A NEW AFFINE INVARIANT FOR POLYTOPES AND SCHNEIDER S PROJECTION PROBLEM

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1 A NEW AFFINE INVARIANT FOR POLYTOPES AND SCHNEIDER S PROJECTION PROBLEM Erwi Lutwak, Deae Yag, ad Gaoyog Zhag Departmet of Mathematics Polytechic Uiversity Brookly, NY 1101 Abstract. New affie ivariat fuctioals for covex polytopes are itroduced. Some sharp affie isoperimetric iequalities are established for the ew fuctioals. These ew iequalities lead to fairly strog volume estimates for projectio bodies. Two of the ew affie isoperimetric iequalities are extesios of Ball s reverse isoperimetric iequalities. If K is a covex body (i.e., a compact, covex subset with oempty iterior) i Euclidea -space, R, the o the uit sphere, S 1, its support fuctio, h(k, ): S 1 R, is defied for u S 1 by h(k, u) = max{u y : y K}, where u y deotes the stadard ier product of u ad y. The projectio body, ΠK, of K ca be defied as the covex body whose support fuctio, for u S 1, is give by h(πk, 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. A importat usolved problem regardig projectio bodies is Scheider s projectio problem: What is the least upper boud, as K rages over the class of origi-symmetric covex bodies i R, of the affie-ivariat ratio ( ) [V (ΠK)/V (K) 1 ] 1/, 1991 Mathematics Subject Classificatio. 5A40. Key words ad phrases. affie isoperimetric iequalities, reverse isoperimetric iequalities, projectio bodies, asymptotic iequalities. Research supported, i part, by NSF Grat DMS Typeset by AMS-TEX

2 where V is used to abbreviate vol. See [S1], [S], [SW] ad [Le]. Scheider [S1] cojectured that this ratio is maximized by parallelotopes. I [S1], Scheider also preseted applicatios of such results i stochastic geometry. However, a couterexample was produced i [Br] to show that this is ot the case. We will preset a modified versio of Scheider s cojecture that has a affirmative aswer. I additio, we will obtai a iequality that gives a upper boud for the affie ratio ( ). While our upper boud is ot sharp for ay, evertheless it is asymptotically optimal. To be more specific, i this paper, we itroduce a ew cetro-affie fuctioal U, defied o the class of polytopes, which is closely related to the volume fuctioal V. While i geeral U(K) < V (K), if K is a radom polytope (with may faces), the U(K) is very close to V (K). We shall prove the followig variatio of Scheider s projectio cojecture: Theorem. If K is a origi-symmetric covex polytope i R, the V (ΠK) U(K) V (K) 1 ( ) 1! with equality if ad oly if K is a parallelotope. The iequality of the theorem immediately provides a asymptotically optimal boud for the affie ratio ( ): Corollary 4.7. If K is a covex body i R that is symmetric about some poit, the V (ΠK)/V (K) 1 ( ) 1.! While the iequality of Corollary 4.7 is ot sharp for ay value of, it is asymptotically optimal i the sese that a weakeed form of Corollary 4.7 is: Corollary 4.7. If K is a covex body i R that is symmetric about some poit, the [V (ΠK)/V (K) 1 ] 1/ e. If K is take to be the cube, the the affie ratio (*) is a costat (to be specific 1) idepedet of the dimesio. Thus, up to a costat multiple, the iequality of Corollary 4.7 is best possible. The fact that there exists a costat, idepedet of the dimesio, that domiates the affie ratio (*) was show by Giaopoulos ad Papadimitrakis [GiPa]. We will also establish a sharp affie isoperimetric iequality (Theorem 4.11) for our ew fuctioal that will immediately give:

3 Corollary 4.1. If K is a covex body i R, the LUTWAK, YANG, AND ZHANG 3 V (ΠK)/V (K) 1 ( + 1) +1 (!) 3. Agai, while this iequality is ot sharp for ay value of, it is asymptotically optimal. Yet a third sharp affie isoperimetric iequality (Theorem 4.8) for our fuctioal will yield a asymptotically optimal boud for a ope problem regardig polar projectio bodies (Corollary 4.9). I the ext-to-last sectio, we itroduce a family of affie fuctioals, U 1,..., U, for which U 1 = V ad U = U. Two sharp affie isoperimetric iequalities (Theorems 5. ad 5.3) will be preseted for these fuctioals. These iequalities geeralize Ball s reverse isoperimetric iequality. 1. Backgroud ad otatio I this sectio we preset the termiology ad otatio we shall use throughout. For quick referece we collect some kow results that will be the igrediets of the proofs give i subsequet sectios. For geeral referece the reader may wish to cosult the books of Garder [G], Leichtweiß [Le], Scheider [S], ad Thompso [T]. If K is a covex body that cotais the origi i its iterior, the write K for the polar of K; i.e., K = {x R : x y 1 for all y K}. Let P be a covex polytope i R that cotais the origi i its iterior. Let u 1,..., u N deote the outer uit ormals of P. Let h 1,..., h N deote the correspodig distaces from the origi to the faces ad a 1,..., a N the areas (i.e. ( 1)-dimesioal volumes) of the correspodig faces. I [LYZ] the ellipsoid Γ P was defied as the ellipsoid whose polar, Γ P, has its support fuctio give by (1.1) h(γ P, u) = 1 V (P ) N u u i a i h i, for u S 1. Note that we use Γ P rather tha (Γ P ) to deote the polar of Γ P. The ew ellipsoid is i a sese a dual of the Legedre ellipsoid of classical

4 4 mechaics. (See e.g., Leichtweiß [Le], Lidestrauss ad Milma [LiM], Milma ad Pajor [MPa1, MPa], ad Petty [P1] for referece regardig the Legedre ellipsoid.) We shall make use of the fact that the operator Γ is a cetro-affie operator i the sese that (1.) Γ φp = φγ P, for all φ GL(), where φp = {φx: x P }. This fact was established i [LYZ]. We shall also require a similar fact, first established by Petty [P] (see e.g. [BoLi] ad [L] for alterate proofs), regardig the operator Π: (1.3) ΠφP = φ t ΠP, for all φ SL(), where φ t deotes the iverse of the traspose of φ. We shall write Π P for the polar of ΠP, rather tha (ΠP ). Recall that McMulle s itrisic volumes, V 0 (P ),..., V (P ), of the polytope P ca be defied [Mc] as coefficiets i the Steier polyomial: V (P + λb) = λ i ω i V i (P ), i=0 where ω i is the i-dimesioal volume of the uit ball i R i ad ω 0 = 1. Thus V (P ) = V (P ). Suppose u 1,..., u N S 1 ad λ 1,..., λ N > 0. If K is a covex body whose support fuctio, for u S 1, is give by h(k, u) = N λ i u u i, the K is called a zootope. Obviously the projectio bodies of polytopes are zootopes. Although we shall make o use of this fact, it ca be show that all zootopes are projectio bodies of origi-symmetric polytopes. We will eed the McMulle-Mathero-Weil formula (see [Sh], [Ma], ad [W]) for the itrisic volume, V k (P ), for 1 k, of the zootope K: (1.4) V k (K) = k k! 1 i 1,...,i k N λ i1 λ ik [u i1,..., u ik ], where [u i1,..., u ik ] deotes the k-dimesioal volume of the k-dimesioal parallelotope {c 1 u i1 + + c k u ik : 0 c i 1}. (See [SW] ad [GoW] for surveys about zooids ad zootopes.)

5 LUTWAK, YANG, AND ZHANG 5 The Joh ellipsoid of a covex body is the largest (i volume) ellipsoid that is cotaied i the body. The Joh poit of a covex body is the ceter of the Joh ellipsoid of the body. A covex body i R is said to be i Joh positio if its Joh ellipsoid is the stadard uit ball i R. Obviously, every covex body i R may be GL()-trasformed ito Joh positio.. A basic idetity The followig basic fact is critical for our mai results. Lemma.1. Suppose u 1,..., u N S 1 ad λ 1,..., λ N > 0. If N λ i u u i = 1, for all u S 1, the for each k such that 1 k, 1 i 1,...,i k N λ i1 λ ik [u i1,..., u ik ] =! ( k)!. To prove Lemma.1 we shall make use of some basic facts regardig mixed discrimiats. Recall that for positive semi-defiite matrices Q 1,..., Q N ad real λ 1,..., λ N 0, the determiat of the liear combiatio λ 1 Q λ N Q N is a homogeeous polyomial of degree i the λ i, det(λ 1 Q λ N Q N ) = 1 i 1,...,i N λ i1 λ i D(Q i1,..., Q i ), where the coefficiet D(Q i1,..., Q i ) depeds oly o Q i1,..., Q i (ad ot o ay of the other Q j ) ad thus may be chose to be symmetric i its argumets. The coefficiet D(Q i1,..., Q i ) is called the mixed discrimiat of Q i1,..., Q i. The mixed discrimiat D(Q,..., Q, I,..., I), with k copies of Q ad k copies of the idetity matrix, I, will be abbreviated by D k (Q). Note that the elemetary mixed discrimiats D 0 (Q),..., D (Q) are thus defied as the coefficiets of the polyomial ( ) det(q + λi) = λ i D i (Q). i Obviously D (Q) = det(q) while D 1 (Q) is the trace of Q. i=0

6 6 We require the followig easily-established (see e.g., Petty [P1]) fact: Suppose y ij R, 1 i N, 1 j, ad let the positive semi-defiite matrices Q j, 1 j, be defied by x Q j x = N x y ij, for all x R, the the mixed discrimiat of Q 1,..., Q is give by (.) D(Q 1,..., Q ) = 1! [y i1 1,..., y i ]. 1 i 1,...,i N It follows immediately from (.) that if for o-egative measures µ 1,..., µ o S 1, the positive semi-defiite matrices Q j, 1 j, are defied by u Q j u = u v dµ j (v), S 1 for all u S 1, the the mixed discrimiat of Q 1,..., Q is give by (.3) D(Q 1,..., Q ) = 1 [v 1,..., v ] dµ 1 (v 1 ) dµ (v ).! S 1 S 1 From this we obtai: Lemma.4. Suppose u 1,..., u N S 1 ad λ 1,..., λ N > 0. If Q is a positive defiite matrix so that, u Qu = N λ i u u i, for all u S 1, the, for 1 k, D k (Q) = ( k)!! 1 i 1,...,i k N λ i1 λ ik [u i1,..., u ik ].

7 LUTWAK, YANG, AND ZHANG 7 To prove this take µ 1 = = µ k i (.3) to be the measure that is cocetrated o u 1,..., u N with weights λ 1,..., λ N, ad let dµ i (v) = ω 1 dv, for k + 1 i. (Note that Q i = I, for k + 1 i ), ad get (.5) D k (Q) = c,k [u i1,..., u ik ] λ i1 λ ik,! where c,k is give by c,k [v 1,..., v k ] = ω k 1 i 1,...,i k N [v 1,..., v k, v k+1,..., v ] dv k+1 dv. S 1 S 1 Sice c,k above is idepedet of our choice of Q we ca compute c,k most easily by i (.5) choosig {u 1,..., u N } to be the stadard orthoormal basis, {e 1,..., e }, i R, ad all the λ i = 1 (ad thus Q = I). This immediately shows that ad completes the proof. c,k = ( k)!, Obviously Lemma.1 is the special case of Lemma.4 whe Q = I. 3. The ew affie fuctioal ad a ew affie class of polytopes Defiitio 3.1. If P is a covex polytope i R which cotais the origi i its iterior, ad u 1,..., u N are the outer ormal uit vectors to the faces of P, with h 1,..., h N the correspodig distaces of the faces from the origi ad a 1,..., a N the correspodig areas of the faces, the defie U(P ) by U(P ) = 1 u i1 u i 0 h i1 h i a i1 a i. Obviously the fuctioal U is cetro-affie ivariat i that, (3.) U(φP ) = U(P ), for all φ SL(). Sice V (P ) = 1 N a ih i, it follows immediately that (3.3) U(P ) < V (P ). As a aside, we observe that U(P ) is sigificatly less tha V (P ) oly if P is highly symmetric ad has few faces. For a radom polytope with a large umber of faces U(P ) is very close to V (P ). It is this property of the fuctioal U which will make it so useful.

8 8 It will be helpful to itroduce a ew class of covex polytopes i R. A covex polytope is said to be i the class P if for ay two o-coplaar sets of vertices of the polytope, say v 1,..., v ad v 1,..., v, the simplices whose vertices are 0, v 1,..., v ad 0, v 1,..., v have idetical volumes. Obviously, this is a cetroaffie ivariat class i that for P P ad φ GL(), we have φp P. It is easily see that both the regular simplex, whose cetroid is at the origi, ad the regular cross-polytope are i P. As a aside, we ote that it is easily see that the umber of sides, N, of a body i P is such that 3 N 6, with all values betwee 3 ad 6 actually assumed. I fact all the bodies i P are easily characterized. However, for larger, o trivial descriptio of the bodies i P seems likely. Let P deote the class of polars of the polytopes i P. Obviously, this is a cetro-affie ivariat class as well. We shall establish: 4. Iequalities for Scheider s problem Lemma 4.1. If P is a covex polytope i R that cotais the origi i its iterior, the ( ) 1 ω [U(P )V (P )] / V (Γ P )V (ΠP ),! with equality if ad oly if P P. To prove the lemma, suppose P is a covex polytope i R that cotais the origi i its iterior ad u 1,..., u N deote the outer uit ormals of P, with h 1,..., h N deotig the correspodig distaces from the origi to the faces ad a 1,..., a N the areas of the correspodig faces. Obviously, the support fuctio of ΠP is give by h(πp, u) = 1 N u u i a i, for all u S 1, ad thus by the McMulle-Mathero-Weil formula (1.4) we have (4.) V (ΠP ) = 1! 1 i 1,...,i N a i1 a i [u i1,..., u i ]. Sice volume is a SL()-ivariat fuctioal, i light of (3.), (1.), ad (1.3), we see that i order to establish the lemma we may assume, without loss of geerality, that Γ P is a ball; i.e., ( ) 1 V (Γ P ) (4.3) Γ P = B, ω

9 LUTWAK, YANG, AND ZHANG 9 where B deotes the uit ball cetered at the origi ad, as before, ω = V (B). From (4.3) ad defiitio (1.1) of Γ, we have Now Lemma.1, with ( ) ω 1 = V (Γ P ) V (P ) λ i = a i h i ( V (Γ P ) ω N u u i a i h i. ) 1 V (P ) gives (4.4) ( ) ω = V (Γ P ) 1!V (P ) 1 i 1,...,i N a i1 h i1 ai h i [u i1,..., u i ]. Now (4.4), together with the Hölder iequality, ad (4.) give:!v (P ) ( ω U(P ) V (Γ P ) = 1 U(P ) ) u i1 u i 0 ( ) [ui1,..., u i ] h h a a i1 i i1 i h i1 h i 1 U(P ) [u i1,..., u i ]a i1 a i u i1 u i ( ) 0! V (ΠP ) = U(P ), with equality if ad oly if [u i1,..., u i ] h i1 h i is idepedet of the choice of the subscripts wheever u i1 u i 0. But [u i1,..., u i ] h i1 h i = [u i1 ρ i 1,..., u i ρ i ], where u ij /h ij = u ij ρ i j are the vertices of P ad [u i1 ρ i 1,..., u i ρ i ] is equal to! times the volume of the simplex whose vertices are 0, u i1 ρ i 1,..., u i ρ i. Thus equality is possible if ad oly if P P. This completes the proof. The followig lemma, proved i [LYZ], will be eeded.

10 10 Lemma 4.5. If P is a origi-symmetric covex polytope i R, the V (Γ P ) ω V (P ), with equality if ad oly if P is a parallelotope. This together with Lemma 4.1 immediately gives: Theorem 4.6. If P is a origi-symmetric covex polytope i R, the V (ΠP ) U(P ) V (P ) 1 ( ) 1! with equality if ad oly if P is a parallelotope. A immediate cosequece of this ad (3.3) is: Corollary 4.7. If K is a covex body i R that is symmetric about some poit, the V (ΠK) V (K) 1 ( ) 1.! Reiser s iequality [R1], [R], [GMR] states that if K is a projectio body i R, the V (K)V (K ) 4!. The best lower boud for the cetro-affie volume product V (K)V (K ), as K rages over the class of origi-symmetric covex bodies, is ukow. The best results to date are those of Bourgai ad Milma [BoM]. Theorem 4.6 together with Reiser s iequality immediately gives: Theorem 4.8. If P is a origi-symmetric covex polytope i R, the V (Π P )U(P ) V (P ) 1 with equality if ad oly if P is a parallelotope. From Theorem 4.8 ad (3.3) we immediately get:, (!) 1 Corollary 4.9. If K is a covex body i R that is symmetric about some poit, the V (Π K)V (K) 1. (!) 1 The problem of determiig the best lower boud for the affie product [V (Π K)V (K) 1 ] 1/,

11 LUTWAK, YANG, AND ZHANG 11 as K rages over the class of origi-symmetric bodies is ope ad importat. The best upper boud for the affie product V (Π K)V (K) 1, as K rages over the class of all covex bodies, is give by the Petty projectio iequality [P3]. The best lower boud for the affie product V (Π K)V (K) 1, as K rages over the class of all covex bodies, is give by the Zhag projectio iequality [Z]. (See also e.g., the books of Scheider [S], Leichtweiss [Le], ad Garder [G].) That the iequality of Corollary 4.9 provides a asymptotically optimal lower boud for the affie product [V (Π K)V (K) 1 ] 1/, as K rages over the class of origi-symmetric bodies, may be see by takig K to be the cube. The followig result was established i [LYZ]: Lemma If P is a covex polytope i R that has its Joh poit at the origi, the!ω V (Γ P ) V (P ), ( + 1) +1 with equality if ad oly if P is a simplex. Together with Lemma 4.1, this gives: Theorem If P is a covex polytope i R that has its Joh poit at the origi, the V (ΠP ) U(P ) V (P ) 1 +1 ( + 1), (!) 3 with equality if ad oly if P is a simplex. From this ad (3.3) we have: Corollary 4.1. If K is a covex body i R, the V (ΠK)/V (K) 1 ( + 1) +1 /(!) 3. Theorem 4.11 immediately gives Corollary 4.1 for polytopes whose Joh poit is at the origi. But both V ad Π are traslatio ivariat, which shows that the iequality of Corollary 4.1 holds for arbitrary polytopes. Sice both V ad Π are cotiuous o the space of covex bodies, with the Hausdorff topology, a obvious approximatio argumet shows that the iequality of Corollary 4.1 must hold for all covex bodies. That the iequality of Corollary 4.1 provides a asymptotically optimal boud for the affie ratio (*) ca be see by takig K to be the simplex.

12 1 5. Extesios of Ball s reverse isoperimetric iequality Defiitio 5.1. If P is a covex polytope i R which cotais the origi i its iterior, ad u 1,..., u N are the outer ormal uit vectors to the faces of P, with h 1,..., h N the correspodig distaces of the faces from the origi ad a 1,..., a N the correspodig areas of the faces, the for 1 j, defie U j (P ) by: U j (P ) j = 1 j h i1 h ij a i1 a ij. u i1 u ij 0 Obviously, U 1 (P ) = V (P ) ad U (P ) = U(P ). The fuctioal U j is a cetroaffie ivariat: For each polytope P, U j (φp ) = U j (P ), for all φ SL(). Ball [B] proved that a origi-symmetric polytope P i R that has bee GL()- trasformed ito its Joh positio satisfies the followig reverse isoperimetric iequality: ( ) S(P ) V (P ) 1. A covex polytope P was defied i [LYZ] to be i dual isotropic positio if Γ P is a ball ad V (P ) = 1. Note that for each covex polytope P, that cotais the origi i its iterior, there is a GL() trasformatio of P that trasforms P ito a polytope i dual isotropic positio. From the fact that V 1 (ΠP ) = V 1 (P ) = S(P ), for every polytope P, oe immediately sees that the iequality of the ext theorem, for j = 1, is precisely Ball s symmetric reverse isoperimetric iequality. Theorem 5.. If P is a origi-symmetric covex polytope i R that has bee GL()-trasformed ito dual isotropic positio, the V j (ΠP ) U j (P ) j V (P ) j j with equality if ad oly if P is a cube. j (! j ) 1, 1 j <, j! ( j)! Thus Ball s symmetric reverse isoperimetric iequality will hold whe the polytope P is i dual isotropic positio (as well as i Joh positio). Ball [B] proved that each polytope P i R that has bee GL()-trasformed ito its Joh positio satisfies the reverse isoperimetric iequality: S(P ) V (P ) 1 3/ ( + 1) (+1)/ /!. Our ext theorem (for j = 1) shows that this is also the case if the polytope is GL()-trasformed ito dual isotropic positio.

13 LUTWAK, YANG, AND ZHANG 13 Theorem 5.3. If P is a covex polytope i R that has bee traslated so that its Joh poit is at the origi ad GL()-trasformed so that it is i dual isotropic positio, the V j (ΠP ) U j (P ) j V (P ) j j j (!) 1 j ( + 1) (+1)j j![( j)!] 1, 1 j <, with equality if ad oly if P is a regular simplex. Agai, ote that Theorem 5.3 shows that Ball s reverse isoperimetric iequality will hold whe the polytope P is i dual isotropic positio (as well as i Joh positio). To prove Theorems 5. ad 5.3 we first suppose that P is a covex polytope i R that cotais the origi i its iterior with u 1,..., u N the outer uit ormals of P, with h 1,..., h N the correspodig distaces from the origi to the faces ad a 1,..., a N the areas of the correspodig faces. Sice the support fuctio of ΠP is give by h(πp, u) = 1 N u u i a i, from the McMulle-Mathero-Weil formula for the itrisic volume of zootopes, (1.4), we have (5.4) V j (ΠP ) = 1 j! 1 i 1,...,i j N Sice Γ P is defied, for u S 1, by h(γ P, u) = 1 V (P ) [u i1,..., u ij ]a i1 a ij. N u u i a i h i, ad it is assumed that ad V (P ) = 1, we have ( ) 1 V (Γ P ) Γ P = B, ω ( ω V (Γ P ) ) = N u u i a i h i.

14 14 Now Lemma.1, with gives (5.5) ( ω V (Γ P ) ) j λ i = a i h i = ( j)!! ( V (Γ P ) ω 1 i 1,...,i j N ) a i1 h i1 ai j h ij [u i1,..., u ij ]. Now (5.5), together with the Hölder iequality, ad (5.4) give (exactly as i the proof of Lemma 4.1): ( ) j ( ) 1! ω j! V j (ΠP ) j U j (P ) j ( j)! V (Γ P ) j U j (P ) j, with equality if ad oly if [u i1,..., u ij ] h i1 h ij is idepedet of the choice of the subscripts wheever u i1 u ij 0. (Note that if P is a cube cetered at the origi, or P is a regular simplex with its cetroid at the origi, the this certaily is the case.) Lemma 4.10 (or Lemma 4.5 i the origi-symmetric case) together with the last iequality provide the coclusios of Theorems 5. ad Ope problems Two obvious questios regardig the fuctioals V ad U beg to be asked. Questio 6.1. If P is a origi-symmetric covex polytope i R, the is it the case that U(P ) 1 (!) 1/ V (P ), with equality if ad oly if P is a parallelotope? Questio 6.. Suppose P is a covex polytope i R with its Joh poit at the origi. Is it the case that U(P ) [( + 1)!] 1 ( + 1) with equality if ad oly if P is a simplex? V (P ), We ote that the domai of defiitio of the fuctioal U may be exteded (i a atural maer) to iclude all covex bodies. While we have chose to preset our iequalities oly for covex polytopes, all of the iequalities preseted i this ote hold for arbitrary covex bodies.

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