INEQUALITIES FOR MIXED INTERSECTION BODIES

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1 Chi. A. Math. 26B:3(2005), INEQUALITIES FOR MIXED INTERSECTION BODIES YUAN Shufeg LENG Gagsog Abstract I this paper, some properties of mixed itersectio bodies are give, ad iequalities from the dual Bru-Mikowski theory (such as the dual Mikowski iequality, the dual Aleksadrov-Fechel iequalities ad the dual Bru-Mikowski iequalities) are established for mixed itersectio bodies. Keywords Star body, Mixed itersectio body, Dual mixed volume, Spherical rado trasform 2000 MR Subject Classificatio 52A40, 52A20. Itroductio Itersectio body is a basic cocept i the dual Bru-Mikowski theory. The history of itersectio bodies bega with Busema s theorem which has importat implicatios for Busema s theory of area i Fisler spaces (see [3]). Itersectio bodies were first explicitly defied ad amed by Lutwak i the importat paper [4], ad played a key role i the ultimate solutio of Busema-Petty problem (see [4, 7, 8, 0,, 4, 26 28]). It was i [6, 4] that the duality betwee itersectio bodies ad projectio bodies was first made clear. Iterest i projectio bodies was rekidled by three highly ifluetial articles which appeared i the latter half of the 60 s by Bolker [], Petty [9], ad Scheider [22]. Projectio bodies have bee the objects of itese ivestigatio durig the past three decades (see [2, 6, 2, 24]), ad Lutwak established the iequalities for the mixed projectio bodies ad their polar bodies (see [3, 5, 6]). Though the otio of mixed itersectio bodies has also bee raised (see [2, p.25]), their properties have ot bee studied systematically by ow. The correspodig work about the mixed itersectio bodies is doe i this paper. Basic properties of mixed itersectio bodies are give i 2. I 3, the proof of dual Mikowski iequality for mixed itersectio bodies is preseted ad a upper boud estimate about mixed itersectio bodies is give. The dual Aleksadrov-Fechel iequalities ad dual Bru-Mikowski iequalities for mixed itersectio bodies are prove i 4 ad 5. At the same time the equality coditios of these iequalities are obtaied. Mauscript received December 4, Revised Jauary 4, Departmet of Mathematics, Shagyu College, Shaoxig Uiversity, Shagyu 32300, Zhejiag, Chia; Shaghai Istitute of Applied Mathematics ad Mechaics, Shaghai Uiversity, Shaghai , Chia. yuashufeg2003@63.com Departmet of Mathematics, Shaghai Uiversity, Shaghai , Chia. gleg@mail.shu.edu.c Project supported by the Natioal Natural Sciece Foudatio of Chia (No.02707).

2 424 YUAN, S. F. & LENG,G.S. As usual, S deotes the uit sphere, B the uit ball, ad o the origi i Euclidea -space E.Ifuis a uit vector, that is, a elemet of S,wedeotebyu the ()- dimesioal subspace orthogoal to u ad by l u the lie through the origi parallel to u. We write V k for k-dimesioal Lebesgue measure i E,where0 k, ad where we idetify V k with k-dimesioal Hausdorff measure (V 0 is the coutig measure). We also geerally write V istead of V. κ i deotes the volume of i-dimesioal uit ball, where i. AsetLis star-shaped at o if L l u is either empty or a (possibly degeerate) closed lie segmet for each u S.IfLis star-shaped at o, we defie its radial fuctio ρ L by max{c : cu L}, L l u, ρ L (u) = 0, otherwise. A body is a compact set equal to the closure of its iterior. By a star body we mea a body L star-shaped at o such that ρ L, restricted to its support, is cotiuous. We deote the class of star bodies i E by L, ad the subclass of star bodies cotaiig the origi i their iteriors by L o. If x i E, i m, thex + +x m is defied to be the usual vector sum of the poits x i, if all of them are cotaied i a lie through o, ad 0 otherwise. Let L i be a star body i E with o L i,adt i 0, i m. The t L + +t m L m = {t x + +t m x m : x i L i, i m} is called a radial liear combiatio. Moreover, for each u S, ρ tl e +t2l 2 (u) =t ρ L (u)+t 2 ρ L2 (u). (.) The dual mixed volume Ṽ (L,,L )ofstarbodiesl,,l cotaiig the origi i E is defied by Ṽ (L,,L ) ρ L (u) ρ L (u)du, (.2) S where du sigifies itegratio o S with respect to V,whichiS is idetified with spherical Lebesgue measure. Whe L = L 2 = = L i = K, L i+ = = L = K 2 i (.2), we write Ṽi(K,K 2 )forṽ(k, i; K 2,i). For 0 i, the dual volume Ṽi(L) ad dual quermassitegral W i (L) are defied by Ṽ i (L) = W i (L) =Ṽ (L, i; B, i) ρ L (u) i du. (.3) S I particular, Ṽ(L) =V (L), i.e., the polar coordiate formula for volume of star body L with o L is V (L) ρ L (u) du. (.4) S

3 INEQUALITIES FOR MIXED INTERSECTION BODIES 425 Let L j be a star body i E with o L j, j, ad suppose that i. Lutwak proved the dual Aleksadrov-Fechel iequality (see [6, Sectio B.4]) Ṽ (L,,L ) i i Ṽ (L j,i; L i+,,l ), (.5) j= i which the equality holds if ad oly if L,,L i are dilatates of each other. The iequality has the same form as the classical Aleksadrov-Fechel iequality. Two special cases of (.5) are worthy of ote. If L i L,witho L i,ad i, the ad Ṽ (L,,L ) V (L )V (L 2 ) V (L ), (.6) Ṽ i (L,L 2 ) V (L ) i V (L 2 ) i, (.7) i which the equalities hold if ad oly if the bodies are dilatates of each other. If L L, 2, with ρ L C(S ), the the itersectio body I(L) ofl is a star body defied by ρ IL (u) =V (L u ) ρ L (v) dv. (.8) S u Suppose that f is a bouded Borel fuctio o S. The spherical Rado trasform Rf of f is defied by (Rf)(u) = f(v)dv S u for all u S. The trasform R is self-adjoit (see, for example, [6, Theorem C.2.6]), that is, f(u)(rg)(u)du = S (Rf)(u)g(u)du, S (.9) for bouded Borel fuctios f ad g o S. Let φ SO,whereSO is the special orthogoal group of rotatios about the origi. The (see [25, p.635]) φ(rf)(u) =(Rf)(φ u)= f(v)dv. S (φ u) (.0) Suppose L L o ad u S.Thei-chord fuctio ρ i,l of L at o is defied by ρ L (u) i + ρ L ( u) i, i 0, ρ i,l (u) = ρ L (u)ρ L ( u), i =0, (.) ad the i-chordal symmetral i L of L is a cetered set defied by ρ i, e il (u) =ρ i,l(u). (.2)

4 426 YUAN, S. F. & LENG,G.S. 2. Properties of Mixed Itersectio Bodies Defiitio 2.. (see [2, 4]) Let K,,K L, 2. The mixed itersectio body I(K,,K ) of K,,K is defied by ρ I(K,,K )(u) =ṽ(k u,,k u ), (2.) where u S, ṽ is ()-dimesioal dual mixed volume. The defiitio implies that I(K,,K ) is a cetered star body. Takig K = = K = K, we otice that the diagoal form of the mixed itersectio body reduces to the itersectio body, i.e., I(K,,K)=I(K). Specially, I(B,,B)=κ B. By (.2), we ca rewrite (2.) as the equivalet itegral form, restricted to the star bodies cotaiig the origi i their iteriors, ivolvig the spherical Rado trasform R: ρ I(K,,K )(u) ρ K (v) ρ K (v)dv (2.2) S u ( ) = R ρ K ρ K (u). Now, we develop some basic properties of the mixed itersectio operator I : L o L o }{{} L. L o. Of course, most of the followig results remai valid for I : L L }{{} Propositio 2.. The mixed itersectio operator is positively homogeeous, i.e., if K,,K L o,adα,,α 0, the I(α K,,α K )=α α I(K,,K ). Propositio 2.2. The mixed itersectio operator is multiliear with respect to the radial liear combiatios, i.e., if K,K,,K L o,ad α, β 0, the I(αK +βk,k 2,,K )=αi(k,k 2,,K ) +βi(k,k 2,,K ). Propositio 2.3. The mixed itersectio operator is mootoe odecreasig with respect to set iclusio, i.e., if K,K,K 2,,K L o, K K,the I(K, K 2,,K ) I(K,K 2,,K ). These properties of the mixed itersectio operator follow immediately from (2.2) ad (.). The itersectio bodies of liearly equivalet star bodies are liearly equivalet (see [7]), i.e., if φ GL,theI(φL) = detφ φ t (IL), where φ t is the traspose of the iverse of φ. For the special orthogoal group SO (with determiat oe) ad the mixed itersectio operator, we have Propositio 2.4. If K,,K L o,φ SO,the I(φK,,φK )=φ t I(K,,K ).

5 INEQUALITIES FOR MIXED INTERSECTION BODIES 427 Proof. Let u S. By the property of liear trasformatio: x φy = φ t x y, we kow that x u =0ifadolyifφ x φ t u = 0. Therefore, if ω is the uit vector i the directio of φ t u,theω = φ u. If E is a V -measurable subset of ()-dimesioal subspace u i E,witho E, ad ψ is a liear trasformatio i E, the we have (see [6, p.274]) V (ψe) = S u ρ ψe (u)du = ψ t u detψ V (E). It is easy to see that, if L,,L are V -measurable star bodies cotaied i u, with o L,,L, the ṽ(ψl,,ψl ) ρ ψl (v) ρ ψl (v)dv S u = ψ t u detψ ṽ(l,,l ). Hece ρ I(φK,,φK )(u) =ṽ(φk u,,φk u ) =ṽ(φ(k φ u ),,φ(k φ u )) = φ t u ṽ(k φ u,,k φ u ) φ t u ρ K (v) ρ K (v)dv S (φ u ) φ t u S ( φ t u ) ρ K (v) ρ K (v)dv φ t u φ t u ρ K (v) ρ K (v)dv. S (( φ t u φ t ) u) From the property (.0) of spherical Rado trasform, we have ρ I(φK,,φK )(u) = φ t u φ t u φ t( ) ρ K (v) ρ K (v)dv s u = φ t (ρ I(K,,K )(u)). The proof is completed. Propositio 2.5. Let M =(K,,K i ), i, adwritei i (M,K) for I(M,K,,K). IfK, L, K }{{},,K i L o,the i I i (M,K L) +I i (M,K L) =I i (M,K) +I i (M,L). Proof. Sice ρ K L (u) =max{ρ K (u), ρ L (u)}, ρ K L (u)=mi{ρ K (u),ρ L (u)}, ad together with (.), we get ρ Ii(M,K L)e +Ii(M,K L) (u) =ρ I i(m,k)e +Ii(M,L) (u), u S.

6 428 YUAN, S. F. & LENG,G.S. Thus, it is easy to obtai Propositio 2.5. Take K = = K i = K, K i = = K = L i I(K,,K ), ad I(K,,K,L,,L) will be writte as I }{{}}{{} i (K, L). If L = B, thei i (K, B) is called the ith i i itersectio body of K ad is usually writte as I i K. Specially, IK=I 0 K. Propositio 2.6. If K is a cetered star body, ad L L o, 0 <i, the I i (K, i L)=I i (K, L). Proof. Let u S. From (.) ad (.2), it follows that ρ e il (u) = ( ρl (u) i + ρ L ( u) i 2 Sice K is cetered, ρ K ( u) i = ρ K (u) i.thus ρ Ii(K, il) e (u) ρ K (v) i ρ e il (v)i dv s u ( ) = ρ K (v) i ρ L (v) i dv + ρ K (v) i ρ L ( v) i dv 2() s u s u ( ) = ρ K (v) i ρ L (v) i dv + ρ K ( v) i ρ L ( v) i dv 2() s u s u ρ K (v) i ρ L (v) i dv s u = ρ Ii(K,L)(u). The proof is completed. K Propositio 2.7. If K,,K,K,,K L o, I(K,,K )=I(K,, ),adm is a cetered star body, the ) i. Ṽ (K,,K,M)=Ṽ (K,,K,M). Proof. Sice M is a cetered star body, ρ M C e (S ). Thus, there is f C e (S ), such that ρ M = Rf (see [4, 23]). Sice for each u S, ρ I(K,,K )(u) =ρ I(K,,K ) (u), we have R(ρ K ρ K )=R(ρ K ρ K ). The, by (.2), (.9) ad the above equalities, it follows that Ṽ (K,,K,M) ρ K (u) ρ K (u)ρ M (u)du S ρ K (u) ρ K (u)rf(u)du S R(ρ K ρ K )(u)f(u)du S R(ρ K ρ K )(u)f(u)du S

7 INEQUALITIES FOR MIXED INTERSECTION BODIES 429 ρ K (u) ρ K (u)ρ M (u)du S = Ṽ (K,,K,M). 3. Dual Mikowski Iequality for Mixed Itersectio Bodies I this sectio, the followig dual Mikowski iequalities for mixed itersectio bodies will be established. Theorem 3.. If K, K L o,ad0 i, the V (I i (K, K )) V (IK) i V (IK ) i, (3.) ad the equality holds if ad oly if K ad K are dilatates. Proof. Suppose u S. Let L,L 2 be star bodies cotaied i ()-dimesioal subspace u, o L,L 2, ad write ṽ i (L,L 2 )forṽ(l, i ; L 2,i). From (.7), we have ṽ i (L,L 2 ) V (L ) i V (L 2 ) i. Now usig the polar coordiate formula for volume (.4), the above iequality ad Hölder itegral iequality, we have V (I i (K, K )) ρ Ii(K,K )(u) du ṽ i (K u,k u ) du S S V (K u ) ( i ) /() V (K u ) i /() du S ( ) ( i )/() ( ) i/() V (K u ) du V (K u ) du S S ( ( i )/()( ) i/() = ρ IK (u) du) ρ IK (u) du S S = V (IK) ( i )/() V (IK ) i/(). From the equality coditios of the iequality (.7) ad Hölder itegral iequality, this implies that K u ad K u must be dilatates. It is well kow that if K u ad K u are dilatates for all u S,theK ad K are dilatates (see [20]). Hece, we get that Theorem 3. holds with equality if ad oly if K ad K are dilatates. Corollary 3.. If K is a covex body whose cetroid is at the origi, K is the polar body of K, ad 0 i, the V (I i (K,K)) V (K) 2i κ κ 2i, ad the equality holds if ad oly if K is a cetered ellipsoid. Proof. Takig K = K, K = K i Theorem 3., we have V (I i (K,K)) V (I(K )) i V (IK) i.

8 430 YUAN, S. F. & LENG,G.S. Notice the followig two well-kow iequalities: Busema itersectio iequality (see [6, p.333]) states that if K L o,the V (IK) (κ /κ 2 )V (K), ad the equality holds if ad oly if K is a cetered ellipsoid; the Exteded Blaschke-Sataló iequality (see [8]) states that if K is a covex body whose cetroid is at the origi, the V (K )V (K) κ 2, ad the equality holds if ad oly if K is a cetered ellipsoid. Hece, we get V (I i (K,K)) (κ () (κ () = κ () /κ ()( 2) /κ ()( 2) )V (K ) ()( i ) V (K) ()i ) (κ 2()( i ) /V (K) ()( i ) )V (K) ()i κ ()( 2i) V (K) ()(2i +). It is easy to obtai the equality coditio of Corollary Dual Aleksadrov-Fechel Iequalities for Mixed Itersectio Bodies The dual Aleksadrov-Fechel iequality, for mixed itersectio bodies, which will be prove is: If K,,K L o, i, the V (I(K,,K )) i i V (I(K j,i; K i+,,k )), (4.) j= ad the equality holds if ad oly if K,,K i are dilatates. This is the special case i =0ofTheorem4.. Theorem 4.. If K,,K L o, 0 i, 0 <m, the W i (I(K,,K )) m m W i (I(K j,m; K m+,,k )), j= ad the equality holds if ad oly if K,,K m are dilatates. Proof. By (.3) ad (2.2), we have W i (I(K,,K )) ρ I(K,,K )(u) i du S ( ) idu. ρ K (v) ρ K (v)dv S S u

9 INEQUALITIES FOR MIXED INTERSECTION BODIES 43 A extesio of Hölder iequality states that (see [9, p.40]) m m ( /m, f i (u)du f i (u) du) m Ω i= i= ad the equality holds if ad oly if f i are proportioal. Applyig the extesio of Hölder iequality, we have W i (I(K,,K )) m ( ( m ( )) ( i)/mdu ) m ρ Kj (v) m ρ Km+ (v) ρ K (v)dv S j= S u ( ( m ( i)/mdu ) m. = ρ I(Kj,m;K m+,,k )(u)) S j= Applyig the extesio of Hölder iequality agai, we obtai m ( ) W i (I(K,,K )) m ρ I(Kj,m;K m+,,k )(u) i du j= S m = W i (I(K j,m; K m+,,k )). j= From the equality coditio of the first Hölder iequality, this implies that ρ Kj (v) m ρ Km+ (v) ρ K (v) (j =,,m) must be proportioal. Sice K i L o (i =,,), K,,K m are dilatates. It follows that there exist a i 0(i = 2,,m) such that K = a i K i.the ρ I(K,m;K m+,,k )(u) =ṽ(k u,m; K m+ u,,k u ) Ω =ṽ(a i (K i u ),m; K m+ u,,k u ) = a m i ṽ((k i u ),m; K m+ u,,k u ) = a m i ρ I(K i,m;k m+,,k )(u). That K = a i K i (i =2,,m) implies that ρ I(Kj,m;K m+,,k ) (j =,,m)are proportioal, which is also the equality coditio of the secod Hölder iequality. So the equality of Theorem 4. holds if ad oly if K,,K m are dilatates. Takig i = i (4.), we have Corollary 4.. If K,,K L o,the V (I(K,,K )) V (I(K )) V (I(K )), ad the equality holds if ad oly if K,,K are dilatates. Takig K = = K 2 j = K, K j = L, K j = = K = B i (4.), where j 2, we obtai

10 432 YUAN, S. F. & LENG,G.S. Corollary 4.2. If K, L L o, j< 2, the V (I(K,,K,L,B,,B)) j V (I }{{}}{{} j K) 2 j V (I j L), 2 j j ad the equality holds if ad oly if K ad L are dilatates. Remark 4.. Take K = = K i = K, K i = = K = K, i = i (4.), ad Theorem 3. follows immediately. 5. Dual Bru-Mikowski Iequalities for Mixed Itersectio Bodies First, i this sectio, we will cosider the dual Bru-Mikowski iequality for the mixed itersectio bodies ad the radial Blaschke liear combiatio. Let K ad K be star bodies i E with o K, K, α, α 2 0, ad defie the radial Blaschke liear combiatio α K +α 2 K by ρ α K b +α2 K (u) = α ρ K (u) + α 2 ρ K (u). (5.) Theorem 5.. If K, K, L L o, α, α 2 0, the V (I (α K +α 2 K,L)) / α V (I (K, L)) / + α 2 V (I (K,L)) /, ad the equality holds if ad oly if I (K, L),I (K,L) are dilatates. Proof. Let u S.SiceK u,k u are star bodies cotaiig the origi i u, by (5.), we have ρ α (K u )b +α2 (K u ) (v) 2 = α ρ K u (v) 2 + α 2 ρ K u (v) 2. The, from (2.2) ad the above equality, we get ρ I(α K +α2 K b,l) (u) ρ (α K +α2 K b ) u (v) 2 ρ L u (v)dv S S ρ (α (K u )b +α2 (K u )) (v) 2 ρ L u (v)dv S S = α ρ K u (v) 2 ρ L u (v)dv + α 2 ρ S S K u (v) 2 ρ L u (v)dv S S = α ρ K (v) 2 ρ L (v)dv + α 2 ρ K (v) 2 ρ L (v)dv S u S u = α ρ I(K,L)(u)+α 2 ρ I(K,L)(u). Usig (.4), the above result ad Mikowski itegral iequalities, we have V (I (α K +α 2 K,L)) (α ρ I(K,L)(u)+α 2 ρ I(K,L)(u)) du S

11 INEQUALITIES FOR MIXED INTERSECTION BODIES 433 (( / ( ) / ) α ρ I (K,L)(u) du) + α 2 S ρ I (K,L)(u) du S =(α V (I (K, L)) / + α 2 V (I (K,L)) / ). From the equality coditio of Mikowski itegral iequality, it shows that ρ I(K,L)(u) = aρ I(K,L)(u), where a 0. Thus, Theorem 5. holds with equality if ad oly if I (K, L), I (K,L) are dilatates. Secod, the dual Bru-Mikowski iequalities for the radial liear combiatio ad itersectio bodies, which will be established is: If K, L L o,the V (I(K +L)) /(()) V (IK) /(()) + V (IL) /(()), ad the equality holds if ad oly if K ad L are dilatates. This is the special case j =0,i=0ofTheorem5.2. Theorem 5.2. If 0 i<,0 j<, K, L, M,,M i, M,,M j L o, C =(M,,M i ), ad D =(M,,M j ), the Ṽ i (I j (K +L, D),C) /(( i)( j )) Ṽi(I j (K, D),C) /(( i)( j )) + Ṽi(I j (L, D),C) /(( i)( j )), ad the equality holds if ad oly if K, L are dilatates. I order to prove Theorem 5.2, we eedthefollowigtwolemmas. Lemma 5.. If K,,K,L,,L L o,the Ṽ (K,,K,I(L,,L )) = Ṽ (L,,L,I(K,,K )). Proof. Let u S.The Ṽ (K,,K,I(L,,L )) ρ K (u) ρ K (u)ρ I(L,,L )(u)du S ρ K (u) ρ K (u) ρ L (v) ρ L (v)dvdu. S S u Let f = ρ K ρ K, g = ρ L ρ L. The Ṽ (K,,K,I(L,,L )) f(u)rg(u)du. S I the same way, Ṽ (L,,L,I(K,,K )) g(u)rf(u)du. S From the property (.9) of the spherical Rado trasformatio, we obtai Lemma 5.. Lemma 5.2. If 0 i<, K, L, K,,K i L o,adc =(K,,K i ), the Ṽ i (K +L, C) /( i) Ṽi(K, C) /( i) + Ṽi(L, C) /( i), (5.2)

12 434 YUAN, S. F. & LENG,G.S. ad the equality holds if ad oly if K ad L are dilatates. Proof. Let ρ C = ρ K ρ Ki. By (.3), (.) ad Mikowski itegral iequality, we obtai Ṽ i (K +L, C) /( i) ( ) /( i) = ρ K+L e (u) i ρ C (u)du S ( ) /( i) = (ρ K (u)ρ C (u) /( i) + ρ L (u)ρ C (u) /( i) ) i du S ( /( i) ( ρ K (u) i ρ C (u)du) + ρ L (u) i ρ C (u)du S S = Ṽi(K, C) /( i) + Ṽi(L, C) /( i). ) /( i) From the equality coditio of Mikowski itegral iequality, we get that (5.2) holds with equality if ad oly if ρ K (u)ρ C (u) /( i) = aρ L (u)ρ C (u) /( i),wherea 0. This implies that ρ K (u) =ρ al (u), i.e., K = al. The proof is completed. Proof of Theorem 5.2. If j = 2, the from Propositio 2.2, it follows that I 2 (K +L, D) =I 2 (K, D) +I 2 (L, D). Hece, for j = 2, the iequality of Theorem 5.2 reduces to the oe of Lemma 5.2. For i =, usig Lemma 5. ad Lemma 5.2, we obtai Ṽ (I j (K +L, D),C) /( j ) = Ṽ (K +L,,K +L,D,,D,I(C,,C)) /( j ) }{{}}{{} j j Ṽ (K,,K,D,,D,I(C,,C)) /( j ) }{{}}{{} j j + Ṽ (L,,L,D,,D,I(C,,C)) /( j ) }{{}}{{} j j = Ṽ(I j (K, D),C) /( j ) + Ṽ(I j (L, D),C) /( j ). Thus, oly the cases where j< 2, ad i< eed to be treated. Suppose Q L o. Usig Lemma 5. ad Lemma 5.2, we get Ṽ (Q, i ; C; I j (K +L, D)) /( j ) = Ṽ (K +L, j ; D; I i (Q, C)) /( j ) (5.3) Ṽ (K, j ; D; I i(q, C)) /( j ) + Ṽ (L, j ; D; I i(q, C)) /( j ). (5.4)

13 INEQUALITIES FOR MIXED INTERSECTION BODIES 435 Usig Lemma 5. agai ad the dual Aleksadrov-Fechel iequality, we have Ṽ (K, j ; D; I i (Q, C)) /( j ) = Ṽ (Q, i ; C; I j(k, D)) /( j ) = Ṽ (Q, i ; I j(k, D); C) /( j ) Ṽi(Q, C) ( i)/(( j )( i)) Ṽ i (I j (K, D),C) /(( j )( i)). (5.5) I exactly the same way, it ca be see that Ṽ (L, j ; D; I i (Q, C)) /( j ) Ṽi(Q, C) ( i)/(( j )( i)) Ṽ i (I j (L, D),C) /(( j )( i)). (5.6) Combiig (5.3) (5.6), we obtai Ṽ (Q, i ; C; I j (K +L, D)) /( j ) Ṽ i (Q, C) ( i)/(( j )( i)) Ṽi(I j (K, D),C) /(( j )( i)) + Ṽi(I j (L, D),C) /(( j )( i)). Take Q=I j (K +L, D), ad the iequality of Theorem 5.2 is obtaied. Cosider the equality coditio of Theorem 5.2. From the equality coditio of (5.4), it follows that K ad L are dilatates. There exists a 0, such that K = al. Let ρ D (u) =ρ M (u) ρ M j (u). Whe K = al, ρ Ij(K +L,D) e (u) ρ K+L e (v) j ρ D (v)dv S u ρ al+l e (v) j ρ D (v)dv S u ρ (a+)l (v) j ρ D (v)dv S u =(a +) j ρ L (v) j ρ D (v)dv S u =(a +) j ρ Ij(L,D)(u). I the same way, whe K = al, ρ Ij(K e +L,D) (u) =(+ a ) j ρ Ij (K,D)(u). Sice (5.5) holds with equality if ad oly if I j (K +L, D) adi j (K, D) are dilatates, ad (5.6) holds with equality if ad oly if I j (K +L, D) adi j (L, D) are dilatates, this shows that Theorem 5.2 holds. Corollary 5.. If K, L, M, M,,M 2 L o, D =(M,,M 2 ),the Ṽ (M,; I(D, K +L)) V (M) ()/ (V (I(D, K)) / + V (I(D, L)) / ), ad the equality holds if ad oly if M,I(D, L),I(D, L) are dilatates. Proof. Takig i =, j= 2, M = = M i = M i Theorem 5.2, we get Ṽ (M,; I(D, K +L)) Ṽ (M,; I(D, K)) + Ṽ (M,; I(D, L)). Accordig to (.7), Corollary 5. follows immediately.

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