Minimal surface area position of a convex body is not always an M-position

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1 Miimal surface area positio of a covex body is ot always a M-positio Christos Saroglou Abstract Milma proved that there exists a absolute costat C > 0 such that, for every covex body i R there exists a liear image T of with volume, such that T + D / C, where D is the Euclidea ball of volume. T is the said to be i M-positio. Giaopoulos ad Milma asked if every covex body that has miimal surface area amog all its affie images of volume is also i M-positio. We prove that the aswer to this questio is egative, eve i the -ucoditioal case. Itroductio Let C > 0. We say that a covex body of volume i R is i M-positio with costat C if () + D / C, where D is the Euclidea ball of volume. Here, = deotes -dimesioal volume ad A + B = {x + y : x A, y B} is the Mikowski sum of the sets A ad B. The startig poit of this paper is a famous result of Milma [7] statig that there exists a absolute positive costat C such that every covex body has a liear image of volume which is i M-positio with costat C. I what follows, we will refer to the M-positio without specifyig some precise value for this umerical costat. It follows from the Bru Mikowski iequality + 2 / / + 2 / for o-empty compact subsets, 2 of R, that if is i M-positio the { } + D / mi T + D / : T SL(). Thus, () may be viewed as a iverse form of the Bru Mikowski iequality. The otatio a b meas that the ratio a/b is bouded by absolute costats (from above ad from below). We say that is i miimal surface area positio if the surface area () of is miimal amog those of its affie images of the same volume. Petty [9] gave a characterizatio of the miimal surface area positio: has miimal surface area if ad oly if the fuctio x x, y 2 ds (y) S is costat o S, where S deotes the surface area measure of. It is also well-kow that the miimal surface area positio is uique up to orthogoal trasformatios. Giaopoulos ad Milma [4] gave characterizatios of the same type for problems that ivolve other quermassitegrals. I the same paper, they asked if the miimal surface area positio is also a M-positio. Our mai result is the followig.

2 Theorem.. There exists a absolute costat c 0 > 0 with the followig property: for every positive iteger, there exists a -ucoditioal covex body of volume i R which is i miimal surface area positio ad, at the same time, + D / c 0 /8. Theorem. shows that, if is large eough the the miimal surface area positio may be far from beig a M-positio, eve i the -ucoditioal case. Recall that is called -ucoditioal if it is symmetric with respect to all coordiate hyperplaes. Let us also ote that, i the case = 2, it was proved i [4] that every covex body with miimal surface area has the property + td = mi{ T + td : T SL()} for every t > 0. Theorem. shows that this is o loger true i higher dimesios, at least whe c /8 t c /8, where c > 0 is a small eough absolute positive costat. 2 Backgroud Let ad L be two covex bodies i R. It is well kow by a classical result of Mikowski that, for t > 0, the volume of the covex body L + t is a polyomial of degree, as a fuctio of t. More precisely, oe ca write L + t = j= t j ( j ) V ([j], L[ j]), where V ([j], L[ j]) are o-egative quatities, called i commo mixed volumes of ad L. I this paper we are mostly iterested i the case j =. We will write for simplicity V (, L) := V (, L,..., L) = V ([], L[ ]). Note that V (, ) =. Also, if B 2 is the Euclidea uit ball i R the V (, B 2 ) ad V (B 2, ) are (up to a costat depedig oly o ) the mea width ad the surface area of respectively. A fudametal fact cocerig V (, L) is Mikowski s iequality V (, L) L, with equality if ad oly if ad L are homothetic. Assume, ow, that both ad L have the origi as a iterior poit. The quatity V(, L) ca be expressed as: V (, L) = h (x) ds L (x), S where h (x) = max y x, y is the support fuctio of ad S L is the surface area measure of L (S L is a measure o S ). Recall the defiitio of S L : If ω is a Borel subset of S, the { S L (ω) = x bd(l) : (u, t) ω R, so that (tu + u ) L = {x}}. Here, bd(l) deotes the boudary of L. Let be a covex body i R with the origi as a iterior poit. The polar body of is defied by = {x R : x, y, y }. We will use the asymptotic formula b ( ) / ω 2/ b, 2

3 which holds true if the cetroid of is at the origi. Here, ω deotes the volume of the Euclidea uit ball B 2 ad b > 0 is a absolute costat. The right had side iequality is the classical Blaschke Sataló iequality ad it is sharp; equality holds if ad oly if is a ellipsoid cetered at the origi. The left had side iequality was proved much later by Bourgai ad Milma (see [2] ad [7]) ad it is ofte called the iverse Blaschke Sataló iequality. We close this Sectio with three facts that will be eeded i the sequel. The first oe, roughly speakig, states that is i M-positio if ad oly if is i M-positio. This follows from the proof of Milma s theorem (see [7, Sectio 4, Remark 3]) o the existece of the M-positio (see also [3, Theorem 5.3]). Fact I. There exists a absolute costat b 2 > 0 such that, if is a covex body with cetroid at the origi i R, the + D / b 2 + D /, where A = A / A. For the ext two, we eed the defiitio of the coverig umber of by L. If ad L are compact subsets of R with o-empty iteriors, we defie { } k N(, L) = mi k N : (x i + L) for some x,..., x k R. Fact II. If is compact ad L is a cetrally symmetric covex body i R, the i= + L + L 2 N(, L) 2. L L Fact III. There exists a absolute costat b 3 > 0 such that if ad L are covex bodies i R, the L + D b 3 + L + D. Fact II is a easy cosequece of the defiitios. The iequality o the left is trivial while a short proof for the iequality o the right is give i [7, sectio 5] (actually, this is stated i [7] i the special case where is covex ad L is a ball; however the same proof works for our purposes as well). Fact III follows immediately from Fact II i the case where L is cetrally symmetric. The geeral case ca be deduced from the Rogers Shephard iequality [0]. 3 Curvature Images The otio of the curvature image of a star-body was itroduced by Lutwak [6] ad will play a importat role i the proof of our mai result. If is a star-body with cetroid at 0, the curvature image C() of is the uique covex body with cetroid at the origi ad surface area measure ds C() = + ρ+ dλ =: f C()dλ, where ρ is the radial fuctio of ad λ is the Lebesgue measure o S. Existece ad uiqueess of C() are guarateed by Mikowski s existece theorem (see e.g. [], pp ). It ca be proved that C() is affiely associated with. Oe ca check that (2) C(T ) = (T ) C() for every volume preservig liear trasformatio T of R. 3

4 We defie the quatity F = + { } mi x 2 dx : T SL(), T where 2 deotes Euclidea orm. Usig itegratio i polar coordiates, we readily see that the surface area of C() ca be writte i the form (C()) = x 2 dx. Takig ito accout (2) we obtai the ext Lemma. Lemma 3.. Let be a star-body with cetroid at 0. The, C() has miimal surface area if ad oly if F = x + 2 dx. The key step for the proof of Theorem. will be the followig: Theorem 3.2. Let be a star-body of volume i R, with cetroid at 0. Let L be a covex body i R, with cetroid at 0, such that L ad L α for some α. Let T SL(2), such that F D = x 2 dx. 2 T ( D ) If C(T ( D )) is the homothet of C(T ( D )) of volume, the C(T ( D )) + D 2 /2 c 4 F, for some costat c = c(α) > 0 which depeds oly o α. For the proof of Theorem 3.2 we will modify a idea from [, Propositio.4]; for our purposes, we have to deal with the L -case istead of the more coveiet L 2 -case. The proof of Theorem 3.2 will be give i the ext Sectio. We close this Sectio with some iformatio o the relatio of the M-positio of with the M-positio of C(). Propositio 3.3. There exists a absolute costat c > 0 such that, if is as i Theorem 3.2, the Proof. Mikowski s iequality implies that L C() V (L, C()) = = = +. S c C() c c(α). ρ L (x) h L (x)ds C() (x) S (x) + dλ(x) ρ + O the other had, usig the iverse Blaschke Sataló iequality we get L b L b α = b α. S ρ (x) ρ + (x) + dλ(x) 4

5 It follows that C() ( ) b α c (α). + O the other had, from Hölder s iequality we have = = ρ (x) dλ(x) = h [ ] + S C() (x) h + C() (x)ρ (x) dλ(x) S ( ) ( ) S h C() (x) dλ(x) + S ( + ) ( + ) h C()(x)ρ + (x) dλ(x) + = + [( + )] + C () + C() + c C() +, where c > 0 is a absolute costat (i the last step, we have used the Blaschke Sataló iequality for C()). Propositio 3.4. There exists a costat c(α) > 0, which depeds oly o α, such that if is as i Theorem 3.2 ad C() := C() / C(), the Proof. Usig Mikowski s iequality, we have + D / c(α) C() + D /. L + C() C() V (L + C(), C()) = (h L (x) + h C() (x))f C() (x)dλ(x) S = h L (x)f C() (x) dλ(x) + C() S = ρ L (x)f C()(x) dλ(x) + C() S ρ (x)ρ+ (x) dλ(x) + C() S + + C() = + C(). We set L = L / L. From the iverse Blaschke Sataló iequality ad Propositio 3.3 it follows that (3) L + C() / b c L + C() / b c ( + c c(α)). The, usig Facts I ad III for L = L / L, we obtai + D / L + D / α L + D / b 2 α L + D / b 2 b 3 α C() + L / c(α) C() + D /. C() / C() + D / Remark. Oe ca use (3) ad Facts I ad III as i the proof of Propositio 3.4 to derive a iverse form of the Propositio. Thus, if L / α, the is i M-positio if ad oly if C() is i M-positio (with a costat that depeds oly o α). 5

6 4 Proof of Theorem 3.2 Lemma 4.. Let F R ad F 2 R 2 be compact sets with F = F 2 2 =. The, for every T SL( + 2 ), T F T F 2 2. Proof. Approximatig F ad F 2 by uios of o-overlappig cubes, we may assume that F ad F 2 are cubes. I this case, for every T SL( + 2 ) we have T F T F 2 V 2 =, where V = spa(t (F )) ad T F 2 V is the orthogoal projectio of T F 2 oto V. The result follows. Lemma 4.2. Let be a star-body of volume i R such that x 2 dx = F, ad let T SL(2) satisfy The, for some absolute costat c > 0. F D = 2 T D / T ( D ) c F x 2 dx. Proof. For every T SL(2) we ca write z 2 dz = T ( D ) T z 2 dz = D D ( T x 2 + T y 2 ) dy dx. T x + T y 2 dy dx D Sice we ca also write z 2 = T ( D ) T x + T y 2 dy dx = D T x + T y 2 + T x T y 2 D 2 D T x T y 2 dy dx, dy dx 2 D ( T x 2 + T y 2 ) dy dx. I other words, T ( D ) z 2 dz ( T x 2 + T y 2 ) dy dx D = D T x 2 dx + T y 2 dy D = T x 2 dx + T y 2 dy. D Moreover, there exist orthogoal trasformatios U, U 2 O(2) such that U (spa(t )) = spa() ad U 2 (spa(t D )) = spa(d ). 6

7 We set T := U T spa() ad U 2 := U 2 T spa(d). The, T x 2 dx = U T x 2 dx = T x 2 dx = det T / det T T x / dx 2 = det T / x 2 dx, where S := T / det T / (ote that S SL()). Similar calculatios give T x 2 dx = det T 2 / x 2 dx, D S 2 D for some S 2 SL(). Sice S Lemma 4. shows that ( ) / det T / T = = T /, (4) det T / T D /. If ( T := diag,...,, F,..., ) F. F F The, we may choose U = U 2 = I 2, ad hece, S = S 2 = I, det T / = F ad det T 2 / = F. Sice x 2dx = F, we have 2 T ( D ) z 2 dz = F because F D. Therefore, we have show that z 2 dz 2 2 which implies that T ( D ) x 2 dx + F D x 2 dx F F + F F DN F, T ( D ) z 2 dz c F, c F det T / x 2 dx + det T 2 / S det T / S x 2 dx det T / F, where c, c > 0 are absolute costats. It follows from (4) that c F T D / F, S 2 D x 2 dx ad this completes the proof of the Lemma. 7

8 Lemma 4.3. There exists a absolute costat c 4 > 0 such that, for every star-body of volume i R ad T SL(2) so that F D = x 2 dx, 2 oe has T ( D ) c 4 4 F T ( D ) + D 2 /2 2. Proof. We set t = T ( D ) + D 2 /2. Fact II implies that 2 N(T ( D ), D 2 ) /2 2t. Cosequetly, there exist k (2t) 2 ad x,..., x k R 2 such that T ( D ) k (x i + D 2 ), i= which gives ad hece, where V = spa(t D ). It follows that k T D (x i + D 2 ), i= k T D [(x i + D 2 ) V ], i= T D k (x i + D 2 ) V = k D 2 V (2Ct) 2, i= where C > 0 is a absolute costat. Now, we have c 3 F T D / (2Ct) 2, from Lemma 4.2. Proof of Theorem 3.2. From Propositio 3.4 ad Lemma 4.3, we have C(T ( D )) + D 2 /2 2 5 Proof of the mai result c 2 T ( D ) + D 2 /2 2 c(α) c 4 4 F. I order to deduce Theorem. from Theorem 3.2, we eed to costruct -ucoditioal star-bodies which have large F ad, i additio, are almost covex. Propositio 5.. There exists a absolute costat c 5 > 0 such that, for every positive iteger, there exists a -ucoditioal star-body with cov() / c 5 ad F c 5. Proof. Let {e,..., e } be a orthoormal basis of R. We defie v i = e i ad v +i = e i, i =,...,. Sice the radius of D is of order, if is large eough we have that the balls v i + D, i =,..., 2, are disjoit. We set i = cov ({0} (v i + D )) 8

9 ad = 2 i= i. It is clear that is cetrally symmetric ad -symmetric (hece, -ucoditioal), that is ρ ( x e + + x e ) = ρ (x σ() e + + x σ() e ) for all real umbers x,..., x ad every permutatio σ of the idices,...,. It is also clear that C() is -symmetric, ad so, by Petty s characterizatio, C() must have miimal surface area. The, by Lemma 3., We estimate F : we have x 2 dx = + F. i x 2 dx D x + v i 2 dx D ( v i 2 x 2 )dx. Usig the fact that v i 2 x 2 c for all x D, we easily coclude: i x 2 dx c 5. O the other had, where c 5, c 5 are absolute costats. Thus, i v i + D + cov [ {0} ( (D v i ) + v i )] = D + D v i v i c 5, F 2 c 5 /(2c 5 ) +/. It remais to prove that the ratio cov() / is small. Set C = cov({v,..., v 2 }). It is well-kow that C / ad also that C is i M-positio. Sice v + D C + D, it follows that ( cov() / ) / C + D /. Remark. By a classical theorem of F. Joh, the order for F is the maximal possible. Lemma 5.2. Let M be a covex body i R. If M is i miimal surface positio, the the (+)-dimesioal body ) ( ) ( (M) + M = (2) + M (2) + (M) + [ /2, /2] is also i miimal surface area positio ad has the same volume as M. Lemma 4.4 is a simple cosequece of Petty s characterizatio of the miimal surface area positio ad we omit its proof. Proof of Theorem.. Give a positive iteger, we cosider the star-body of Propositio 4.. We may apply Theorem 3.2 to costruct a -ucoditioal body M 2 i R 2, which has volume ad is i miimal surface area positio, ad at the same time, M 2 + D 2 /2 2 c 4 F cc 5 /8. 9

10 We also set M 2+ = M 2, where M 2 is the covex body defied i Lemma 4.4, with M = M 2. The, ) ( ) M 2+ + D 2+ ( (M) 2+ (4) 2+ M + (D 2+ e 2+) (4) 2 2+ (M) 2 2+ [ /2, /2] Sice M 2 has miimal surface area, oe ca easily check that (M) /2+. It easily follows that where c 6 > 0 is a absolute costat. M 2+ + D 2+ /2+ 2+ c 6 ( + ) /8, Let be a covex body which cotais the origi. It is well-kow that there exists a volume preservig liear map T such that the quatity L 2 T := x, y 2 dx +2 is costat as a fuctio of y S. The, T is said to be isotropic ad the umber L T is called isotropic costat of (see [8] for basic results o this cocept). It has bee cojectured that the isotropic costats of all cetrally symmetric covex bodies are uiformly bouded (this would actually imply that the isotropic costat of ay covex body which cotais the origi is uiformly bouded; see [5] for the proof of this result). The class of bodies of elliptic type (the termiology is due to Lutwak) is defied to be the family of all curvature images of covex bodies. As a fial remark, we would like to describe the coectio of the problem we study i this paper with the problem of boudig the isotropic costat. Propositio 5.3. The isotropic costats of all cetrally symmetric covex bodies are uiformly bouded if ad oly if, i the class of cetrally symmetric covex bodies of elliptic type, the miimal surface area positio is also a M-positio. The if part of Propositio 4.2 ca be deduced from Lemma 3., Theorem 3.2 ad the well-kow fact that L F for ay cetrally symmetric covex body (oe may alteratively use Propositio 3.2 ad Propositio.4 from []). The other directio follows from Remark ad Sectio 2.4 from [8]. T. Ackowledgemet. I would like to thak Professor Souzaa Papadopoulou for may helpful discussios cocerig this paper. I also thak the referee for improvig sigificatly the form of this paper. Refereces [] J. Bourgai, B. lartag 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. [2] J. Bourgai ad V. D. Milma, New volume ratio properties for covex symmetric bodies i R d, Ivet. Math. 88 (987), [3] A. Giaopoulos ad V.D. Milma, Asymptotic Covex Geometry: a short overview, Differet faces of geometry (eds S.. Doaldso, Ya. Eliashberg, M. Gromov). Iteratioal Mathematical Series, luwer/pleum Publishers 3 (2004), [4] A. Giaopoulos ad V. D. Milma, Extremal problems ad isotropic positios of covex bodies, Israel J. Math. 7 (2000), [5] B. lartag, A isomorphic versio of the slicig problem, J. Fuct. Aal. 28 (2005), [6] 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),

11 [7] V. D. Milma, Isomorphic symmetrizatio ad geometric iequalities, i Geometric Aspects of Fuctioal Aalysis, Lecture Notes i Mathematics 376 (989), [8] 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), [9] C. M. Petty, Surface area of a covex body uder affie trasformatios, Proc. Amer. Math. Soc. 2 (96), [0] C. A. Rogers ad G. C. Shephard, The differece body of a covex body, Arch. Math. 8 (957), [] R. Scheider, Covex Bodies: The Bru Mikowski theory, Ecyclopedia of Mathematics ad its Applicatios 44, Cambridge Uiversity Press, Cambridge, 993. Ch. Saroglou: Departmet of Mathematics, Uiversity of Crete, Heraklio, Crete, Greece. saroglou@math.uoc.gr