Extremum problems for the cone volume functional of convex polytopes

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1 Advaces i Mathematics Extremum problems for the coe volume fuctioal of covex polytopes Ge Xiog 1 Departmet of Mathematics, Shaghai Uiversity, Shaghai , PR Chia Received 24 August 2009; accepted 27 May 2010 Available olie 3 Jue 2010 Commuicated by Gil Kalai Abstract Lutwak, Yag ad Zhag defied the coe volume fuctioal U over covex polytopes i R cotaiig the origi i their iteriors, ad cojectured that the greatest lower boud o the ratio of this cetro-affie ivariat U to volume V is attaied by parallelotopes. I this paper, we give affirmative aswers to the cojecture i R 2 ad R 3. Some ew sharp iequalities characterizig parallelotopes i R are established. Moreover, a simplified proof for the cojecture restricted to the class of origi-symmetric covex polytopes i R is provided Elsevier Ic. All rights reserved. MSC: 52A40 Keywords: Covex polytope; Parallelotope; Cetro-affie ivariat; Coe volume fuctioal; Projectio body 1. Itroductio A covex body K i.e., a compact, covex subset with oempty iterior i Euclidea -space R, is determied by its support fuctio, hk,, o the uit sphere S 1, where hk, u = max{u y: y K} ad where u y deotes the stadard ier product of u ad y. The projectio body, ΠK,ofK is the covex body whose support fuctio, for u S 1,is give by address: xiogge@shu.edu.c. 1 Research of the author was supported by NSFC No ad Iovatio Program of Shaghai Muicipal Educatio Commissio No. 11YZ /$ see frot matter 2010 Elsevier Ic. All rights reserved. doi: /j.aim

2 G. Xiog / Advaces i Mathematics hπk, u = vol 1 K u, where vol 1 deotes 1-dimesioal volume ad K u deotes the image of the orthogoal projectio of K oto the codimesio 1 subspace orthogoal to u. Projectio bodies were itroduced by Mikowski at the tur of the previous cetury i coectio with Cauchy s surface area formula. They have bee the objects of itese ivestigatio durig the past two decades. May importat results for projectio bodies ad their dual aalogs, itersectio bodies, have bee obtaied see, e.g., [2 4,6,10,8,11,17,19 23,28,31 34,37,38]. I recet years, their geeralizatios to L p -settigs are attracted much attetio ad acquired remarkable advaces [5,13 15,24,26,36,39]. A importat usolved problem regardig projectio bodies is Scheider s projectio problem see, e.g., [7,9,18,29 31] ad [35]: what is the least upper boud, as K rages over the class of origi-symmetric covex bodies i R, of the affie-ivariat ratio [ VΠK/VK 1 ] 1, 1 where V is used to abbreviate the -dimesioal volume. A effective tool to study Scheider s projectio problem is the coe volume fuctioal U itroduced by Lutwak, Yag ad Zhag [25]: If P is a covex polytope i R which cotais the origi o i its iterior, the defie UP by UP = 1 u i1 u i 0 h i1 h i a i1 a i, 2 where u 1,...,u N are the outer ormal uit vectors to the correspodig facets F 1,...,F N of P, ad the facet with outer ormal vector u i has area i.e. 1-dimesioal volume a i ad distace h i from the origi. Let V i = 1 h ia i. The V i is the volume of the coe covo, F i, ad UP = u i1 u i 0 Obviously the fuctioal U is cetro-affie ivariat i that, V i1 V i. 3 UφP= UP, φ SL. 4 Sice VP= 1 Ni=1 a i h i, it follows immediately that UP/VP 1. It is oted that for a radom polytope with a large umber of facets, UP is very close to VP. It is this importat property of the fuctioal U which makes it so useful. For istace, with the fuctioal U, LYZ [25] preseted a modified versio of Scheider s projectio problem VΠP 1 2 UP 2 VP 2 2 1, 5! ad the gave a asymptotically optimal boud for the affie ratio 1.

3 3216 G. Xiog / Advaces i Mathematics As a aside, we observe that the coe volume fuctioal U has strog coectio with the coe measure: For every star-shaped body K R, the coe measure of a subset A of K is the volume of [0, 1]A ={ta: a A, 0 t 1}, i.e. the coe with base A ad cusp o. The coe measure appears i the Gromov Milma theorem [12] o the cocetratio of Lipschitz fuctios o uiformly covex bodies. I [27], Naor established the precise relatio betwee the surface measure ad coe measure o the sphere of l p. Oe fudametal, but still remais ope extremum problem, o the ratio of U to V is posed by LYZ [25]. Cojecture. If P is a covex polytope i R with its cetroid at the origi, the UP VP with equality if ad oly if P is a parallelotope.!1/, The first progress o LYZ s cojecture was attributed to He, Leg ad Li [16]. They proved that the cojecture is true whe restricted to the class of origi-symmetric covex polytopes. Theorem 1.1. If P is a origi-symmetric covex polytope i R, the UP VP with equality if ad oly if P is a parallelotope.!1/, 6 This paper is devoted to the study of LYZ s cojecture. We give affirmative aswers to the cojecture i R 2 ad R 3. Theorem 1.2. Let P be a covex polytope i R with its cetroid at the origi. If is equal to 2 or 3, the UP VP with equality if ad oly if P is a parallelotope.!1/, Iequality 6 is a reverse isoperimetric type iequality see, e.g., [1]. Let,itgives 1 UP 1, e VP which is ot depedet o the covex polytope ad space dimesio. This property will make it useful i the local theory of Baach spaces see [4]. Here, parallelotopes as the extremal bodies, have uderlyig importace i covex ad discrete geometry [40 42]. The characterizatio of parallelotopes i the symmetric cases, or simplices i the o-symmetric cases, as extremal bodies of classical fuctioals is a cetral problem

4 G. Xiog / Advaces i Mathematics i covex geometry. I this article, some ew sharp iequalities characterizig parallelotopes i R are established Lemmas 2.3, 3.4 ad 4.1, which are closely related to the fuctioal U. At the same time, it is surprisig to see that parallelotopes are the oly miimizers of fuctioal U. It is oted that we adopt techiques of the geometric symmetrizatio ad methods of projectio to tackle the cojecture, which are readily applicable to symmetric covex polytopes. So, a mostly simplified proof for Theorem 1.1 is available. I fact, the techiques egaged i this paper are applicable to all covex bodies. However, the methods used i [16] rely o the symmetry ad caot pass to o-symmetric case. We work i -dimesioal Euclidea space R, 2, with origi o, basis e 1,...,e, ad use coordiates x = x 1,...,x t for x R.LetB j be the cetered uit ball i R j, whose volume is deoted by ω j. The surface of B j, that is the j 1-dimesioal uit sphere, is deoted by S j 1. This paper, except for the itroductio, is divided ito three sectios. The proofs of Theorem 1.2 ad Theorem 1.1 are preseted i Sectio 3 ad Sectio 4, respectively. 2. Estimate of u i1 u i 1 u k =0 V k, where {u i1,...,u i 1 } {u 1,...,u N } ad u i1 u i 1 0 From the well-kow Bru s cocavity priciple, it follows immediately that Lemma 2.1. Let K R be a covex body ad L R be a j-dimesioal subspace, 1 j 1.If the f 1 j is cocave o K L. f : L R, fx= vol j K L + x, Remark. If covex body K is origi-symmetric, the f is a eve fuctio o K L, ad cosequetly f is mootoously decreasig o ay ray that starts from the origi. This property will be used i Lemma 4.1. Lemma 2.2. Let K R j R j, 1 j 1, be a covex body with its cetroid at the origi. Suppose D = K R j ad fx= vol j K R j + x, x D. The where the equality holds if ad oly if fxis costat o D. Proof. Sice the cetroid of K is at the origi, it has x, y dx dy = 0. f0 vol j D VK, 7 K Cosequetly, it gives 0 = K xdxdy= D xf x dx.

5 3218 G. Xiog / Advaces i Mathematics From Lemma 2.1 ad Jesse s iequality, we have f j 1 0 = f j 1 1 VK D xf x dx D f +1 j j x dx. VK From the well-kow Hölder iequality, it follows that that is, f0 D f +1 j j VK x dx j D fxdx+1 j vol j DV K j = VK vol j D, f0 vol j D VK. If fx is costat o D, it follows immediately that the equality of 7 holds. O the other had, if f0 vol j D = VK, the all the equalities i the argumets have to be attaied. From the equality coditio of Hölder iequality, it follows that fxis costat o D. This completes the proof. Lemma 2.3. Let P be a covex polytope i R with its cetroid at the origi. For ay fixed {u i1,...,u i 1 } {u 1,...,u N } with u i1 u i 1 0, if the ormal vector u k of F k satisfies u i1 u i 1 u k = 0, the u i1 u i 1 u k =0 V k 1 VP. 8 If P is a parallelotope, the the equality of 8 holds. Coversely, if the equalities of 8 hold for all subsets {u i1,...,u i 1 } {u 1,...,u N } with u i1 u i 1 0 simultaeously, the P is a parallelotope. Proof. For ay fixed {u i1,...,u i 1 } {u 1,...,u N },u i1 u i 1 0, let L = spa{u i1,...,u i 1 }, fx= vol 1 P L + x. The fxis cocave o D = P L, ad u i1 u i 1 u k =0 V k = D = 1 1 x h fxdsx x D [ ] 1 x fx 1 h dsx x where dsx is the 2-dimesioal Lebesgue measure o D.

6 G. Xiog / Advaces i Mathematics Without loss of geerality assume L = R 1. Geometrically, it is ituitively that D fx [ 1 1 h x x dsx] is the volume of the set P = { x, y R 1 R 1 : x 0 D, x 0,y P R 1 + x 0,x [o, x0 ] }, ad P has the same orthogoal projectio oto R 1 as covex polytope P. So, Now, we aim to show u i1 u i 1 u k =0 V k = 1 V P. V P VP. For this aim, we make use of spherical coordiates i the subspace L. Suppose the equatio of D is Let ρ = ρ 0 θ 1,θ 2,...,θ 2. Fρ,θ= fρcos θ 1,...,ρsi θ 1 si θ 2, 0 ρ ρ 0. The Fρ,θis cocave with respect to ρ,i.e. So that is, Fρ,θ ρ ρ 0 Fρ 0,θ+ ρ 0 ρ ρ 0 F0, 0. ρ 0 ρ 0 V P VP= dθ ρ 2 Fρ 0,θdρ dθ ρ 2 Fρ,θdρ S 2 0 S V P F0, 0dθ, S 2 ρ VP 1 V P+ 1 F0, 0 vol 1D. From Lemma 2.2, it follows that VP V P. Fially, we prove the equality coditio i 8. Suppose that P is a parallelotope with its cetroid at the origi. Sice

7 3220 G. Xiog / Advaces i Mathematics V i = V cov F i {o} = 1 2 VP, i= 1,...,N, ad u i1 u i 1 0, u i1 u i 1 u k = 0 if ad oly if u k =±u ir, r = 1,..., 1, it follows that u i1 u i 1 u k =0 V k = 1 1 VP 2 1 = 2 VP. Coversely, for ay {u i1,...,u i 1 } {u 1,...,u N }, u i1 u i 1 0, suppose the equality of 8 holds simultaeously. From the above argumets, it gives V P= VPad VP= F0, 0 vol 1 D. From Lemma 2.2, it follows that fx is costat o D, which thereby implies that P has oe pair of opposite outer ormal uit vectors ±v 1, ad all other outer ormal uit vectors of P are i the great subsphere H 1 spaed by u i1,u i2,...,u i 1. Replace u i1 by v 1. Obviously, v 1,u i2,...,u i 1 are 1 liearly idepedet outer ormal uit vectors of P. Similarly, for these outer ormals, the equality i 8 implies that P has aother pair of opposite outer ormal uit vectors ±v 2, v 2 v 1, ad all other outer ormal uit vectors are i the great subsphere H 2 spaed by v 1, u i2,...,u i 1.SoP has two pairs of outer ormals ±v 1, ±v 2, ad all other outer ormals are i the subsphere H 1 H 2. Usig repeatedly this argumet 2 times, the we obtai 1 pairs of liearly idepedet outer ormals ±v 1, ±v 2,...,±v 1,ofP, ad all other outer ormals are i the subsphere H 1 H 2 H k H 1, where H k is the great subsphere spaed by v 1,v 2,...,v k 1,u ik,...,u i 1,2 k 1. However, from the costructio, we kow that H 1 H 2 H k H 1 is a subsphere spaed oly by u i 1.For 1 liearly idepedet outer ormals, v 1,v 2,...,v 1,ofP, the equality i 8 implies that ±u i 1 is also a pair of opposite outer ormals of P.So±v 1, ±v 2,...,±v 1, ±u i 1 are precisely the pairs of opposite outer ormals of P, which meas that P is exactly a parallelotope. This completes the proof. 3. Estimate of u i u k =0 V k ad proof of Theorem 1.2 Lemma 3.1. Suppose Q is a -dimesioal frustum of a coe i R with cetroid C. Let F 1 ad F 2 be the upper base ad lower base of Q, respectively. The V covc, F 1 + V covc, F 2 1 VQ, 9 where the equality holds if ad oly if Q is a cylider. Proof. Without loss of geerality assume Q R 1 R 1, whose lower base is o the R 1 - coordiate plae ad cetroid o the x -axis with coordiate z 0. Suppose the radii of upper base F 1 ad lower base F 2 are r ad R, respectively, ad r<r.letr x deote the radius of the cross-sectio of Q at height x,0 x h, where h is the height of Q. Suppose the frustum of a coe is formed from a coe M with height b. The

8 G. Xiog / Advaces i Mathematics r x = b x ta θ, b h = r R r h, where θ is the half agle of the coe M. Thex -coordiate z 0 of cetroid C is Sice z 0 = h 0 x ω 1 rx 1 1 dx +1 h 0 ω 1rx 1 = R+1 r +1 R r r dx R r h. V covc, F 1 + V covc, F 2 1 VQ = ω [ 1 z 0 R 1 r 1 + r 1 h 1 R r ] R r h, substitutig z 0 ito the formula, the it is sufficiet to prove Sice fuctio 1 R +1 r R r r R r 1 R r R r r 1 R 1 r g = 1 +1 a+1 1 a 1 1 a = 1 a t 1 1 a 1 a dt, a > 1, 1 1 is strictly mootoously decreasig o, it follows that iequality 10 holds. If Q is a cylider, it follows immediately that the equality of 9 holds. O the cotrary, if Q is ot a cylider, sice g is strictly decreasig, the the equality i 10 will ot hold, ad cosequetly the equality i 9 will ot hold. This completes the proof. Lemma 3.2. Let P be a covex polytope i R with its cetroid at the origi. For ay {u i } {u 1,...,u N },lets be the Schwartz symmetrizatio of P with respect to u i. Suppose the cetroid of S is C S, ad the upper base ad lower base of S are F 1 ad F 2, respectively. The V covo, F 1 + V covo, F 2 = V covc S,F 1 + V covc S,F Proof. Without loss of geerality, for ay give {u i } {u 1,...,u N }, we assume that u i is parallel to the x -axis. Let A P x ad A S x be the areas of cross-sectios of P ad S at height x, h 1 x h 2, respectively. The the x coordiate of C S is h 2 x C S = x A S x dx = h 1 h 2 h 1 x A P x dx = 0. Hece, the equality 11 ca be derived immediately. This completes the proof. We will eed the followig elemetary geometrical fact:

9 3222 G. Xiog / Advaces i Mathematics Lemma 3.3. Let Q be a -dimesioal frustum of a coe i R with upper base F 1 ad lower base F 2, respectively. Suppose vol 1 F 1 vol 1 F 2. For ay two poits C 1,C 2 it Q, if the distace from C 1 to F 1 is ot greater tha the distace from C 2 to F 1, the V covc 1,F 1 + V covc 1,F 2 V covc 2,F 1 + V covc 2,F Proof. For ay poit C it Q, suppose the distaces from C to F 1 ad F 2 are h 1 ad h 2, respectively. Assume the height of Q is h. The V covc, F 1 + V covc, F 2 = 1 h vol 1F 2 1 h 1[ vol 1 F 2 vol 1 F 1 ]. It implies that V covc, F 1 + V covc, F 2 is strictly decreasig o h 1. This completes the proof. Lemma 3.4. Let P be a covex polytope i R with its cetroid at the origi. For ay fixed {u i } {u 1,...,u N }, if the ormal vector u k of F k satisfies u k u i = 0, the u k u i =0, u k {u 1,...,u N } V k 1 VP. 13 If P is a parallelotope, the the equality of 13 holds. Coversely, if the equalities of 13 hold for all u i s simultaeously, the P is a parallelotope. Proof. Without loss of geerality, for ay fixed {u i } {u 1,...,u N }, we assume that u i is parallel to the x -axis. Suppose S is the Schwartz symmetrizatio of covex polytope P with respect to the x -axis, h 1 x h 2, ad T is the frustum of a coe iscribed iside S with x -axis as its rotatio axis. Suppose the upper base ad lower base of T are F 1 ad F 2, respectively, ad vol 1 F 1 vol 1 F 2. Let Q be the frustum of a coe with the same volume, the same height, ad the same lower base F 2 of S. The frustum of a coe Q is also a body of revolutio about the x -axis. Suppose the upper base of Q is F 3. The vol 1 F 1 vol 1 F 3. If S = Q, it meas that the Schwartz symmetrizatio itself is a frustum of a coe, the the iequality 13 ca be derived from Lemmas 3.2 ad 3.1 immediately. If S Q, we will show that there exists a uique umber l, h 1 <l<h 2, such that the lateral boudary of S the boudary of S takig away the top ad the base ad the lateral boudary of Q itersect at the height l. For this aim, let H be a 2-plae that cotais the x -axis. The the itersectio of H with the lateral boudary of Q is a lie segmet N, ad the itersectio of H with the lateral boudary of S is a covex curve C. The lie segmet N ad the covex curve C itersect at a boudary poit o the lower base F 2 of Q ad S. IfN ad C itersect at oly oe poit or at more tha two poits, the the covex curve C must be o oe side of the lie segmet N. This implies that either S Q or Q S because both S ad Q are bodies of revolutio. I view of VS= VQ, the case S Q gives VS<VQthat is impossible, ad

10 G. Xiog / Advaces i Mathematics the case Q S gives S = Q. Therefore, the lie segmet N ad the covex curve C itersect at exactly two poits if S Q. This shows the uiqueess of the height l if S Q. Next, we will show that the distace from the cetroid C Q of Q to F 3 is ot greater tha the distace from the cetroid C S of S to F 3. For this aim, we oly eed to compare the x coordiates of C Q ad C S. Suppose VP= VS= VQ= V. Let A Q x, A S x be the area of cross-sectio of Q ad S at height x, h 1 x h 2, respectively. The x C Q x C S = 1 V = 1 V [ h2 h 1 [ l [ 1 l V = 0, h 1 l h 1 x A Q x dx h 2 h 1 x A S x dx x AQ x A S x dx + AQ x A S x dx + l h 2 l h 2 l ] x AQ x A S x dx ] AQ x A S x dx ] that is, x C Q x C S. The, from Lemma 3.2, followed by Lemmas 3.3 ad 3.1, we have V k = V covo, F 1 + V covo, F 2 that is, u k u i =0, u k {u 1,...,u N } = V covc S,F 1 + V covc S,F 2 V covc Q,F 1 + V covc Q,F 2 V covc Q,F 3 + V covc Q,F 2 1 VQ= 1 VP, 14 u k u i =0, u k {u 1,...,u N } V k 1 VP. Suppose that P is a parallelotope cetered at the origi. Sice V i = V cov F i {o} = 1 2 VP, i= 1,...,N, ad u k u i = 0 if ad oly if u k =±u i,wehave u k u i =0, u k {u 1,...,u N } V k = VP= 1 VP.

11 3224 G. Xiog / Advaces i Mathematics Coversely, suppose the equality u k u i =0, u k {u 1,...,u N } V k = 1 VP holds for ay u i {u 1,...,u N } simultaeously. The all the equalities i 14 have to hold. From Lemma 3.1, it follows that the Schwartz symmetrizatio of P with respect to each u i is a cylider. So u i ad u i both are outer ormal uit vectors to the facets of P. With the coditio that the cetroid of P is at the origi, it follows that V i = 2 1 VP, i = 1,...,N.SoP has exactly 2 facets ad ±u 1,...,±u are its outer ormal uit vectors, which implies that P is a parallelotope. This completes the proof. With these lemmas i had, we ca complete the proof of Theorem 1.2. Proof of Theorem 1.2. Firstly, suppose that is equal to 2. From the defiitio of UP, followed by Lemma 3.4 or Lemma 2.3, we have that is, UP 2 = u i1 u i2 0 u i1 0 V i1 V i2 = u i1 0 V i1 V V i1 V 1 2 V = 1 2 V 2, UP 2 VP 2. u i1 u k =0 V k The, suppose that is equal to 3. From the defiitio of UP, followed by Lemma 2.3, the Lemma 3.4, we have that is UP 3 = u i1 u i2 u i3 0 u i1 u i2 0 = 1 3 V u i1 0 = 2! 3 2 V 2 u i1 0 V i1 V i2 V i3 = u i1 u i2 0 V i1 V i2 V V i1 V i2 V 2 3 V = 1 3 V V i1 V u i1 u k =0 V i1 = 3! 3 3 V 3, u i1 u i2 0 V k 1 3 V u i1 0 V i1 V i2 u i1 u i2 u k =0 V i1 V 1 3 V V k

12 G. Xiog / Advaces i Mathematics UP VP 3!1/3. 3 The equality coditio ca be derived from Lemma 3.4 immediately. This completes the proof. 4. Estimate of u i1 u ij u k =0 V k for symmetrical covex polytopes, where {u i1,...,u ij } {u 1,...,u N },u i1 u ij 0, 2 j 1 If P is a origi-symmetric covex polytope i R, we ca give a uified estimate for the sums of coe volumes o ay fiite 2 j 1 outer ormal uit vectors, which is also obtaied i [16] but through log ad complicated argumets. Lemma 4.1. Let P be a origi-symmetric polytope i R with iterior poits. For ay fixed {u i1,...,u ij } {u 1,...,u N }, such that u i1 u ij 0, 2 j 1, if the ormal vector u k of F k satisfies u i1 u ij u k = 0, the u i1 u ij u k =0 V k j VP. 15 If P is a parallelotope, the the equality of 15 holds. Coversely, if the equalities of 15 hold for all subsets {u i1,...,u ij } {u 1,...,u N } with u i1 u ij 0 simultaeously, 2 j 1, the P is a parallelotope. Proof. For ay fixed {u i1,...,u ij } {u 1,...,u N }, u i1 u ij 0, let L = spa{u i1,...,u ij }, fx= vol j P L + x. From Lemma 2.1, f 1 j x is cocave o D = P L. The u i1 u ij u k =0 V k = D = j 1 x h fxdsx x D [ ] 1 x fx j h dsx x where dsx is the j 1-dimesioal Lebesgue area measure o D. Geometrically, it is ituitively that D fx[ j 1 h x x dsx] is the volume of the set P = { x, y L L : x 0 D, x 0,y P L + x 0,x [o, x0 ] }, ad P has the same orthogoal projectio oto L as covex polytope P.

13 3226 G. Xiog / Advaces i Mathematics So, Now, we aim to show u i1 u ij u k =0 V k = j V P. V P VP. For this aim, we make use of spherical coordiate i the subspace L. Suppose the equatio of D is: Let The ρ = ρ 0 θ 1,θ 2,...,θ j 1. Fρ,θ= fρcos θ 1,...,ρsi θ 1 si θ j 1. V P VP= S j 1 dθ ρ 0 0 ρ j 1[ Fρ 0,θ Fρ,θ ] dρ 0, the iequality holds sice P is origi-symmetric. If P is a parallelotope cetered at the origi, it follows that u i1 u ij u k =0 V k = 1 2 VP 2j = j VP. Coversely, for ay {u i1,...,u ij } {u 1,...,u N }, with u i1 u ij 0, 2 j 1, suppose that u i1 u ij u k =0 V k = j VP. Set j = 1. From the equality coditio i Lemma 2.3, it follows that P is a parallelotope. This completes the proof. The proof of Theorem 1.1 is basically similar to the proof of He Leg Li Theorem [16]. To make the paper self-cotaied, we preset it here. Proof of Theorem 1.1. From the defiitio of UP, followed by Lemma 4.1, the Lemma 3.4, we have

14 UP = G. Xiog / Advaces i Mathematics = u i1 u i 0 u i1 u i 1 0 u i1 u i 1 0 = 1 V V i1 V i u i1 u i ! = 2 V 2 = V i1 V i 1 V u i1 u i 1 u k =0 V i1 V i 1 V 1 V V i1 V i 1 u i1 u i2 0 V i1 V i2 2! 2 V 2 V i1 V u i1 0 1! 1 V 1 V i1 =! V, u i1 0 u i1 u k =0 V k V k that is UP!1/ VP. The coditio of equality ca be derived from Lemma 4.1 immediately. This completes the proof. Combied 6 with 5, it gives VΠK UK 1 2 1! 1, 16 where the equality holds if ad oly if K is a parallelotope. It also ca be regarded as a modified versio of Scheider s projectio problem. Ackowledgmets This work was doe whe I was visitig Polytechic Istitute of NYU durig I was partially supported by CSC Chia Scholarship Coucil ad Polytechic Istitute of NYU. I am very grateful for academic guidace ad hospitality of professors Moika Ludwig, Erwi Lutwak, Deae Yag ad Gaoyog Zhag. I thak professor Deyi Li for may discussios. I also thak the referee for careful readig ad helpful commets o the paper.

15 3228 G. Xiog / Advaces i Mathematics Refereces [1] K. Ball, Volume ratios ad a reverse isoperimetric iequality, J. Lod. Math. Soc [2] K. Ball, Shadows of covex bodies, Tras. Amer. Math. Soc [3] K. Bör czky Jr., The stability of the Rogers Shephard iequality ad of some related iequalities, Adv. Math [4] J. Bourgai, J. Lidestrauss, Projectio bodies, i: Geometric Aspects of Fuctioal Aalysis 1986/1987, i: Lecture Notes i Math., vol. 1317, Spriger-Verlag, Berli, 1988, pp [5] S. Campi, P. Grochi, The L p -Busema Petty cetroid iequality, Adv. Math [6] G.D. Chakeria, E. Lutwak, Bodies with similar projectios, Tras. Amer. Math. Soc [7] R.J. Garder, Geometric Tomography, secod editio, Cambridge Uiversity Press, Cambridge, [8] R.J. Garder, Itersectio bodies ad the Busema Petty problem, Tras. Amer. Math. Soc [9] R.J. Garder, The Bru Mikowski iequality, Bull. Amer. Math. Soc. N.S [10] R.J. Garder, G. Zhag, Affie iequalities ad radial mea bodies, Amer. J. Math [11] P.R. Goodey, G. Zhag, Characterizatios ad iequalities for zooids, J. Lod. Math. Soc [12] M. Gromov, V. Milma, Geeralizatio of the spherical isoperimetric iequality to uiformly covex Baach spaces, Compos. Math [13] C. Haberl, L p itersectio bodies, Adv. Math [14] C. Haberl, M. Ludwig, A characterizatio of L p itersectio bodies, It. Math. Res. Not. 2006, Art. ID10548, 29 pp. [15] C. Haberl, F. Schuster, Geeral L p affie isoperimetric iequalities, J. Differetial Geom [16] He Biwu, Leg Gagsog, Li Kaghai, Projectio problems for symmetric polytopes, Adv. Math [17] A. Koldobsky, Fourier Aalysis i Covex Geometry, Math. Surveys Moogr., vol. 116, America Mathematical Society, Providece, RI, [18] K. Leichtweiß, Affie Geometry of Covex Bodies, J.A. Barth, Heidelberg, [19] M. Ludwig, Projectio bodies ad valuatios, Adv. Math [20] E. Lutwak, Mixed projectio iequalities, Tras. Amer. Math. Soc [21] E. Lutwak, O some affie isoperimetric iequalities, J. Differetial Geom [22] E. Lutwak, Itersectio bodies ad dual mixed volumes, Adv. Math [23] E. Lutwak, Iequalities for mixed projectio bodies, Tras. Amer. Math. Soc [24] E. Lutwak, D. Yag, G. Zhag, L p affie isoperimetric iequalities, J. Differetial Geom [25] E. Lutwak, D. Yag, G. Zhag, A ew affie ivariat for polytopes ad Scheider s projectio problem, Tras. Amer. Math. Soc [26] E. Lutwak, D. Yag, G. Zhag, Sharp affie L p Sobolev iequalities, J. Differetial Geom [27] A. Naor, The surface measure ad coe measure o the sphere of lp, Tras. Amer. Math. Soc [28] C. Petty, Isoperimetric problems, i: Proceedigs, Coferece o Covexity Combiatorial Geometry, Uiversity of Oklahoma, , pp [29] R. Scheider, Radom ploytopes geerated by aisotropic hyperplaes, Bull. Lod. Math. Soc [30] R. Scheider, Covex Bodies: The Bru Mikowski Theory, Cambridge Uiversity Press, Cambridge, [31] R. Scheider, W. Weil, Zooids ad related topics, i: P.M. Gruber, J.M. Wills Eds., Covexity ad Its Applicatios, Birkhäuser, Basel, 1983, pp [32] F.E. Schuster, Covolutios ad multiplier trasformatios of covex bodies, Tras. Amer. Math. Soc [33] F.E. Schuster, Valuatios ad Busema Petty type problems, Adv. Math [34] C. Steieder, Subword complexity ad projectio bodies, Adv. Math [35] A.C. Thompso, Mikowski Geometry, Cambridge Uiversity Press, Cambridge, [36] E. Werer, D. Ye, New L p affie isoperimetric iequalities, Adv. Math [37] G. Zhag, Restricted chord projectio ad affie iequalities, Geom. Dedicata [38] G. Zhag, Cetered bodies ad dual mixed volumes, Tras. Amer. Math. Soc [39] G. Zhag, The affie Sobolev iequality, J. Differetial Geom [40] G. Ziegler, Lectures o Polytopes, Spriger-Verlag, New York, [41] C. Zog, What is kow about uit cubes, Bull. Amer. Math. Soc. N.S [42] C. Zog, The Cube A Widow to Covex ad Discrete Geometry, Cambridge Uiversity Press, Cambridge, 2006.