ω, such that x,u = hk (u) }), Ω is a Borel subset of S, where H is the ( )-dimesioal Haussdorff measure. If S K is absolutely cotiuous with respect to the Lebesgue measure, its desity is deoted by f K ad it is called curvature fuctio of K. A zooid i R is a origi symmetric covex body that ca be approximated (i the Hausdorff metric) by fiite Mikowski sums of lie segmets (called zootopes). It is iterestig to ote that every covex symmetric body i R 2 is a zooid, ad this statemet is o loger true i R, 3. We refer to [S, Sectio 3.2] for more iformatio o zooids ad zootopes. Oe of the equivalet defiitios of zooids, leads to a otio of a projectio body of a covex body K. A symmetric covex body ΠK is the projectio body of K if its support fuctio is give by (3) h ΠK (u) = K u = 2 S x u ds K (x) for all u S. 2

3 If K,...,K are compact covex sets i R, we deote their mixed volume by V(K,...,K ). We refer to [S] or [Ga2] for a extesive discussio o the theory of mixed volumes. Note that if S(K) is the surface area of K, the (4) S(K) = V(K[ ],B 2 ) = V(B 2,K[ ]) = S ds K. I geeral, V(K[ ],L) = h L ds K S ad K = h K ds K. S The Mikowski iequality for mixed volumes states: (5) V(K[ ],L) K ( )/ L /, with equality if ad oly if K ad L are homothetic. As F. Joh [J] (see also [AGM], page 50) proved, there exists a uique ellipsoid JK of maximal volume cotaied i K, the so-called Joh ellipsoid of K. Sice, for T GL() we have that J(TK) = T(JK), there always exists T SL(), such that J(TK) is a ball. The quatity vr(k) := ( K / JK ) / is called the volume ratio of K. Cosider a covex body K, with K = ad such that JK is a Euclidea ball. The, = K = S h JK ds K = JK / S(K). S h K ds K We kow by Stirlig s formula that ω / /, hece (6) S(K) c JK = c vr(k). / ω / Defie also the quatity S(TK) (K) := mi T GL() TK. ( )/ We say that K is i miimal surface area positio if S(K) = (K) ad K = (see [AGM], Sectio 2.3). With this defiitio, (6) gives: (7) (K) c vr(k). Parameters (K) ad vr(k) are affie ivariats. A useful characterizatio of the miimal surface area positio due to Petty [Pe] states that a covex body K of volume is i miimal surface area positio if ad oly if its surface area measure S K is isotropic, i.e. (8) x,u 2 ds K (x) = S S(K), for all u S. We say that the covex body K of volume is i miimal mea width positio (see [AGM, Sectio 2.2]) if S h K (u)du S S h TK (u)du S for every T SL(). A very useful result which follows from Figiel, Tomczak-Jaegerma, Lewis ad Pisier estimates (see [AGM, Corollary 6.5.3]) gives that for a symmetric covex body K R i miimal mea width positio ad of volume we get (9) S h K (u)du C log, S for some absolute costat C > 0. If K cotais the origi i its iterior, the polar body K of K is defied to be the covex body K = {x R : x,y, y K}. 3

4 If T GL() ad H G,k, the followig formulas hold: (TK) = T K ad (K H) = K H. Here, G,k deotes the Grassmaia maifold of k-dimesioal subspaces of R ad the otatio A H deotes the orthogoal projectio of A oto the subspace H. Let us assume that the origi is the cetroid of K. The Blaschke-Sataló iequality (see [Sa, RZ]) together with its reverse (see [BM, RZ]) give: (0) ( K K ) / /. Set N(K,D ) to be the coverig umber of K by traslates of D, i.e. N(K,D ) = mi{n N : x,...,x N R, such that K N (x i +D )}. Milma ([M], see also [AGM, Chapter 8]) proved that there exists a absolute costat C > 0 ad a liear image K of K of volume, such that { max N(K,D ),N( K / K,D ), K D, ( K / K ) D } C. If the previous iequality holds for K, we say that K is i M-positio. We will also eed to use the otio of isotropic positio of a covex body (see [MP, BGVV]): there exists T SL(), such that x,y 2 dy = L 2 K K (+2)/ x 2, x R TK we will call TK a isotropic covex body or a body i isotropic positio. The parameter L K is called the isotropic costat of K. L K depeds oly o K ad it is ivariat uder ivertible liear maps. It turs out that () L 2 K = mi T SL() x 2 dx = K (+2)/ TK mi T GL() x 2 dx. TK (+2)/ TK It is a major problem to show that L K is uiformly bouded from above by a absolute costat (the fact that L K > c ca be proved by compariso with a Euclidea ball see [MP, BGVV]). The best estimate curretly is due to Klartag [K]: L K C /4 who removed the logarithmic term i the previous estimate of Bourgai [Bou2]. It should be oted that (see [MP, BGVV]): ( /2 (2) K u x,u du) 2, u S. K Thus, the problem of boudig L K is equivalet to Bourgai s slicig problem. 3. Surface area vs surface area of projectios of zooids The goal of this sectio is to prove the followig: Theorem 3.. Let 3. There exists a sequece {Z m } of zooids i R ad a uit vector u S, such that lim m R u (Z m ) = 2( )/. The followig result (that we thik is of idepedet iterest) will be crucial for the proof of Theorem 3.. Propositio 3.2. Let K be a covex body i R, 3 ad u,v S, u ±v. For t > 0, set K t := K +t[ v,v]. The, lim R u(k t ) = ( )2 t ( 2) K v S(K v ) S ( (K v ) (v v ) ) (K v ) (v v ), where v := u v. 4

5 Proof of Theorem 3.. We first otice that if w S ad K is a covex body i w, the lim R (2t K )S(K) w(k +t[ w,w]) = lim t t (2 K +2tS(K)) K =. I particular, for ay ε > 0, there exists a symmetric covex body i R 2 (thus a zooid) ad w S 2, such that R w (Z) 3ε/4. We will show the assertio of the theorem by iductio i. Let ε > 0. Assume that there exists a zooid Z i R, such that R w (Z) 2( 2) ε( 2) ( ) 2, for some w R. The, there exists a zooid Z i R R R, such that Z e = Z ad a vector u S, such that w = u e. By Propositio 3.2, we get: if 4, while lim R u(z +t[ e,e ]) = ( )2 t ( 2) R w(z e ) 2( ) ε, lim R u(z +t[ e,e ]) = 4 t 3 R w(z e ) 4 3 ε, if = 3. Note that Z +t[ e,e ] is also a zooid for ay t > 0. Sice ε was chose to be arbitrary, this proves our claim. Proof of Propositio 3.2. The followig coectig the mixed volume of the bodies ad their projectios ca be foud i [S, Sectio 5.3]: (3) V([ u,u],m,...,m ) = 2 V u (M u,...,m u ) for ay u S. Here V u deotes the mixed volume fuctioal i u. Let t > 0. Usig (4) ad (3), we have: K +t[ v,v] = t i V(K[i],[ v,v][ i]) (4) Also, (5) Moreover, = K +tv(k[ ],[ v,v]) = K + 2t K v. S(K +t[ v,v]) = V ( (K +t[ v,v])[ ],B2 ) = t i V ( K[i],[ v,v][ i ],B2 ) ( = V(K[ ],B2 2t ) )+ V v (K[ 2],[ v,v],b 2 ) ( = S(K)+2tV v (K v )[ 2],B2 v ) = S(K)+ 2t S(K v ). (K +t[ v,v]) u = 2 V( (K +t[ v,v])[ ],[ u,u] ) = t i V(K[i],[ v,v][ i],[ u,u]) 2 = ( ) V(K[ ],[ u,u])+tv(k[ 2],[ v,v],[ u,u]) 2 = K u ( +tv v (K v )[ 2],[ v,v ] ) = K u + t v (6) (K v 5 ) (v v ).

6 Fially, S ( (K +t[ v,v]) u ) ( ((K = ( )V +t[ v,v]) u ) [ 2],B2 u ) = ( ) V ( (K +t[ v,v])[ 2],B2 2,[ u,u]) = ( ) 2 t i V(K[ i 2],[ v,v][i],b2 2,[ u,u]) = ( ) ) (V(K[ 2],B2 2,[ u,u])+tv(k[ 3],[ v,v],b 2,[ u,u]) = S(K u ( )+( )tv v (K v )[ 3],B2 v,[ v,v ] ) = S(K u )+ ( )t v ( ((K v V 2 v v ) (v v ) ) [ 3],(B2 v ) (v v ) = S(K u )+ t v (7) 2 S( (K v ) (v v ) ). Combiig (4), (5), (6), (7) ad lettig t, we easily arrive to the proof of our assertio. 4. A remark o the reverse Loomis-Whitey iequality The mai goalof this sectio is to provide a sharp asymptotic estimate for the quatity Λ / Loomis-Whitey iequality. Theorem 4.. There exists a absolute costat c > 0, such that (8) Λ (c ). Moreover, there exists a symmetric covex body L i R, such that (9) mi L e i (c ) L, {e,...,e } F where c > 0 is aother absolute costat. i the reverse TheproofofTheorem4.followsfromtwotheoremsdue tok.ball. The firstisball sreverseisoperimetric iequality (for symmetric covex bodies): Theorem A. [Ba2] Let K be a covex body ad be a simplex i R. The, (K) ( ) c. The secod is a remarkable example of a covex body whose projectios all have large volumes(comparig to its volume): Theorem B. [Ba] There exists a symmetric covex body L i R of volume, such that L u, for all u S. ) Proof of Theorem 4.: A example of a covex body that satisfies (9) will be the covex body L from Theorem B. To prove (8), we may assume that K is of volume. Let us first cosider a covex body M i miimal surface area positio (thus M = ). It was observed i [GP], that (8), Theorem A ad the Cauchy-Swartz iequality gives (20) ( M u = x,u ds M (x) x,u ds M (x) 2 S 2 S = ( ) /2 ( ) /2 x 2 ds M (x) ds M (x) 2 S S = 2 (M) M ( )/ c, 6 ) /2 ( S ds M (x) ) /2

7 for all u S. Notice that for every covex body K, K = there exists T SL(), such that TK is i miimal surface area positio. Sice both (8) ad (8) are ivariat uder orthogoal trasformatios, we may assume that T is a diagoal matrix with strictly positive etries. The, T = diag(λ,...,λ ) for some λ,...,λ > 0 with λ i =. Cosequetly, if {e,...,e } is the stadard orthoormal basis i R, it follows by (20) ad the fact that T is diagoal that: K e i = T T(K e i ) = TK e i = j {,...,}\{i} TK e i (c ) = (c ). λ i λ i 5. Curvature measures ad compariso of volumes Let us cotiue i aother directio with the followig observatio: Propositio 5.. Let K, L be cetrally symmetric covex bodies i R 2. If S K S L, the K L. Proof: Usig the defiitio of projectio body of a covex body (see equatio (3) we get that ΠK ΠL. However oe ca otice that for ay symmetric covex body M R 2 we get that ΠM = 2OM, where O is the rotatio by π/2. The result follows. Remark 5.2. After a suitable traslatio, Propositio 5. remais true i the o-symmetric plaar case as well. This is due to the additivity of the curvature measures of plaar covex bodies. Ideed, set µ := S L S K. Mikowski s Existece ad Uiqueess Theorem (see [S, Sectio 8.2]) states that there exists a uique (up to traslatio) compact covex set M, such that S M = µ. Assumig that K, L ad M cotai the origi, we get: L = K +M K, where we used the fact (see [S, Sectio 8.3]) that i the plae S K+M = S K +S M ad agai Mikowski s Existece ad Uiqueess Theorem. Oe caot aturally expect Propositio 5. to hold true i ay dimesio. Ideed, let K = [,] 3 [ /0,/0] 2 [ 2,2] ad L = [,], 3, the S K S L, but K L. From aother poit of view, oe ca always guaratee the compariso of volumes: Propositio 5.3. Let K, L be cetrally symmetric covex bodies i R. If S K S L, the K L. Proof: Ideed, from S K S L we get S h L S K S h L S L, ad ca use Mikowski iequality (5) to fiish the proof. Thus it is iterestig to ask for compariso with a Euclidea Ball : If K is a symmetric covex body of volume, such that JK is a Euclidea ball ad S K S cd, for some absolute costat c > 0, is it true that K has bouded volume ratio? Actually, oe might ask a weaker (as (7) shows) versio of the previous questio: Questio 5.4. Let K be a symmetric covex body, such that JK is a ball. If S K S cd, is it true that the quatity (K)/ is bouded from above by a absolute costat? A extra motivatio for Questio 5.4 is Propositio 5.8 (see below). As we show the aswer to this questio is egative. Recall that a covex body is called -symmetric if its symmetry group cotais the symmetry group of the stadard -cube. Theorem 5.5. There exists a absolute cotat c > 0, such that for each poitive iteger, there exists a -symmetric covex body L of volume (thus JL is a Euclidea ball ad (L) = S(L)) i R, such that S L > cs D, but (L)/ c. 7 λ j

8 Remark Ball s reverse isoperimetric iequality shows that the factor / gives the worst possible order i Theorem Note that cs D = S c /( ) D, so the assumptio i Theorem 5.5 is much stroger tha the assumptio i Questio 5.4. Proof of Theorem 5.5: Let C be the stadard -cube of volume. Cosider the covex body K defied as the Blaschke average of C ad D : S K = 2 S C + 2 S D. Note that C is i M-positio (this, follows, for example from coverig C by copies of D, see Lemma 8..3 i [AGM]), thus: C D C. Usig the Mikowski iequality (5), we obtai: C K ( )/ C D / K ( )/ h C D ds K S = h C D 2 ds C + h C D S 2 ds D S h C ds C + h D ds D 2 S 2 S This shows that = 2 C + 2 D =. (2) K ( )/ C. Moreover, usig the fact that S(D ) ad S(C ) = 2, we get: therefore 2+c S(C )+S(D ) S(K) 2 C 2C K, ( )/ S(K) K ( )/ c. Set L := (/ K / )K. The, L = ad S L 2 K ( )/S D [/(2C)]S D. However, (L) = S(K) K ( )/ c, as claimed. Let K 0 be a cetrally symmetric star body i R. Recall the defiitio of the curvature image C(K 0 ) of K 0 by its surface area measure (see [L] for more iformatio): f C(K0)(x) = ds C(K 0)(x) = ρ+ K 0 (x), x S. dx + A covex body L is called body of elliptic type if there exists a covex body K 0, such that L = C(K 0 ). Questio 5.7. Is Questio 5.4 true for symmetric bodies of elliptic type? Below we metio the followig related result. The proof follows the idea of [Sa, Propositio 5.3] ad we omit it. Propositio 5.8. (Coditioal) If Questio 5.7 had a affirmative aswer, the the slicig problem would have a affirmative aswer as well. 8

9 As show i [Sa], for ay symmetric covex body K, L K (C(K)) (so the opposite of Propositio 5.8 obviously holds). Hece, it follows by (7) that if all cetrally symmetric bodies of elliptic type have uiformly bouded volume ratios, the the isotropic costat would be uiformly bouded. It would be, therefore, atural to ask if the reverse iequality of (7) is true. Questio 5.9. Is there a absolute costat c > 0. such that for all cetrally symmetric bodies of elliptic type K? c vr(k) (K), If Questio 5.9 had a affirmative aswer, the for ay symmetric covex body K, the followig would hold true: L K is bouded if ad oly if vr(c(k)) is bouded. This provides a good motivatio for our ext result: a reverse iequality to (7) caot hold true for geeral symmetric covex bodies. Theorem 5.0. There exists a covex body K i R (actually K ca be take to be -symmetric), such that (K) c vr(k) log(+). Remark 5.. It follows by Ball s volume ratio iequality [Ba2] ad the classical isoperimetric iequality that Theorem 5.0 gives the worst possible case up to a logarithmic factor. Proof of Theorem 5.0: We first otice that the mea width of B (the uit ball of l ) is of the order (log(+))/ (this ca be calculated by passig to the itegral over the gaussia measure ad usig stadard estimates for the sequece of stadard ormal variables, see [AGM] ad [LT, Chapter 3, page 79] for more details). Therefore, itegratig i polar coordiates, we get: D x dx = + + = ω x ρ + D dx S dx x S S + S ω / x dσ(x) S (log(+))/ = log(+), where σ is the Haar probability measure o S. We ote that x dx = D { x > s} ds = D 0 Let s 0 > 0 be such that (22) D s 0 B = D \s 0 B = /2. D x dx = s0 0 s0 It follows that s 0 C log(+). Take, ow, > 0 D \sb ds+ D \s 0 B ds+ 0 D \sb ds. s 0 D \sb ds K := D s 0 B. Note that K is -symmetric, with K = /2 ad thus (K) = S(K) S(D ) C 2. s 0 D \sb ds > s 0 /2. Moreover, sice K is -symmetric, JK is a Euclidea ball ad the largest Euclidea ball cotaied i K. Usig the defiitio of K we get that JK is ot larger the the largest Euclidea ball cotaied i s 0 B, so JK s 0 B 2 (/c ) log(+)b 2, 9

10 for some absolute costat c > 0. Thus, ( ) / K vr(k) c /2 log(+)b2 c / log(+)ω c 2 Remark 5.2. By a well kow lemma of C. Borell [Bor], we have D \sb C 0e c0s, c 2 (K). log(+) C 2 log(+) for s 2s 0, where c 0,C 0 > 0 are absolute costats ad s 0 is defied by (22). Therefore, ad 2s 0 D \sb ds 2s0 0 0 C 0 e c0s ds C 0 D \sb ds 2s 0 D = 2s 0. Thus, s 0 c 3 log(+), which shows that the logarithmic factor i the example of Theorem 5.0 caot be removed. Remark 5.3. Note that if K is the example from Theorem 5.0 (recall that K = /2), by (8) ad (20), we have K u c 4,where we used the fact that sice K is -symmetric, S K is isotropic. Also, K u K u c 5 K ( )/ c 6, sice K is i isotropic positio ad it has bouded isotropic costat, as -symmetric. Cosequetly (after a suitable dilatio), we may take K i Theorem 5.0 such that all its projectios have volumes of costat order, whe it is i miimal surface area positio. The ext two theorems cotiue the discussio started i Propositio 5.3. Our goal is to provide bouds o the differece of volumes of covex bodies via its curvature fuctios. Theorem 5.. Let K,L R be covex bodies with absolutely cotiuous surface area measure, such that f K (θ) f L (θ), for all θ S. The L K ω Proof: Cosider two covex bodies K,L R such that (23) f K (θ) f L (θ) ǫ, for some ǫ 0 ad all θ S. The or S h L (θ)f K (θ)dθ mi θ S (f L(θ) f K (θ)). S h L (θ)f L (θ)dθ ǫ V(K[ ],L) L ǫv(b 2 [ ],L). S h L (θ)dθ Applyig the Mikowski iequality (5) to V(K[ ],L) ad V(B2 [ ],L) we get Thus L K L L ǫ L B 2. K ǫ B2. To fiish the proof of the theorem we ote that (23) is always true with ǫ = mi θ S (f L (θ) f K (θ)). We ote that if we ca cosider K to be rb 2 L i the above theorem ad sed r 0 to get that ω for ay L with absolutely cotiuous surface area measure. mi θ S (f L(θ)), 0

11 Theorem 5.2. There exists a absolute costat C > 0 such that for ay K,L R, two covex bodies with absolutely cotiuous surface area measure, with f L (θ 0 ) f K (θ 0 ), for some θ 0 S ad such that K is i the miimal width positio, we have L K Clog ω Proof: Cosider two covex bodies K,L R such that (24) f L (θ) f K (θ)+ǫ, for some ǫ 0 ad all θ S. The max θ S (f L(θ) f K (θ)). h K (θ)f L (θ)dθ h K (θ)f K (θ)dθ +ǫ h K (θ)dθ S S S or V(L[ ],K) K +ǫ h K (θ)dθ. S Now we ca apply the Mikowski iequality (5) to V(L[ ],K). We ca also use the fact that K is i miimal width positio ad apply iequality (9) to claim that h K (θ)dθ C S K log = Cω K log S Thus K L K +ǫ Cω K log. or L K ǫc B2 log. To fiish the proof of the theorem we ote that (24) is always true for ǫ = max θ S (f L (θ) f K (θ)). 6. A ote o the slicig problem Recall that he otio of a itersectio body of a star body was itroduced by E. Lutwak [L2]: IL is called the itersectio body of L if the radius of IL i every directio is equal to the ( )-dimesioal volume of the cetral hyperplae sectio of L perpedicular to this directio: ξ S, ρ IL (u) = L u, As show i [L2] (usig a result of Busema s theorem [Bu]), if L happes to be a symmetric covex body, the IL is also a symmetric covex body. A more geeral class of itersectio bodies was defied by R. Garder [Ga] ad G. Zhag [Zh] as the closure of itersectio bodies of star bodies i the radial metric d(k,l) = sup ξ S ρ K (ξ) ρ L (ξ). We refer reader to books [Ga2, Ko, KoY, RZ] for more iformatio o defiitio ad properties of Itersectio body ad their applicatios i Covex Geometry ad Geometric Tomography. Defie the quatity L := max{l K : K is a covex body i R }. For,k N, k ad c > 0, defie the class C,k,c of covex bodies K i R, that have a l-dimesioal projectio P, with l k, for which there exists a l-dimesioal itersectio body, such that P L ad L / P c. Fially, defie I,k to be the smallest costat t, for which C,k,t cotais all cetrally symmetric covex bodies i R. The followig was proved i [Ko2, Corollary ]: Theorem C. There exists a absolute costat C 0 > 0, such that if L is a symmetric covex body i R, such that L C,,a, for some a > 0 ad µ is ay eve measure with cotiuous desity i R, the (25) µ(l) (C 0 /a) max u S µ(l u ) L /. It follows by Theorem C (for µ = ) ad (2) that L C 0I,, for some absolute costat C 0 > 0. Our goal is to replace I, with I,k, where k is at least proportioal to.

12 Theorem 6.. There exists a absolute costat C > 0, such that for ay N ad for ay λ (0,], it holds L C /λ I,[λ], where [ ] deotes the iteger part fuctio. For istace, if K has bouded outer volume ratio (it is the trivial of course that K has bouded isotropic costat), we kow that K has projectios of proporsioal dimesio which are almost spherical. The followig questio is therefore atural: Questio 6.2. Is it true that ay covex body K R has projectios of at least proportioal dimesio that are close to some itersectio body i the sese of the Baach-Mazur distace? Before provig Theorem 6., we will eed some geometric statemets. Lemma 6.3. Let K be a symmetric covex body of volume, i M positio. There exists a absolute costat C 0 > 0, such that for ay subspace H of R, it holds /C 0 K H / K H / C 0. Proof: By assumptio, there exist poits x,...,x N R, with N < C, such that Let H G,k. The, thus K H K N (x i +D ). N (x i +D ) H N D H N c B2 k = N(/k) k c kb k 2 C (/k) k c k D k ( C(+c ) ) (/k) k, K H / C (/k) k/ C. O the other had, if we replace K by (/ K ) / K ad sice (/ K ) / K =, we get: ( K /K ) H / C, which gives K H C K k/2. Usig (0) ad the fact that (K H) = K H, we obtai or as required. (c /k) k/ K H / K H C K k/2 C(c 2 /) k/ K H / (c /(c 2 k)) k/ c, The ext propositio is based o the existece of a M-positio. Propositio 6.4. Let K be a symmetric covex body of volume, be such that K H / C, for all subspaces H of R, for some absolute costat C > 0. The, there exists a absolute costat C, such that K H / C, for all subspaces H of R. 2

13 Proof: Oe ca fid a covex body K of volume, i M-positio, such that K = TK, for some T SL(). As i the proof of Theorem 4., we may assume that T is diagoal; actually we are oly goig to eed the fact that T is symmetric, i.e. T = T. We will make use of a special case of a result from [F] (see also [S, (5.28)]), statig that if M is a covex body i R, H G,k ad P is a covex body i H, the (26) M H = ( k Let k {,..., } ad H G,k. By (0), we get: (27) ad similarly, ) P V( M[k],P[ k] ). ( (TK) H ) / (c /k) k/ (TK) H / (c /k) k/ /c (28) K H / (c 2 /k) k/ K H / (c 2 /k) k/ /c. Moreover, for ay covex body P i H, usig (26) we obtai: (TK) H ( ) = k P V( (T K )[k],p[ k] ) ( ) = k P V( K [k],(t P)[ k] ) ( ) = k P V( K [k],(tp)[ k] ) = TP ( ) P k TP V( K [k],(tp)[ k] ) = TP (29) K (T H), P where we used the fact that TP is a covex body i (T H). Thus, (27), (28) ad (29) imply Replacig P by T P, we get: ( ) / TP c. P (30) P / c T P /, for all covex bodies P i H, for all subspaces H of R. O the other had, as before oe ca write: ( ) (TK) H = k P V( (TK)[k],P[ k] ) ( ) = k P V( K[k],(T P)[ k] ) ( ) = T P P k T P V( K[k],(T P)[ k] ) = T P P K (T H). Therefore, the previous idetity, Lemma 6.3 ad (30) give: K H / = (TK) H / ( T P / P ) / K (T H) / C /c, for ay subspace H of R. 3

14 Propositio 6.5. Let λ (0,] ad K be a 0-symmetric covex body i R of volume, such that K H / A, for some A > 0 ad for all subspaces H of R. If, i aditio, K C,k,t, for some t > 0, k λ, the there exists a ( )-dimesioal subspace F, such that for some absolute costat c > 0. Let H G,k. By Fubii s Theorem, we have: (3) = K = K F c A /λ t, K H K (H +x) dx. Let G be a codimesio subspace of H. The, the subspace F := spa{g H } has dimesio. The followig claim is the key step i the proof of Propositio 6.5: Claim. (K F) H = (K H) G ad (K F) (H +x) = K (H +x), for all x G. Proof of the claim: To prove the first part, let x (K H) G. Sice x K H, there exists y H, such that x + y K. Sice x G, x + y F, it follows that x + y K F, so x (K F) H. Thus, (K F) H (K H) G. Coversely, let x (K F) H. The, there exists y H, such that x+y K F. Thus, x + y K ad x + y F, so x K H ad x F H = G. Cosequetly, x (K H) G, which shows that (K H) G (K F) H, as required. To prove the secod part, ote that for ay x G, x+h spa(g H ) = F, thus K (x+h ) (K F) (x+h ). Sice the opposite iclusio is trivial, our claim is poved. Proof of Propositio 6.5: Usig agai Fubii s Theorem ad the previous claim, we get: (32) K F = K F (H +x) dx = K F (H +x) dx = K (H +x) dx. (K F) H (K H) G (K H) G Assume, ow, that K H L ad L / K H < t, where L is a itersectio body i some k-dimesioal subspace H of R. Set µ to be the measure with desity H x K (H +x) i H ad L := K H H. Usig Theorem C, we coclude that there exists a codimesio- subspace G of H, such that: (33) µ((k H) G) K H /k ( c/t)µ(k H), for some absolute costat c > 0. Sice K H /k A /λ, if we set F = spa{g H }, applyig (3) ad (32) to (33), we obtai: as claimed. K F = µ((k H) G) c 0 A /λ t, Proof of Theorem 6.. As show i [K], there exists a symmetric covex body K, such that L K c 0 L. We may assume that K is of volume ad i isotropic positio. The, as it was proved i [BKM, Corollary 3.5], K H / is bouded from above by a absolute costat. It follows immediately by Propositios 6.4, 6.5 ad the defiitio of I,k that there exists a ( )-dimesioal subspace F, such that K F c C /λ I,[λ], for some absolute costat C > 0. Our claim follows immediately from the previous iequality ad (2). We remark that our method ca be used to icrease the class of symmetric covex bodies that are kow to satisfy (25) (with a absolute costat). Ideed, oe ca replace the Lebesgue measure i Propositio 6.5 by ay eve measure (oe has to repeat the steps of the proof) to obtai the followig: 4

15 Propositio 6.6. Let µ be a eve measure i R, with cotiuous desity, λ (0,] ad K be a 0- symmetric covex body i R, such that K H /k A K /, for some A > 0 ad for all k λ ad H G,k. If, i aditio, K C,k0,t, for some t > 0 ad for some k 0 λ, the there exists a ( )- dimesioal subspace F, such that µ(k F) K / c A /λ t µ(k), for some absolute costat c > 0. I particular, it follows from the previous statemet ad from Lemma 6.3 that if K C,k,c, for some k λ ad K is i M-positio, the (25) holds for K (with a absolute costat) for all eve measures µ with cotiuous desity. As a example, take K = ab 2 L, where L is ay covex body i M-positio, diml = ad L = ab 2. Refereces [AGM] S. Artstei-Avida, A. Giaopoulos, V. D. Milma, Asymptotic geometric aalysis. Part I, Mathematical Surveys ad Moographs, 202. America Mathematical Society, Providece, RI, 205. [Ba] K. Ball, Shadows of covex bodies, Tras. Amer. Math. Soc., 327 (99), [Ba2] K. M. Ball, Volume ratio ad a reverse isoperimetric iequality, Joural of the Lodo Math. Soc., 44(2) (99), [Bor] C. Borell, Covex measures o locally covex spaces, Ark. Mat. 2 (974), [Bou] J. Bourgai, Geometry of Baach spaces ad harmoic aalysis, I Proceedigs of the Iteratioal Cogress of Mathematicias (Berkeley, Calif., 986), 2, Amer. Math. Soc., Providece, RI, 987, [Bou2] J. Bourgai, O the distributio of polyomials o high-dimesioal covex sets, i Geometric aspects of fuctioal aalysis (989-90), Lecture Notes i Math., 469, Spriger, Berli, (99), [BM] J. Bourgai ad V.D. Milma, New volume ratio properties for covex symmetric bodies, R d, Ivet. Math., 88 (987), [BKM] J. Bourgai, B. Klartag ad V. D. Milma, Symmetrizatio ad isotropic costats of covex bodies, i Geometric Aspects of Fuctioal Aalysis, Lecture Notes i Mathematics 850 (2004), 0 6. [BGVV] S. Brazitikos, A. Giaopoulos, P. Valettas, ad B. Vritsiou, Geometry of isotropic covex bodies, Mathematical Surveys ad Moographs, 96. America Mathematical Society, Providece, RI, 204. [Bu] H. Busema, A theorem of covex bodies of the Bru-Mikowski type, Proc. Nat. Acad. Sci. U.S.A. 35, (949), [CGaG] S. Campi, R. J. Garder, ad P. Grochi, Reverse ad dual Loomis-Whitey-type iequalities, Tras. Amer. Math. Soc. 368 (206), [CGG] S. Campi, P. Gritzma ad P. Grochi, O the reverse Loomis-Whitey iequality, 206, preprit, DOI: arxiv: [F] V. P. Fedotov, O the sum of pth surface fuctios (i Russia), Ukrai. Geom. Sb., 2 (978), [FGM] M. Fradelizi, A. Giaopoulos ad M. Meyer, Some iequalities about mixed volumes, Israel J. Math. 35 (2003), [Ga] R.J. Garder, Itersectio bodies ad the Busema - Petty problem, Tras. Amer. Math. Soc. 342 (994), [Ga2] R. J. Garder, Geometric Tomography, Secod editio. Ecyclopedia of Mathematics ad its Applicatios, 58. Cambridge Uiversity Press, Cambridge, [GHP] A. Giaopoulos, M. Hartzoulaki ad G. Paouris, O a local versio of the Aleksadrov-Fechel iequality for the quermassitegrals of a covex body, Proc. Amer. Math. Soc., 30 (2002), [GKV] A. Giaopoulos, A. Koldobsky ad P. Valettas, Iequalities for the surface area of projectios of covex bodies, 206, preprit. [GP] A. Giaopoulos, M. Papadimitrakis, Isotropic surface area measures, Mathematika 46 (999) 3. [J] F. Joh, Extremum problems with iequalities as subsidiary coditios, Studies ad Essays Preseted to R. Courat o his 60th Birthday, Jauary 8, 948, Itersciece Publishers, Ic., New York, N. Y., 948, pp [K] B. Klartag, A isomorphic versio of the slicig problem, J. Fuct. Aal. 28 (2005), [Ko] A. Koldobsky, Fourier Aalysis i Covex Geometry, Math. Surveys ad Moographs, AMS, Providece RI [Ko2] A. Koldobsky, Slicig iequalities for measures of covex bodies, Adv. Math. 283 (205), [KoY] A. Koldobsky, V. Yaski, The Iterface betwee Covex Geometry ad Harmoic Aalysis, CBMS Regioal Coferece Series, 08, America Mathematical Society, Providece RI, [LT] M. Ledoux, M. Talagrad, Probability i Baach spaces. Isoperimetry ad processes. Classics i Mathematics. Spriger- Verlag, Berli, 20 [LW] L.H. Loomis ad H. Whitey, A iequality related to the isoperimetric iequality, Bull. Amer. Math. Soc., 55 (949), [L] E. Lutwak, Selected affie isoperimetric iequalities, i Hadbook of Covex Geometry (P. M. Gruber ad J. M. Wills eds.), Vol. A, Elsevier Sciece Publishers, North Hollad (993), [L2] E. Lutwak, Itersectio bodies ad dual mixed volumes, Adv. i Math. 7 (988), [M] V. D. Milma, Isomorphic symmetrizatio ad geometric iequalities, i Geometric Aspects of Fuctioal Aalysis, Lecture Notes i Mathematics 376 (989),

16 [MP] V. D. Milma ad A. Pajor, Isotropic positio ad iertia ellipsoids ad zooids of the uit ball of a ormed -dimesioal space, i Geometric Aspects of Fuctioal Aalysis, Lecture Notes i Mathematics 376 (989), [Pe] C. M. Petty, Surface area of a covex body uder affie trasformatios, Proc. Amer. Math. Soc. 2 (96), [RZ] D. Ryabogi ad A. Zvavitch, Aalytic methods i covex geometry, IM PAN Lecture Notes, Vol. 2, Warsaw 204. [Sa] L.A. Sataló, A affie ivariat for covex bodies of -dimesioal space, Portugaliae Math. (I Spaish) 8 (949), [Sa] C. Saroglou, Miimal surface area positio of a covex body is ot always a M-positio, Israel J. Math., 95 (203), [S] R. Scheider, Covex Bodies: The Bru Mikowski theory, Secod Expaded Editio, Ecyclopedia of Mathematics ad its Applicatios, 5, Cambridge Uiversity Press, Cambridge, 204. [Zh] G. Zhag, Cetered bodies ad dual mixed volumes, Tras. Amer. Math. Soc. 345 (994), Departmet of Mathematics, Uiversity of Missouri, Columbia, MO 652, USA address: koldobskiya@missouri.edu Departmet of Mathematics, Ket State Uiversity, Ket, OH 44242, USA address: csaroglo@ket.edu & christos.saroglou@gmail.com Departmet of Mathematics, Ket State Uiversity, Ket, OH 44242, USA address: zvavitch@math.ket.edu 6