Several properties of new ellipsoids

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1 Appl. Math. Mech. -Egl. Ed (7): DOI /s y c Shaghai Uiversity ad Spriger-Verlag 008 Applied Mathematics ad Mechaics (Eglish Editio) Several properties of ew ellipsoids SHEN Ya-ju ( ) 1 YUAN Ju ( ) (1. Departmet of Mathematics Shaghai Uiversity Shaghai P. R. Chia;. School of Mathematics ad Computer Sciece Najig Normal Uiversity Najig P. R. Chia) (Commuicated by ZHOU Zhe-wei) Abstract We first characterize a polytope whose ew ellipsoid is a ball. Furthermore we prove some properties for the operator Γ ad obtai some iequalities. Key words ew ellipsoid polytopes mootoicity Chiese Library Classificatio O Mathematics Subject Classificatio 5A0 5A40 Itroductio For each covex subset i R it is well-kow that there is a uique ellipsoid with the followig property: The momet of iertia of the ellipsoid ad that of the covex set are the same i every 1-dimesioal subspace of R. This ellipsoid is called the Legedre ellipsoid of the covex set. The Legedre ellipsoid ad its polar (the Biet ellipsoid) are well-kow cocepts from classical mechaics. See [1 3] for historical refereces. It has slowly come to be recogized that alogside the Bru-Mikowski theory is a dual theory. A atural questio is whether there is a dual aalog to the classical Legedre ellipsoid i the Bru-Mikowski theory. Applyig the L p -curvature theory [4 5] Lutwak Yag ad Zhag precisely demostrated the existece of this dual object. Further some beautiful ad deep properties for this dual aalog of the Legedre ellipsoid have bee discovered [6 7]. Let deote the -dimesioal volume of the covex body K i R ad Γ K deote the ew ellipsoid which is defied by Lutwak Yag ad Zhag. The aim of this paper is to study the ew ellipsoid further. We first characterize the polytope whose ew ellipsoid is a ball. Theorem 1 If P is a polytope i R with the origi as its iterior ad whose faces have outer uit ormals u 1 u u N let a i deote the area of the face with outer ormal u i ad h i deote the distace from the origi to this face = N a i h i. The Γ K is a ball if ad oly if (u u i ) a i = h i Received Jul / Revised May Project supported by the Natioal Natural Sciece Foudatio of Chia (Nos ) ad the Sciece ad Techology Research Item of Zhejiag Provicial Departmet of Educatio (No ) Correspodig author SHEN Ya-ju Associate Professor syju168@163.com

2 968 SHEN Ya-ju ad YUAN Ju hold for all u S 1. Geerally Γ K Γ L does ot imply. But if K Z where Z = {Γ K : K K o } the the mootoicity of operator Γ is true. Theorem Let K Z L K o. If Γ K Γ L the 1 Notatio ad prelimiary works. Let K deote the set of a covex body(compact covex subsets with o-empty iteriors) ad K o deote the subset of K that cotais the origi as the iterior poit i Euclidea space R. B deotes the uit ball ad S 1 the uit sphere i R ω deotes volume of B. For K K its support fuctio h K = h(k ) : R (0 ) is defied by h(k u) = max{(u x) : x K} u S 1. (1) Here (u x) deotes the stadard ier product of u ad x h(k u) is sometimes writte as h K (u). If K is a star body that cotais the origi as its iterior poit its radial fuctio ρ(k ) is defied o S 1 by ρ(k u) = max{ 0 : u K} u S 1. () The polar body of K (see [8]) deoted by K is defied by K := {x R (x y) 1 y K}. (3) If K Ko the it follows from the defiitios of support ad radial fuctios ad the defiitio of polar bodies that h K = 1/ρ K ad ρ K = 1/h K. (4) For K L K ad ε > 0 the Firey L -combiatio K + ε L is defied as a covex body whose support fuctio is give by [5 6] h(k + ε L ) = h(k ) + εh(l ). (5) For K L K the L -mixed volume V (K L) of the K ad L defied i [4] is give by V V (K + ε L) (K L) = lim. (6) ε 0 + ε The existece of this limit was demostrated i [4]. It was show that correspodig to each origi-symmetric covex body K there is a positive Borel measure S (K ) o S 1 such that V (K Q) = 1 h Q (v) ds (K v) (7) S 1 for each Q K. It was show i [5] that S (K ) is absolutely cotiuous with respect to the classical surface area measure S K ad that the Rado-Nikodym derivative: ds (K ) ds K = 1 h K. (8)

3 Several properties of ew ellipsoids 969 For each covex body K the ew ellipsoid Γ K is defied by [6] Γ (u) = 1 K (u v) ds (K v) (9) S 1 for u S 1. If P is a polytope with the origi as its iterior poit {u i } N {a i} N ad {h i} N are as i Theorem 1 the the measure S (P ) is cocetrated at the poits u 1...u N S 1 ad S (P u i ) = a i /h i. Thus for the polytope P we have for u S 1 that Γ P (u) = 1 (u u i ) a i h i. (10) We shall require the followig basic iequalities regardig the L mixed volume. The L aalog of the classical Mikowski iequality states that for K L K o V (K L) (11) equality holds if ad oly if K is a dilatate of L. Let K ad L be star bodies the L dual mixed volume Ṽ (K L) of K ad L is defied by [4] Ṽ (K L) = 1 ρ + K S 1 The basic iequality for L dual mixed volumes is (v)ρ (v)dsv. (1) L Ṽ (K L) + (13) with equality if ad oly if K is a dilatate of L. This iequality is a immediate cosequece of the Hölder iequality ad (1). A characterizatio of polytope I this sectio we first give a characterizatio of a polytope. The we establish a iequality for the volume of its ew ellipsoid. Theorem.1 If P is a polytope i R with the origi i its iterior ad {u i } N {a i} N {h i } N ad are as i Theorem 1 the Γ P is a ball if ad oly if ad this holds for all u S 1. Proof If (14) holds the from the defiitio (10) we get (u u i ) a i h i = (14) Γ (u) = P for all u S 1 which meas that Γ P is a ball. Coversely if Γ P is a ball the there exists a real umber R such that Γ P (u) = R (15)

4 970 SHEN Ya-ju ad YUAN Ju for all u S 1. Choosig u = e i for i = 1 i (10) ad summig all iequalities we get R = 1 which together with (15) gives (14). Theorem. l=1 (e i u l ) a l = Uder the hypothesis of Theorem.1 the iequality ( h i a i V (Γ P) ω holds with equality if ad oly if Γ P is a ball. ) (16) Proof Without loss of geerality let e 1 e e be orthogoal uit vectors i -dimesioal space. Let P be a polytope such that the ceter of Γ P is at the origi ad all axes of Γ P are i the directios of e 1 e e (which meas the ellipsoid Γ P is i its stadard positio) the we have By the defiitio of Γ P we have whiceads to V (Γ P) = ω ρ Γ P(e 1 )ρ Γ P(e )...ρ Γ P(e ). (17) 1 Γ P (e i) = 1 Γ P (e i) = 1 Ad further sice u l = (e i u l ) = 1 1 = 1 l=1 (e i u l ) a l l=1 Γ P (e i) = By the arithmetic geometric mea iequality Γ P (e i) l=1 (e i u l ) a l (e i u l ) a l.. (18) 1 ρ Γ (e P 1)ρ Γ (e P )...ρ Γ (e P ) that is ρ Γ P(e 1 )ρ Γ P(e )...ρ Γ P(e ) ( Γ P (e i) )

5 Several properties of ew ellipsoids 971 with equality if ad oly if ρ Γ P(e 1 ) = ρ Γ P(e ) = = ρ Γ P(e ). Therefore Γ P is a ball. By (17) ad (18) we get which gives ( ) ρ Γ P(e 1 )ρ Γ P(e )...ρ Γ P(e ) ( ) V (Γ P) ω. But for a polytope P = 1 h i a i hece with equality if ad oly if Γ P is a ball. 3 Mootoicity of operator Γ ( h i a i ) V (Γ P) ω Let Z deote the class of cetered covex bodies that is the rage of the operator Γ o Ko ; i.e. Z = {Γ K : K K o }. I this sectio we establish the mootoicity of operator Γ. Our mai result is the followig: Theorem 3.1 for all Q Z. ad Let K L K o. If Γ K Γ L the V (K Q) Proof By (7) (10) ad Fubii s theorem we immediately get V (K Γ L) V (L Q) (19) = V (L Γ K). (0) Sice Q Z there exists a M K o such that Q = Γ M. Hece from (0) we have V (K Q) = V (K Γ M) V (L Q) Sice Γ K Γ L the Γ K Γ L which implies that By (7) agai we have = V (M Γ K) (1) V (M) = V (M Γ L). () V (M) h Γ K(u) h Γ L(u) for all u S 1. V (M Γ K) V (M Γ L) which togather with (1) ad () leads to (19).

6 97 SHEN Ya-ju ad YUAN Ju Corollary 3. Let K Z L K o. If Γ K Γ L the. Proof Sice K Z let Q = K i (19); from (11) we obtai 1 V (L Q) which implies that. Theorem 3.3 Let K L K o. If V (K Q) V (L Q) for all Q K o the (i) V (Γ K) V (Γ L) (3) (ii) V (Γ K) V (Γ L). (4) Each of the equalities hold if ad oly if K = L. To prove Theorem 3.3 we eed the followig lemma: Lemma 3.4 [7] Let K L K o. The V (L Γ K) = Ṽ (K Γ L). Proof of Theorem 3.3 (i) Sice V (K Q) V (L Q) for all Q K o takig Q = Γ M for ay covex body M K we have the equality holds if ad oly if K = L. By Lemma 3.4 we have Takig M = Γ L ad usig (13) we obtai V (K Γ M) V (L Γ M) (5) Ṽ (M Γ K) Ṽ (M Γ L). (6) V (Γ K) V (Γ L) (7) with equality if ad oly if Γ K is a dilatate of Γ L. We kow by Lemma 3.4 that iequality (5) ad (6) are equivalet with equality if ad oly if K = L implies that the equality holds i (7). Thus we get the coditio for equality of (3). (ii) Sice V (K Q) V (L Q) here takig Q = Γ M for ay covex body M K o we have ad equality holds if ad oly if K = L. Together with (8) ad (0) we get V (K Γ M) V (L Γ M) (8) V (M Γ K) V (M Γ L).

7 Takig M = Γ L ad usig (11) we obtai that Several properties of ew ellipsoids 973 V (Γ K) V (Γ L) (9) with equality if ad oly if Γ K is a dilatate of Γ L. Sice the coditio that the equalities hold i (8) ad (9) is equivalet to that the equality hold i (4) we kow that the equality i (4) holds if ad oly if K = L. Refereces [1] Leichtweiβ K. Affie geometry of covex bodies[m]. Heidelberg: J A Barth [] Lidestrauss J Milma V D. Local theory of ormal spaces ad covexity[m]. I: Gruber P M Wills J M (eds). Hadbook of Covex Geometry Amsterdam: North-Hollad [3] Milma V D Pajor A. Isotropic positio ad iertia ellipsoids ad zooids of the uit ball of a ormal -dimesioal space[m]. I: Lidestrauss J Milma V D (eds). Geometric Aspect of Fuctioal Aalysis Lecture Note i Math. Berli New York: Spriger (1): [4] Lutwak E. The Bru-Mikowski-Firey theory I: mixed volumes ad the Mikowski problem[j]. J Differetial Geom (9): [5] Lutwak E. The Bru-Mikowski-Firey theory II: affie ad geomiimal surface areas[j]. Adv Math (): [6] Lutwak E Yag D Zhag G Y. A ew ellipsoid associated with covex dodies[j]. Duke Math J (3): [7] Yua J Si L Leg G S. Extremum properties of the ew ellipsoid[j]. Tamkag Joural of Mathematics (): [8] Garder R J. Geometric Tomography[M]. d Editio. Cambridge: Cambridge Uiversity Press 006.