On the Dual Orlicz Mixed Volumes

Transcription

1 Chi. A. Math. 36B6, 2015, DOI: /s x Chiese Aals of Mathematics, Series B c The Editorial Office of CAM ad Spriger-Verlag Berli Heidelberg 2015 O the Dual Orlicz Mixed Volumes Haili JIN 1 Shufeg YUAN 2 Gagsog LENG 3 Abstract I this paper, the authors defie a harmoic Orlicz combiatio ad a dual Orlicz mixed volume of star bodies, ad the establish the dual Orlicz-Mikowski mixedvolume iequality ad the dual Orlicz-Bru-Mikowksi iequality. Keywords Covex body, Harmoic Orlicz combiatio, Dual Orlicz mixed volume, Dual Orlicz-Bru-Mikowski iequality 2000 MR Subject Classificatio 52A20, 52A39 1 Itroductio The dual mixed volumes, as a core cocept i the dual Bru-Mikowski theory, were firstly itroduced by Lutwak [1], ad played a importat role i covex geometry. They are closely related to such importat bodies as: Itersectio bodies see [2], cetroid bodies see [3], ad projectio bodies see [4]. I [5], Garder gave some stability results of these iequalities about dual mixed volumes. I [6], Klai preseted a classificatio theorem for homogeeous valuatios o star-sharped sets by dual mixed volumes. Quite recetly, Garder, Hug ad Weil [7] costructed a geeral framework for the Orlicz- Bru-Mikowski theory, which was itroduced by Lutwak, Yag ad Zhag see [8 11], ad they made clear for the first time its relatio to Orlicz spaces ad orms. I [7], Garder, Hug ad Weil gave a reasoable defiitio of Orlicz additio, the obtaied the Orlicz-Bru- Mikowksi iequality, ad i the ed gave the Orlicz mixed volume of covex bodies which cotai the origi i their iteriors ad get the Orlicz mixed volume iequality. I [12], Xi, Ji ad Leg also obtaied the Orlicz-Bru-Mikowski iequality by Steier symmetry ad the Orlicz Mikowksi mixed volume iequality. R deotes the usual -dimesioal Euclidea space. A set A R is said to be starshaped, if 0 A, ad for each lie l passig through the origi i R,thesetA l is a closed iterval. Deote by S0 the set of all star bodies i R, i.e., the set of all star-shaped sets with a positive ad cotiuous radial fuctio. Mauscript received March 27, Revised July 4, Departmet of Mathematics, Suzhou Uiversity of Sciece ad Techology, Suzhou , Jiagsu, Chia. ahjhl@sohu.com 2 Departmet of Mathematics, Shagyu College, Shaoxig Uiversity, Shaoxig , Zhejiag, Chia. yuashufeg2003@163.com 3 Departmet of Mathematics, Shaghai Uiversity, Shaghai , Chia. gleg@staff.shu.edu.c This work was supported by the Natioal Natural Sciece Foudatio of Chia Nos ,

2 1020 H. L. Ji, S. F. Yua ad G. S. Leg I this paper, we defie the harmoic Orlicz sum K + ϕ L of star bodies K ad L i S0 implicitly by ρk + ϕ L, x ϕ, ρk + ϕ L, x =1 1.1 ρk, x ρl, x for x R.Hereϕ Φ 2, ad we have the set of covex fuctios ϕ :[0, 2 [0, thatare strictly icreasig i each variable ad satisfy ϕ0, 0 = 0 ad ϕ1, 0 = ϕ0, 1 = 1. I Sectio 2, we itroduce a ew otio of the Orlicz harmoic combiatio + ϕ K, L, α, β by meas of a appropriate modificatio of 1.1. The particular istace of iterest correspods to usig 1.1 with ϕx 1,x 2 =ϕ 1 x 1 +ɛϕ 2 x 2 forɛ>0adϕ 1,ϕ 2 Φ, the set of strictly icreasig covex fuctios ϕ: [0, [0, thatsatisfyϕ0 = 0 ad ϕ1 = 1, i which case we write K + ϕ,ε L istead of K + ϕ L, ad we obtai the followig equatio. Theorem 1.1 Suppose ϕ Φ 2. For all K, L S 0, we have ϕ 1 l 1 V K + ϕ,ɛ L V K = 1 ϕ 2 S 1 where ɛ 0 meas that ɛ is decreasig ad teds to 0. ρk u ρ K u dsu, 1.2 ρ L u The itegral o the right-had side of 1.2 with ϕ 2 replaced by ϕ, a ew dual Orlicz mixed volume V ϕ K, L is itroduced, ad we see that either side of the equatio 1.2 is equal to V ϕ2 K, L ad establish the followig dual Orlicz-Mikowski iequality ad the harmoic Orlicz additio versio of the Bru-Mikowski iequality. Theorem 1.2 Suppose ϕ Φ, K, L S 0,adthe 1 V K Ṽ ϕ K, L V Kϕ V L 1 with equality if ad oly if K ad L are dilates. Theorem 1.3 Suppose ϕ Φ 2 such that ϕx 1,x 2 =ϕ 1 x 1 +ϕ 2 x 2 ad ϕ i Φ, i = 1, 2, x i R. IfK, L S 0,the 1 ϕ 1 V K + ϕ L 1 V K 1 with equality if ad oly if K ad L are dilates. V K + ϕ L 1 + ϕ 2 V L 1 2 Preiaries We shall deote the 1-dimesioal uit sphere i R by S 1. A star body K is determied uiquely by its radial fuctio ρ K = ρk, :S 1 R, defied for u S 1 by ρk, u =max{λ 0:λu K}. Suppose that K, L S 0, ad the radial Hausdorff metric δk, L is defied by δk, L = max ρ K u ρ L u. u S 1

3 O the Dual Orlicz Mixed Volumes 1021 If K S0, the the polar coordiate formula for volume V K is V K = 1 ρk, u dsu, 2.1 S 1 where dsu is the spherical Lebesgue measure of S 1. Throughout the paper, Φ m, m N, ad deote the set of covex fuctios ϕ :[0, m [0, that are strictly icreasig i each variable ad satisfy ϕ0 = 0 ad ϕe j =1> 0, j= 1,,m. Whe m = 1, we shall write Φ istead of Φ 1. Let m 2adK j S0. The harmoic Orlicz sum of K 1,,K m, deoted by + ϕ K 1,,K m is defied by ρ + ϕ K 1,,K x =sup m {λ>0:ϕ λ ρ K1 x,, λ } 1 ρ Km x for all x R. Equivaletly, the harmoic Orlicz sum + ϕ K 1,,K m ca be defied implicitly ad uiquely by ρ + ϕ ϕ K 1,,K x m,, ρ + ϕ K 1,,K x m = ρ K1 x ρ Km x A importat special case is obtaied whe ϕx 1,,x m = m ϕ j x j for some fixed ϕ j Φ, j =1,,m such that ϕ 1 1 = = ϕ m 1 = 1. We the write + ϕ K 1,,K m =K 1 + ϕ + ϕ K m. This meas that K 1 + ϕ + ϕ K m is defied either by { ρ + ϕ K 1,,K =sup λ λ m λ>0:ϕ ϕ m ρ K1 x ρ Km x j=1 } 1 for all x R, or by the correspodig special case of 2.3. Let m =2adϕx 1,x 2 =x p 1 + xp 2, p 1, ad we get the harmoic L p sum K + p L see [13]. Suppose that α j 0adϕ j Φ, j =1,,m. If K j S0, j =1,,m, we defie the Orlicz liear combiatio + ϕ K 1,,K m,α 1,,α m by m ρ + ϕ K 1,,K m,α 1,,α x =sup λ m {λ>0: α j ϕ j ρ Kj x j=1 } 1 for all x R. Ulike the L p case, it is ot geerally possible to isolate a harmoic Orlicz scalar multiplicatio, sice there is a depedece ot just o oe coefficiet α j, but o all K 1,,K m ad α 1,,α m. For our purposes, it suffices to focus o the case m = 2. The harmoic Orlicz combiatio + ϕ K, L, α, β fork, L S0 ad α, β 0 ot both zero, is defied equivaletly via the implicit equatio ρ + ϕ K, L, α, β,x αϕ 1 ρk, x ρ + ϕ K, L, α, β,x + βϕ 2 =1 2.6 ρl, x

4 1022 H. L. Ji, S. F. Yua ad G. S. Leg for all x R. It is easy to verify that whe ϕ 1 t =ϕ 2 t =t p, p 1, we get that the harmoic Orlicz liear combiatio + ϕ K, L, α, β equals the harmoic L p combiatio α K + p β L see [13]. I [14], He ad Leg got a strog law of large umbers o the harmoic L p combiatio. Heceforth we shall write K + ϕ,ɛ L istead of + ϕ K, L, 1,ɛ, ad assume throughout that this is defied by 2.6, where α =1,β= ɛ, ad ϕ j Φ, j=1, 2. The left derivative of a real-valued fuctio f is deoted by f l. Suppose that μ is a probability measure o a space X ad g : X I R is a μ-itegrable fuctio, where I is a possibly ifiite iterval. Jese s iequality states that if ϕ : I R is a covex fuctio, the ϕgxdμx ϕ gxdμx. X X If ϕ is a strictly covex, the equality holds if ad oly if gx is costat for μ-almost all x X. 3 Dual Orlicz Mixed Volume We firstly give some properties of the harmoic Orlicz additio. Lemma 3.1 Let ϕ Φ 2. If K, K i,l,l i S0, the the harmoic Orlicz additio + ϕ : S0 2 S0 has the followig properties: 1 Cotiuity K i + ϕ L i K + ϕ L, i.e., ρk i + ϕ L i,x=ρk + ϕ L, x for all x R, i as K i K ad L i L i the radial Hausdorff metric. 2 Mootoicity K 1 + ϕ L 1 K 2 + ϕ L 2 as K 1 K 2 ad L 1 L 2. 3 GL Covariace AK + ϕ L=AK + ϕ AL, A GL. Proof 1 Sice ϕ ρk i + ϕ L i,x ρk i,x, ρki + ϕ L i,x ρl i,x = 1, by the cotiuity of ϕ, wehave ρk i + ϕ L i,x ρk i + ϕ L i,x ϕ i i, =1. ρk, x ρl, x Hece, ρk i + ϕ L i,x=ρk + ϕ L, x. i 2 By the mootoicity of ϕ, 2 is easy to get. 3 Sice ϕ ρak + ϕ AL,x ρak,x Set A 1 x = y. The So A 1 AK + ϕ AL =K + ϕ L., ρak + ϕ AL,x ρal,x =1,wehave ρa 1 AK + ϕ AL,A 1 x ϕ ρk, A 1, ρa 1 AK + ϕ AL,A 1 x x ρl, A 1 =1. x ρa 1 AK + ϕ AL,y ϕ, ρa 1 AK + ϕ AL,y =1. ρk, y ρl, y

5 O the Dual Orlicz Mixed Volumes 1023 Proof of Theorem 1.1 Set K ɛ = K + ϕ,ɛ L. It is easy to see that ρ Kɛ u ρ K u foreach u S 1.The ρ Kɛ u ρ K u = ρ K u 1 ρ K ɛ u ρ K u = ɛ 0 ρ K u From the defiitio of K + ϕ,ɛ L,wehave Let z = ϕ 1 1 ρ Kɛ u ρ K u 1 ɛ = ϕ 2 ρkɛ u ρ L u ρ Kɛ u ρ K u 1 ρkɛ u ρ Ku ϕ 1 1 ɛ = ϕ 1 1. ρkɛ u 1 ɛϕ2 ρ Lu. The ρkɛ u 1 ɛϕ 2 ρ L u ρkɛ u 1 ɛϕ 2 ρ L u 1 1. ρkɛ u 1 ɛϕ2 ρ Lu ad ote that z 1 as ɛ 0. Cosequetly, ρ Kɛ u ρ K u = ρ K u ρkɛ u ϕ 2 ɛ 0 ρ L u = ρk u ϕ 1 l 1ϕ 2 ρ L u ρ K u. z 1 ϕ 1 z ϕ 1 1 Sice ρ Kɛ u is mootoic with respect to ɛ, by Lemma ad Theorem 7.11 of [15], we have ɛ 0 ρ Kɛ u =ρ K u uiformly o S 1. Hece, ρ Kɛ u ρ K u = ϕ 1 l 1ϕ 2 ρ Ku ρ L u ρ Ku uiformly o S 1. Therefore, by the polar coordiate formula for volume 2.1, we obtai ϕ 1 l 1 V K + ϕ,ɛ L V K = 1 S 1 ϕ 2 ρk u ρ K u dsu. ρ L u For ϕ Φ, the dual Orlicz mixed volume ṼϕK, L is defied by Ṽ ϕ K, L = 1 ρk u ρ K u dsu 3.1 ρ L u S 1 ϕ for all K, L S 0. If ϕt =tp, p 1, we get the dual mixed volume Ṽ pk, L see [13, Propositio 1.9]. The followig theorem gives a coectio betwee the dual Orlicz mixed volume ad the harmoic Orlicz combiatio. Theorem 3.1 Suppose ϕ i Φ, i=1, 2 ad ϕx 1,x 2 =ϕ 1 x 1 +ϕ 2 x 2. For all K, L S0, we have Ṽ ϕ2 K, L = ϕ 1 l 1 V K + ϕ,ɛ L V K. 3.2

6 1024 H. L. Ji, S. F. Yua ad G. S. Leg The followig theorems show that the dual Orlicz volume is SL ivariat, cotiuous. Theorem 3.2 Suppose ϕ Φ. IfK, L S 0 ad A SL, the Ṽ ϕ AK, AL =ṼϕK, L. Proof By the same method of the proof of Lemma 3.13, we get AK + ϕ,ɛal = K + ϕ,ɛl for ϕ Φ 2. By Theorem 3.2, we get ṼϕAK, AL =ṼϕK, L. By the cotiuity of ϕ, we have the followig. Theorem 3.3 Suppose ϕ Φ. IfK, L S0 metric, the ṼϕK i,l i ṼϕK, L. ad K i K, L i L i the radial Hausdorff Theorem 3.4 Suppose K, L S0. If ϕ i,ϕ Φ ad ϕ i ϕ, i.e., max ϕ it ϕt 0 t I for every compact iterval I R, theṽϕ i K, L ṼϕK, L. Proof Sice K, L S0 ad ϕ i ϕ, wehaveϕ ρk u i ρ ρk Lu u ϕ ρ K u ρ ρk Lu u uiformly o S 1. By 3.1, we obtai that Ṽϕ i K, L ṼϕK, L. 4 Iequality of the Dual Orlicz Mixed Volume I [13], Lutwak proved the dual L p -mixed volume iequality: If p 1, ad K, L S 0, the Ṽ p K, L V K +p V L p with equality if ad oly if K ad L are dilates. For the dual Orlicz mixed volumes, we also establish the dual Orlicz mixed volume iequalities. The followig lemma will be eeded i the proof of Theorem 1.2, which is easy to be obtaied by the Hölder iequality ad the polar coordiate formula for volumes. Lemma 4.1 see [13, Propositio 1.10] If K, L S0,the 1 ρ K u +1 ρ L u 1 dsu V K +1 V L 1 S 1 with equality if ad oly if K ad L are dilates. ProofofTheorem1.2 By Jese s iequality ad Lemma 4.1, we have Ṽ ϕ K, L = 1 ρk u ϕ ρ K u dsu S ρ 1 L u 1 V Kϕ ρ K u +1 ρ L u 1 dsu S 1 1 V K V Kϕ. V L 1 If the equality holds, by the equality coditios of Jese s iequality ad Lemma 4.1, we have that K ad L are dilates. Coversely, if K ad L are dilates, it is easy to see that the equality holds.

7 O the Dual Orlicz Mixed Volumes 1025 I Theorem 1.2, if we set ϕt =t p, p 1, it leads to the Lutwak s result of the dual L p -mixed volume. By Theorem 1.2, we have the followig uiqueess result which is the Orlicz versio of Propositio 1.11 of [13]. Propositio 4.1 Suppose ϕ Φ, adm S 0 Ṽ ϕ K, Q V K = ṼϕL, Q V L such that K, L M.If for all Q M, the K = L. Proof Takig Q = L gives ṼϕK,L V K = ṼϕL,L V L =1. NowTheorem1.2givesV L V K, with equality if ad oly if K ad L are dilates. Takig Q = K, wegetv K V L. Hece, V K =V L, ad K ad L must be dilates. Thus K = L. I the followig, we give the dual Orlicz-Bru-Mikowski iequality for the harmoic Orlicz additio. ProofofTheorem1.3 By Theorem 1.2, we have V K + ϕ L= 1 ρk + ϕ L, x ρk + ϕ L, x ϕ 1 + ϕ 2 ρ S ρk, x ρl, x K + ϕ L u dsu 1 = Ṽϕ 1 K + ϕ L, K+Ṽϕ 2 K + ϕ L, L V K + ϕ L V K + ϕ L ϕ 1 V K + ϕ L ϕ V K 1 2. V L 1 If the equality holds, by the equality coditio of Theorem 1.2, we have that K ad L are dilates. Coversely, if K ad L are dilates, it is easy to check that the equality holds. Let ϕx 1,x 2 =x p 1 + xp 2, ad we have the followig result. Corollary 4.1 see [13, Propositio 1.12] Suppose K, L S0.Ifp 1, the V K + p L p V K p + V L p with equality if ad oly if K ad L are dilates. Ackowledgemet The authors are grateful to the referees for their valuable suggestios ad commets. Refereces [1] Lutwak, E., Dual mixed volumes, Pacific J. Math., 582, 1975, [2] Lutwak, E., Itersectio bodies ad dual mixed volumes, Adv. Math., 71, 1988, [3] Lutwak, E., Cetroid bodies ad dual mixed volumes, Proc. Lodo Math. Soc., 603, 1990, [4] Zhag, G., Cetered bodies ad dual mixed volumes, Tras. Amer. Math. Soc., 3452, 1994, [5] Garder, R. J., Stability of iequality i the dual Bru-Mikowski theory, J. Math. Aal. Appl., 231, 1999, [6] Klai, D., Star valuatio ad dual mixed volumes, Adv. Math., 121, 1996,

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