VOLUME INEQUALITIES FOR SETS ASSOCIATED WITH CONVEX BODIES

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1 VOLUME INEQUALITIES FOR SETS ASSOCIATED WITH CONVEX BODIES STEFANO CAMPI Dipartimeto di Matematica Pura e Applicata G. Vitali Uiversità degli Studi di Modea e Reggio Emilia Via Campi 213/B Modea, Italy campi@uimo.it PAOLO GRONCHI Istituto per le Applicazioi del Calcolo Sezioe di Fireze Via Madoa del Piao CNR Edificio F, Sesto Fioretio (FI), Italy paolo@fi.iac.cr.it This paper deals with iequalities for the volume of a covex body ad the volume of the projectio body, the L p -cetroid body, ad their polars. Examples are the Blaschke-Sataló iequality, the Petty ad Zhag projectio iequalities, the Busema-Petty iequality. Other iequalities of the same type are still at the stage of cojectures. The use of special cotiuous movemets of covex bodies provides a geeral approach to this subject. A family of iequalities, depedig o a parameter p 1 ad proved by Lutwak for p = 1 ad p = 2, is obtaied. 1. Itroductio ad prelimiaries This paper is devoted to some classical iequalities of Covex Geometry ivolvig the volume of a -dimesioal covex body ad the volume of a further body associated to the give oe. More precisely, our attetio is focused o the projectio body, the L p -cetroid body ad their polar bodies. Our approach comes from the idea that the most part of results coected with these iequalities ca be deduced by the same geeral method, which is based o the use of special cotiuous movemets of the bodies we are dealig with. Let K be a covex body i R, that is a -dimesioal compact covex 1 INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

2 2 set, ad assume that the origi is a iterior poit of K. The support fuctio of the covex body K is defied as h K (u) = max x K u, x, for u R, where, is the usual scalar product i R, ad the radial fuctio of K as ρ K (u) = max{r R : ru K}, for u R. The -dimesioal volume V (K) of K ca be expressed i terms of the radial fuctio by V (K) = 1 ρ K(z) dz, S 1 where S 1 is the uit sphere i R. The polar body K of K ca be defied as K = {x R x, y 1, y K}. The polar body of K depeds o the locatio of the origi. It is easy to see that (K ) = K ad that ρ K (u) = 1 h K (u), for u R. The projectio body of K is the covex body ΠK such that h ΠK (u) = 1 u, v dv, for u R, 2 K where dv deotes the area elemet at the poit o K whose outer uit ormal is v. It is clear from the defiitio that h ΠK (u) is the ( 1)- dimesioal volume of the projectio of K orthogoal to u. For every Borel subset ω of S 1, we defie the area measure σ K (ω) of K as the ( 1)-dimesioal Hausdorff measure of the reverse image of ω through the Gauss map. Recall that the Gauss map seds each poit o K to the set of outward uit ormal vectors to K at that poit. Therefore, the support fuctio of ΠK ca be rewritte as h ΠK (u) = 1 u, v dσ K (v), for u R. 2 S 1 If K is sufficietly smooth ad its Gauss curvature is strictly positive, the h ΠK (u) = 1 u, v 2 γ K (v) dv, for u R, S 1 INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

3 3 beig γ K (v) the Gauss curvature of K at the poit where the outward uit ormal vector is v. Such a represetatio shows that h ΠK is the 1 cosie trasform of γ K. Each projectio body is a zooid, amely it is the limit, i the Hausdorff metric, of a sum of segmets. Here by sum it is meat the Mikowski additio of subsets of R A + B = {x R : x = a + b, a A, b B}. Zooids play a importat role i Covex Geometry, as well as i differet areas of Mathematics. I particular, every full-dimesioal zooid turs out to be the projectio body of a covex body. Give a covex body K, for each real umber p 1, the L p -cetroid body Γ p K of K is the covex body whose support fuctio is where ad { 1 h ΓpK(u) = c,p V (K) c,p = K } 1 u, z p p dz, for u R, κ +p πκ κ p 1, κ r = π r 2 /Γ(1 + r 2 ). Notice that κ is the volume of the uit ball B of R ad the costat c,p is such that Γ p B = B. The above defiitio ca be exteded to compact bodies i R. Up to costats, Γ 1 K is kow i the literature as the cetroid body ΓK of K ad Γ 2 K as the Legedre ellipsoid of K. If K is a origi-symmetric covex body, it turs out that the boudary of ΓK is the locus of the cetroids of all the halves of K obtaied by cuttig K with hyperplaes through the origi. By usig polar coordiates i itegratio, oe has that h ΓK (u) = 1 c,1 V (K) u, z ρ +1 K S 1 (z) dz, which shows that the cetroid body ΓK is, up to a costat, the projectio body of ΛK, the curvature image of K, i.e. the covex body such that γ ΛK (z) = ρ 1 K (z), for every z S 1. The Mikowski Theorem guaratees the existece of such a body (see, for example, 32, Ch. 7.1 ). INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

4 4 If Γ K is iterpreted as a limit of (1), as p, the Γ K = cov(k ( K)), where cov stads for the covex hull. For further details related to the cotet of this prelimiary sectio, we refer to the books of Garder 11 ad Scheider 32 ad to the articles by Goodey ad Weil 12, Lutwak 17, Lutwak ad Zhag 19, Milma ad Pajor 23. As a fial remark, we otice that there are other bodies that ca be associated to a give oe: the itersectio body, the cross-sectio body, the Blaschke body, etc. As Richard Garder writes i the Itroductio of his book 11, geometric tomography houses a zoo of strage geometric bodies, powerful itegral trasforms, ad exotic but highly effective iequalities. 2. Mai volume iequalities I this sectio we list the mai iequalities which ivolve the volume of the bodies itroduced i Sectio 1. At preset, some of them are still cojectures. Further iformatio ad details o iequalities of this type ca be foud i The Blaschke-Sataló iequality For every covex body K, assume that the origi is chose so that V (K ) is miimum. The Blaschke-Sataló iequality states that V (K)V (K ) κ 2, (1) where equality holds if ad oly if K is a origi-symmetric ellipsoid. The quatity o the left-had side of (1) is called volume product of K. Iequality (1) was proved for 3 by Blaschke 1, 2 ad for all by Sataló 31. The equality coditios were proved by Sait Raymod 30 i the symmetric case ad by Petty 26 i the geeral case Mahler s cojecture It was cojectured by Mahler 20 that the miimum of the volume product is attaied whe K is a simplex, that is V (K)V (K ) ( + 1)+1 (!) 2. (2) INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

5 5 Mahler 21 proved that (2) holds if = 2 ad Meyer 22 that i this case equality occurs oly for triagles. For origi-symmetric covex bodies it has bee cojectured that - parallelotopes (ad their polars, i.e. cross-polytopes) miimize the volume product, hece V (K)V (K ) 4!. (3) For = 2, iequality (3) was proved by Mahler 21 ad Reiser 28 showed that parallelograms are the oly miimizers. For > 3 there are bodies, differet tha parallelotopes ad their polars, for which (3) is a equality, as show by Sait Raymod 30. Reiser 27, 28 ad Sait Raymod 30 proved (3) for special classes of covex bodies, amely for zooids ad for the affie images of covex sets symmetric with respect to the coordiate hyperplaes. A simpler proof of Reiser s result was give i 13. Bourgai ad Milma 4 proved that there exists a costat c, ot depedig o the dimesio, such that V (K)V (K ) c κ The projectio body cojectures The problem of fidig miimizers ad maximizers of the fuctioal V (K) 1 V (ΠK) i the class of all covex bodies is ope. I 1972 Petty 25 cojectured that ellipsoids are the oly miimizers. As far as the maximum of V (K) 1 V (ΠK) is cocered, Brae 5 cojectured that, i the class of all covex bodies, simplices are maximizers. The same author cojectured that the largest cetrally symmetric subset of a simplex gives a sharp upper boud of the fuctioal i the class of cetrally symmetric covex bodies The Petty projectio iequality Petty 25 proved that V (K) 1 V (Π K) ( κ κ 1 ), (4) with equality if ad oly if K is a ellipsoid. Notice that Petty s cojecture i 2.3 ad the Blaschke-Sataló iequality (1) imply (4). INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

6 The Zhag projectio iequality The reverse iequality of (4) is due to Zhag 34 : V (K) 1 V (Π K) with equality if ad oly if K is a simplex. ( ) 2, (5) 2.6. The L p -Busema-Petty iequality Let p 1. The V (Γ p K)V (K) 1 1, (6) with equality if ad oly if K is a origi-symmetric ellipsoid. For p = 1, iequality (6) was proved by Petty 24 via the Busema radom simplex iequality (see 11, Theorem 9.2.6). For p = 2, Blaschke 3 proved (6) whe = 3. For p = 2 ad geeral, (6) was proved by Joh 14. For geeral p, iequality (6) was proved by Lutwak, Yag ad Zhag 18 ad, i a differet way, by Campi ad Grochi 8. Cocerig a reverse versio of (6), let us otice that the fuctioal we are cosiderig is ot bouded from above i the class of all covex bodies. A atural assumptio is to restrict ourselves to the bodies cotaiig the origi. It has bee cojectured that i such a class simplices with oe vertex at the origi provide the maximum. I 9 this cojecture is proved for = 2 (see also Sectio 4) The L p -Blaschke-Sataló iequality Let p 1. The V (Γ pk)v (K) κ 2, (7) with equality if ad oly if K is a origi-symmetric ellipsoid, as proved by Lutwak ad Zhag i 19. The ame of this iequality comes from the fact that if K is origi-symmetric ad p teds to ifiity, the (7) gives (1). Eve i this case, it has bee cojectured that suitable simplices are miimizers of the left-had side of (7) i the class of all covex bodies cotaiig the origi. A result i the two-dimesioal case is cotaied i Subsectio 4.3. INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

7 7 3. The Rogers ad Shephard method Most of the results described i the previous sectio ca be obtaied by the use of shadow systems, which were itroduced by Rogers ad Shephard i 29 ad 33. A shadow system alog the directio v is a family of covex bodies K t R that ca be defied by K t = cov{z + α(z)t v : z A R }, (8) where A is a arbitrary bouded set of poits i R, α is a real bouded fuctio o A, ad t rus i a iterval of the real axis. The fuctio α i (8) is called the speed fuctio of the shadow system. As proved i 29, the volume of K t is a covex fuctio of t. The proof is based o the fact that the legth of each chord of K t parallel to v turs out to be a covex fuctio of t. This covexity result was exteded by Shephard 33 to mixed volumes of shadow systems alog the same directio v. We recall that the mixed volume of the covex bodies K 1, K 2,..., K ca be defied by V (K 1, K 2,..., K ) = 1 ( 1) +j V (K i1 +K i2 + +K ij ). (9)! j=1 i 1< <i j As proved by Mikowski (see 32, Ch. 5), mixed volumes are the coefficiets of the homogeeous polyomial V (t 1 K t K ) of degree i the variables t 1, t 2,..., t. Special istaces of mixed volumes are the quermassitegrals W i (K) of a covex set K, which are defied by (9), for i = 0, 1,...,, by takig ( i) copies of K ad i copies of the uit ball B. Notice that every mixed volume ca be expressed by a itegral. A example that we shall use later is give by V (K,..., K, L) = V 1 (K, L) = 1 h L (v) dσ K (v). (10) S 1 Shephard s proof of the covexity of mixed volumes uder shadow systems is based o the remark that every shadow system K t alog v ca be see as the projectio of a ( + 1)-dimesioal covex body oto a fixed hyperplae orthogoal to w with respect to the directio w tv. A special type of shadow system is aturally related to the well kow Steier process of symmetrizatio. Precisely, fix a directio v ad let K = {x + yv R : x K v, y R, f(x) y g(x)} ; INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

8 8 here K v deotes the orthogoal projectio of K oto the hyperplae v through the origi, orthogoal to v, ad f ad g are covex fuctios o K v. The shadow system with speed fuctio α(x) = (f(x) + g(x)) ad t [0, 1] is such that K 0 = K, K 1 = K v, the reflectio of K i the hyperplae v, ad K 1/2 is the Steier symmetral of K with respect to v. I this process, the volume is trivially costat, while the behavior of the other quermassitegrals is described by Shephard s result o mixed volumes. Thus the Steier symmetral of a covex body K ca be obtaied through a shadow system alog a directio v i which the speed fuctio is costat o each chord parallel to v. Shadow systems of this type are called parallel chord movemets. So, a parallel chord movemet alog the directio v is a family of covex bodies K t i R defied by K t = {z + α(x)t v : z K, x = z z, v v}, where α is a cotiuous real fuctio o v. Notice that the above defiitio requires that α must be chose so that, for every t, the set K t is covex. If the speed fuctio α is a affie fuctio (that is, α(x) = x, u + k, for some vector u ad real costat k), the it is easy to see that K t is a affie image of K, for every t i the rage of the movemet. Shadow systems ca be applied successfully accordig to the followig Shephard argumet (see 33 ): If a fuctioal defied i the class of all covex sets is cotiuous, ivariat uder reflectios ad covex with respect to the parameter t of ay parallel chord movemet, the it attais its miimum at the ball amog all sets of give volume. Here the cotiuity refers to the Hausdorff metric. This statemet follows from the well-kow property of the Steier symmetrizatio, that, if suitably repeated, chages every covex set i a ball. Cosequeces of this procedure are, for example, classical isoperimetric type iequalities for quermassitegrals (see 6, p. 144, 11, p. 372). Other geometric fuctioals have the same covex behavior uder shadow systems; see, for example, 33, 7 ad the ext sectio. A questio is whether the same covexity property of the fuctioals ca help i fidig reverse iequalities. We are able to give a positive aswer oly i the case = 2. I order to do this, it is coveiet to itroduce the class Ω of -dimesioal covex bodies K with the followig property: If K t is a parallel chord movemet, t [ 1, 1], ad K = K 0, the the speed of the movemet is a affie fuctio. INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

9 9 Assume ow that F (K) is a fuctioal defied i the class of all - dimesioal covex bodies ad suppose F has a maximum i that class. If F is covex uder parallel chord movemets, the a maximizer of F must belog to Ω. Notice that, if F is bouded from above ad covex uder parallel chord movemets, the F must be ivariat uder affie trasformatios. Moreover, if F is strictly covex, i.e. F (K t ) is strictly covex uless the speed of K t is affie, the all maximizers of F belog to Ω. As show i 7, all simplices are i Ω, triagles are the oly polygos i Ω 2, ad, for 2, i Ω there are other bodies tha simplices. For = 2, by a approximatio argumet, we ca coclude that triagles are maximizers of all cotiuous fuctioals which are covex uder parallel chord movemets (provided the maximum exists). For = 2, if oe cosiders oly cetrally symmetric covex sets, the parallelograms play the same role as triagles i the geeral case. The same method ca be applied also to liear ivariat fuctioals which are ot bouded from above. Sice these are the cases we shall deal with i the ext sectio, we give here all the details. A liear ivariat cotiuous fuctioal is bouded i the class of bodies cotaiig the origi. Ideed, by Joh s theorem ( 15, Theorem III), we ca restrict ourselves to bodies cotaiig a ball of radius oe ad cotaied i a ball of radius, with the same ceter. If F is a covex fuctioal uder parallel chord movemets, the it is covex uder traslatios, hece every maximizer of F has the origi as a extreme poit. I particular, if P is a polytope, we ca assume, without decreasig the value of F, that the origi is oe of its vertices. Let F be a fuctioal defied i the class of all covex bodies cotaiig the origi, which is cotiuous with respect to the Hausdorff metric, liear ivariat ad covex uder parallel chord movemets. Let P be a polygo ad let 0, v 1, v 2,..., v m be its vertices clockwise ordered. Let us cosider the shadow system {P t : t [t 0, t 1 ]}, t 0 < 0 < t 1, alog v 2, with speed 1 at v 1 ad 0 at the other vertices. If t 0 ad t 1 are sufficietly close to 0, the oly the triagle 0v 1 v 2 moves, while the remaiig part of P keeps still. Let us choose [t 0, t 1 ] as the largest iterval such that the area of P t is costat for all t [t 0, t 1 ]. Hece, {P t : t [t 0, t 1 ]} is just a parallel chord movemet ad P t0 ad P t1 have exactly m 1 vertices. By the covexity of F, F (P ) max{f (P t0 ), F (P t1 )}. If m > 4, iteratios of this argumet lead to the coclusio that F (P ) INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

10 10 F (T ), where T is a triagle with a vertex at the origi. The cotiuity of F implies that T is a maximizer i the whole class of plae covex figures cotaiig the origi. 4. Applicatios This sectio cotais some applicatios of two results ivolvig shadow systems ad related ideas. For v S 1, let H v + be the halfspace bouded by v ad cotaiig v. Theorem 4.1. If K t, 0 t 1, is a shadow system alog the directio v, the V (K t H+ v ) 1 is a covex fuctio of t. The proof (see 10 ) is based o a geeralized form of the Prekopa-Leidler iequality. Notice that the theorem deals oly with oe of the halves of K cut off by v. Clearly, if K t is a family of origi-symmetric covex bodies, the V (K t ) 1 is a covex fuctio of t. Theorem 4.2. If K t, 0 t 1, is a parallel chord movemet alog the directio v, the Γ p K t, for p 1, is a shadow system alog v. Hece, V (Γ p K) is covex uder parallel chord movemet. Moreover, it is strictly covex uless the speed fuctio is liear. For the proof of Theorem 4.2 see By applyig Theorem 4.1 to the reciprocal of the volume product V (K)V (K ), we obtai the Blaschke-Sataló iequality (1) for origisymmetric covex bodies (without the characterizatio of ellipsoids as uique maximizers). For = 2, we get the Mahler iequality (3) (agai without characterizatio) Theorem 4.2 immediately gives the L p -Busema-Petty cetroid iequality (6), with the characterizatio of miimizers. The same theorem, for = 2, implies also a reverse form of the iequality. INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

11 11 Namely, all triagles with a vertex at the origi maximize V (Γ p K)V (K) 1 i the class of all plae covex bodies cotaiig the origi Combiig Theorem 4.1 with Theorem 4.2 provides the L p -Blaschke- Sataló iequality (7) (without characterizatio) ad the followig reverse form for = 2: I the class of all plae covex bodies K cotaiig the origi, triagles with a vertex at the origi are miimizers of V (Γ pk)v (K). If oe deals with plae origi-symmetric covex bodies, the cetered parallelograms are maximizers ad miimizers for the fuctioals i 4.2 ad 4.3, respectively Double etry fuctioals I this subsectio we cosider geometric fuctioals depedig o two differet covex bodies. We already oticed i Sectio 3 that mixed volumes are covex alog shadow systems; for example, if K t ad L t, 0 t 1, are shadow systems alog the same directio v, the V 1 (K t, L t ) is a covex fuctio of t. Let us cosider ow the fuctioal where p 1. G p (K, L) = V 1(K, Γ p L), (11) V (K) 1 V (L) 1 Theorem 4.3. If K t ad L t, 0 t 1, are parallel chord movemets alog the same directio v, the G p (K t, L t ), for p 1, is a covex fuctio of t. The theorem follows from Theorem 4.2 ad from the above metioed property of mixed volumes. Theorem 4.4. For every pair of covex bodies K ad L, G p (K, L) 1, (12) where equality holds if ad oly if K ad L are homothetic ellipsoids, with L origi-symmetric. INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

12 12 Proof. The theorem is a cosequece of the Mikowski iequality (see 32, p. 317) ad the L p -Busema-Petty cetroid iequality (6). The equality coditios for these two iequalities imply the secod part of the statemet. The fuctioal G p (K, L) is ot bouded from above. Ideed, if L is moved far from the origi, the Γ p L becomes larger ad larger. Moreover, if K ad L approach to a couple of ( 1)-dimesioal sets lyig o o parallel hyperplaes, the the volumes of K ad L ted to zero, while V 1 (K, L) teds to a positive umber. Nevertheless, Theorem 4.3 provides a two-dimesioal reverse form of iequality (12) whe K = L. Theorem 4.5. For = 2, if K cotais the origi, the G p (K, K) attais its maximum whe K is a triagle with a vertex at the origi. Let us show that iequality (12), for p = 1, implies the Petty projectio iequality (4). We have ( + 1)κ S 1 1 mi G 1 (K, L) = mi L L = mi L = mi L ( + 1)κ L L x, u dxdσ K(u) 2κ 1 V (K) 1 V (L) +1 S 1 x, u dσ K (u)dx 2κ 1 V (K) 1 V (L) +1 ( + 1)κ L h ΠK(x) dx κ 1 V (K) 1 V (L) +1 Π K h ΠK(x) dx = ( + 1)κ = κ 1 V (K) 1 V (Π K) +1 κ κ 1 V (K) 1 V (Π K) 1 where we used the fact that the itegral of a support fuctio h M o a set L of give volume is miimum whe L is just a level set of h M, that is, whe L is homothetic to M. Notice that G 1 (K, L) ca be expressed as G 1 (K, L) = ( + 1)κ K L x, u dxdu. 2κ 1 V (K) 1 V (L) +1 The last expressio remids some fuctioals studied by Lutwak i 16. Precisely, he showed that Petty s projectio body cojecture (see Subsectio 2.3) is equivalet to say that the fuctioal x, u dxdu K L V (K) 1 V (L) 1, INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

13 13 attais its miimum, i the class of all pairs of covex bodies, if ad oly if K ad L are homothetic to origi-symmetric polar reciprocal ellipsoids. I the same paper Lutwak proved also that, for p = 1 ad p = 2, the fuctioal K L x, u p dxdu V (K) +p V (L) +p attais its miimum if ad oly if K ad L are homothetic to origisymmetric polar reciprocal ellipsoids. We are able to exted such a result to the case of a arbitrary p 1. Theorem 4.6. Let p 1, the x, u p κ +p dxdu K L π( + p)κ p 1 κ +2p V (K) +p +p V (L), with equality if ad oly if K ad L are homothetic to origi-symmetric polar reciprocal ellipsoids. Proof. By the defiitio of L p -cetroid body, we have that K L x, u p dxdu c,p V (L) = h p V (K) +p V (L) +p V (K) +p V (L) +p Γ pl (v) dv. K Therefore, for ay fixed L, the miimum of the fuctioal we are dealig with is attaied if ad oly if K is homothetic to a level set of h ΓpL. Sice the fuctioal is ivariat uder rescalig, the miimum is attaied whe K = Γ pl. Thus, K L x, u p dxdu V (K) +p V (L) +p + p [V (L)V (Γ pl)] p c,p κ +p π( + p)κ p 1 κ +2p where we used the L p -Blaschke-Sataló iequality (7). The above argumet provides also the equality coditios., Refereces 1. W. Blaschke, Über affie Geometrie VII: Neue Extremeigeschafte vo Ellipse ud Ellipsoid, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 69 (1917), W. Blaschke, Affie Geometrie IX: Verschiedee Bemerkuge ud Aufgabe, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 69 (1917), INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

14 14 3. W. Blaschke, Affie Geometrie XIV: Eie Miimumaufgabe für Legedres Trägheitsellipsoid, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 70 (1918), J. Bourgai ad V. Milma, New volume ratio properties for covex symmetric bodies i R, Ivet. Math. 88 (1987), N. S. Brae, Volumes of projectio bodies, Mathematika 43 (1996), Yu. D. Burago ad V. A. Zalgaller, Geometric Iequalities, Spriger-Verlag, Berli Heidelberg, S. Campi, A. Colesati ad P. Grochi, A ote o Sylvester s problem for radom polytopes i a covex body, Red. Ist. Mat. Uiv. Trieste 31 (1999), S. Campi ad P. Grochi, The L p -Busema-Petty cetroid iequality, Adv.Math. 167 (2002), S. Campi ad P. Grochi, O the reverse L p -Busema-Petty cetroid iequality, Mathematika 49 (2002), S. Campi ad P. Grochi, O volume product iequalities for covex sets, preprit. 11. R. J. Garder, Geometric Tomography, Cambridge Uiversity Press, Cambridge, P. Goodey ad W. Weil, Zooids ad Geeralisatios, i Hadbook of Covex Geometry (eds. P. M. Gruber ad J. M. Wills), North-Hollad, Amsterdam, 1993, Y. Gordo, M. Meyer ad S. Reiser, Zooids with miimal volume-product A ew proof, Proc. AMS 104 (1988), F. Joh, Polar correspodace with respect to covex regios, Duke Math. J. 3 (1937), F. Joh, Extremum problems with iequalities as subsidiary coditios, i: Courat Aiversary Volume (Itersciece, New York), 1948, pp E. Lutwak, O a cojectured projectio iequality of Petty, Cotemp. Math. 113 (1990), E. Lutwak, Selected affie isoperimetric iequalities, i Hadbook of Covex Geometry (eds. P. M. Gruber ad J. M. Wills), North-Hollad, Amsterdam, 1993, E. Lutwak, D. Yag ad G. Zhag, L p affie isoperimetric iequalities, J. Differetial Geom. 56 (2000), E. Lutwak ad G. Zhag, Blaschke-Sataló iequalities, J. Differetial Geom. 47 (1997), K. Mahler, Ei Übertragugsprizip für kovexe Körper, Časopis Pěst. Mat. Fys. 68 (1939), K. Mahler, Ei Miimalproblem für kovexe Polygoe, Mathematica (Zutphe) B 7 (1939), M. Meyer, Covex bodies with miimal volume product i R 2, Moatsh. Math. 112 (1991), V. D. Milma ad A. Pajor, Isotropic positio ad iertia ellipsoids ad zooids of the uit ball of a ormed -dimesioal space, i Geometric INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.

15 15 Aspects of Fuctioal Aalysis (eds. J. Lidestrauss ad V. D. Milma), Lecture Notes i Mathematics 1376, Spriger, Heidelberg, 1989, C. M. Petty, Cetroid surfaces, Pacific J. Math. 11 (1961), C. M. Petty, Isoperimetric problems, i Proc. Cof. Covexity ad Combiatorial Geomerty, Uiv. Oklahoma, (1971), C. M. Petty, Affie isoperimetric problems, A. N. Y. Acad. Sc. 440 (1985), S. Reiser, Radom polytopes ad the volume product of symmetric covex bodies, Math. Scad. 57 (1985), S. Reiser, Zooids with miimal volume product, Math. Z. 192 (1986), C. A. Rogers ad G. C. Shephard, Some extremal problems for covex bodies, Mathematika 5 (1958), J. Sait-Raymod, Sur le volume des corps covexes symétriques, i Sémiaire Choquet - Iitiatio à l Aalyse 1980/81 Exp. No. 11, Uiversité Pierre et Marie Curie, Paris, 1981, L. A. Sataló, U ivariate afi para los cuerpos covexos del espacio de dimesioes, Portugal. Math. 8 (1949), R. Scheider, Covex bodies: the Bru-Mikowski theory, Cambridge Uiversity Press, Cambridge, G. C. Shephard, Shadow systems of covex bodies, Israel J. Math. 2 (1964), G. Zhag, Restricted chord projectio ad affie iequalities, Geom. Dedicata 39 (1991), INTEGRAL GEOMETRY AND CONVEXITY - Proceedigs of the Iteratioal Coferece World Scietific Publishig Co. Pte. Ltd.