Dual Orlicz geominimal surface area

Transcription

1 Ma ad Wag Joural of Iequalities ad Applicatios :56 DOI /s R E S E A R C H Ope Access Dual Orlicz geomiimal surface area Togyi Ma 1* ad Weidog Wag 2 * Correspodece: matogyi@126com 1 College of Mathematics ad Statistics, Hexi Uiversity, Zhagye, , PR Chia Full list of author iformatio is available at the ed of the article Abstract The L p -geomiimal surface area was itroduced by Lutwak i 1996, which exteded the importat cocept of the geomiimal surface area Recetly, Wag ad Qi defied the p-dual geomiimal surface area, which belogs to the dual Bru-Mikowski theory I this paper, based o the cocept of the dual Orlicz mixed volume, we exted the dual geomiimal surface area to the Orlicz versio ad give its properties I additio, the isoperimetric iequality, a Blaschke-Sataló type iequality, ad the mootoicity iequality for the dual Orlicz geomiimal surface areas are established MSC: 52A30; 52A40 Keywords: covex bodies; star bodies; dual Orlicz mixed volume; dual Orlicz geomiimal surface area 1 Itroductio Let K deote the set of covex bodies compact, covex subsets with o-empty iteriors i Euclidea space R For the set of covex bodies cotaiig the origi i their iteriors, the set of covex bodies whose cetroids lie at the origi, ad the set of covex bodies havig their Sataló poit at the origi i R,wewriteKo, K c,adk s,respectively So deotes the set of star bodies about the origi i R LetVK deotethe -dimesioal volume of a body K, adletb deote the stadard Euclidea uit ball i R ad write ω = VB for its volume, as well as let S 1 deote the uit sphere for B The otio of geomiimal surface area was itroduced by Petty [1] For K Ko,the geomiimal surface area, GK,of K is defied by ω 1 GK=if { V 1 K, QV Q : Q K Here Q deotesthepolarofbodyq,adv 1 K, Q deotes the mixed volume of K, Q Ko see [2] This cocept has already attracted cosiderable iterest; see, for example, [3 5] The extesio of the classical Bru-Mikowski theory is the L p -Bru-Mikowski theory iitialized by Lutwak [6], which has had a eormous impact, providig stroger affie isoperimetric iequalities tha the classical couterparts see, eg, [7, 8] I particular, Lutwak [6] itroducedthep-geomiimal surface area: For K Ko, p 1, the p- geomiimal surface area, G p K, of K is defied by ω p G p K=if { V p K, QV Q p : Q K o Here V p K, Qdeotesthep-mixed volume of K, Q Ko see [6] 2016 Ma ad Wag This article is distributed uder the terms of the Creative Commos Attributio 40 Iteratioal Licese which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial authors ad the source, provide a lik to the Creative Commos licese, ad idicate if chages were made

2 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 2 of 13 Combiig the homogeeousess of the volume with the p-mixed volume, the p- geomiimal surface area could also be defied by G p K=if { V p K, Q:Q K o ad V Q = ω The recet extesio of the Bru-Mikowski theory is the Orlicz-Bru-Mikowski theory lauched by Lutwak et al [9, 10] This theory is far more geeral tha the L p - Bru-Mikowski theory, ad has already attracted cosiderable iterest; see, for example, [9 14] I particular, Yua et al [15] itroduced the Orlicz geomiimal surface area G φ KofK Ko Let deote the set of covex fuctios φ :[0, [0, thatsatisfy φ0 = 0, φ1 = 1, ad lim t φt=+ ForK Ko ad φ, G φ K=if { V φ K, Q:Q K o ad V Q = ω, where V φ K, Q deotes Orlicz mixed volume of K ad Q see [16] A dual theory to the L p -Bru-Mikowski theory was also developed by Lutwak see [17, 18] More recetly, Wag ad Qi [19] gave a defiitio of the p-dual geomiimal surface area which is a dual cocept for L p -geomiimal surface area ad belogs to the dual L p - Bru-Mikowski theory: For K So,thep-dual geomiimal surface area, G p K, of K is defied by ω p G p K=if { Ṽ p K, QV Q p : Q K o Here Ṽ p M, Ndeotesthep-dual mixed volume of M, N So see [6] Further, they [19] established the affie isoperimetric iequality, the Blaschke-Sataló iequality ad the mootoe iequality for the p-dual geomiimal surface area For K So ad p 1, G p K ω p VK +p, 11 with equality if ad oly if K is a ellipsoid cetered at the origi If K Kc ad p 1, the G p K G p K ω 2, 12 with equality if ad oly if K is a ellipsoid If K So,1 p < q,the G p K VK +p p G q K VK +q q 13 Combiig the homogeeousess of the volume ad the p-dual mixed volume, the p- dual geomiimal surface area could also be defied by G p K=if { Ṽ p K, Q:Q K o ad V Q = ω

3 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 3 of 13 A step toward a dual Orlicz-Bru-Mikowski theory for star sets has already bee made by Garder et al [20 22] adzhuet al [23] I some respects, the dual Orlicz- Bru-Mikowski theory is far more geeral tha the Orlicz-Bru-Mikowski theory for covex bodies The purpose of this article is to defie a ew operator G φ KofK So, called the dual Orlicz geomiimal surface area, by G φ K=if { Ṽ φ K, Q:Q K o ad V Q = ω, where Ṽ φ K, Q deotes the dual Orlicz mixed volume of K, Q see Sectio 2 Our mai results are as follows Theorem 11 If K So ad φ, the there exists a uique body K Ko such that G φ K=Ṽ φ K, K ad V K = ω 14 Theorem 12 Let K S o ad φ, the VK G φ K VKφ, 15 ω with equality if ad oly if K is a ellipsoid Whe φt =t p,withp 1, the above affie isoperimetric iequality for the dual Orlicz geomiimal surface area reduces to affie isoperimetric iequality 11forthep-dual geomiimal surface area Theorem 13 Let K Kc ad φ, the G φ K G φ K ω 2, with equality if ad oly if K is a ellipsoid Whe φt=t p,withp 1, the above Blaschke-Sataló type iequality for the dual Orlicz geomiimal surface area reduces to the Blaschke-Sataló type iequality 12forthe p-dual geomiimal surface area Theorem 14 Let φ 1, φ 2 ad φ 1 φ 2 If K S o, the φ 2 1 VK G φ2 K φ 2 VK 1 φ 1 1 VK G φ1 K φ 1 VK 1 Whe φ 1 t=t p, φ 2 t=t q,with1 p q <, the above iequality reduces to Wag s mootoicity iequality 13forthep-dual geomiimal surface area ratio

4 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 4 of 13 2 Backgroud material The support fuctio h K = hk, :S 1 R of a compact covex set K R is defied, for u S 1,by h K u=max{u x : x K, ad it uiquely determies the compact covex set Here u x deotes the stadard ier product of u ad x i R For K, L K,adλ, μ 0 ot both zero, the Mikowski liear combiatio λ K + μ L K is defied by hλ K + μ L, =λhk, +μhl, The classical Bru-Mikowski iequality see [24] states that for covex bodies K, L K ad real λ, μ 0 ot both zero, the volume of the bodies ad of their Mikowski liear combiatio λ K + μ L K are related by Vλ K + μ L λvk + μvl, 21 with equality if ad oly if K ad L are homothetic For real p 1, K, L Ko,adλ, μ 0 ot both zero, the Firey liear combiatio λ K + p μ L,isdefiedbysee[25] hλ K + p μ L, p = λhk, p + μhl, p Firey [25] also established the followig L p Bru-Mikowski iequality If p >1,λ, μ 0 ot both zero, ad K, L K o,the Vλ K + p μ L p λvk p + μvl p, with equality if ad oly if K ad L are dilates The radial fuctio of K, ρ K : S 1 [0,, is defied by ρ K u=max{λ : λu K AsetK R is said to be a star body about the origi, if the lie segmet from the origi to ay poit x K is cotaied i K ad K has cotiuous ad positive radial fuctio ρ K Note that K S o ca be uiquely determied by its radial fuctio ρ K advice versa If λ >0,wehave ρ K λx=λ 1 ρ K x ad ρ λk x=λρ K x More geerally, from the defiitio of the radial fuctio it follows immediately that for GL the radial fuctio of the image K = { y : y K of K So is give by see [26] ρ K, x=ρ K, 1 x, for all x R 22

5 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 5 of 13 TwostarbodiesK, L So are said to be dilates of each other if there is a costat λ >0 such that L = λk,adtheequatioρ L u=λρ K u for all u S 1 Clearly, for K, L So, K L if ad oly if ρ K u ρ L u, for all u S 1 The atural metric o So is the radial metric δ, :So S o R defied as δk, L= ρ K ρ L = sup u S 1 ρk u ρ L u, fork, L S o A sequece of star bodies {K j So is said to be coverget to K S o i δ if δk j, K 0 as j, ad equivaletly, ρ Kj is uiformly coverget to ρ K o S 1 If K Ko,thepolarbody,K,ofK is defied by K = { x R : x y 1, for all y K Obviously, we have K = K The Blaschke-Sataló iequality [27] is oe of the fudametal affie isoperimetric iequalitiesitstatesthatifq Kc the VQV Q ω 2, 23 with equality if ad oly if Q is a ellipsoid For φ ad λ, μ 0 ot both zero, we defie the Orlicz radial sum λ K + φ μ L of two star bodies K, L So,by { t t ρλ K + φ μ L, u=sup t >0:λφ + μφ 1, 24 ρ K u ρ L u for all u S 1 Whe φt=t p,withp 1, it is easy to show that the Orlicz radial sum reduces to Lutwak s p-harmoic radial combiatio see [6]: ρμ K + p μ L, p = λρk, p + μρl, p If K, L K o,the λ K + p μ L = λ K + p μ L We deote the right derivative of a real-valued fuctio f by f r Forφ,thereisφ r 1 > 0becauseφ is covex ad strictly icreasig Let φ By the Orlicz radial sum 24, we defie the dual Orlicz mixed volume Ṽ φ K, L of covex bodies K, L S o by VK + φ ε L VK φ r 1Ṽ φk, L= lim 25 ε 0 + ε

6 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 6 of 13 From 25 we easily obtai the followig itegral formula of the dual Orlicz mixed volume: Ṽ φ K, L= 1 Apparetly, we have S 1 φ ρk u ρk udsu 26 ρ L u Ṽ φ K, K=VK 27 For φt =t p with p 1, the dual Orlicz mixed volume Ṽ φ K, L reducestolutwak s p-dual mixed volume formula see [6]: Ṽ p K, L= 1 ρ +p S 1 K uρ p L udsu, for all K, L So Usig the same argumet as i [23] we establish the followig dual Orlicz-Mikowski iequality: Suppose φ IfK, L So,the VK Ṽ φ K, L VKφ, 28 VL with equality if ad oly if K ad L are dilates of each other Whe φt=t p,withp 1 The 28 reduces to the followig L p -dual Mikowski iequality see [6]: Ṽ p K, L VK +p VL p, with equality if ad oly if K ad L are dilates of each other We also establish the dual Orlicz-Bru-Mikowski iequality as follows: Suppose K, L So,adλ, μ >0Ifφ,the Vλ K +φ μ L λφ + μφ VK Vλ K +φ μ L VL 1, with equality if ad oly if K ad L are dilates of each other Whe φt =t p,withp 1, the above iequality reduces to Lutwak s L p -dual Bru- Mikowski iequality see [6]: Vλ K + p μ L p λvk p + μvl p, with equality if ad oly if K ad L are dilates of each other The followig results are required i the proofs of our mai results Lemma 21 If φ, ad K, L So, the for SL, Ṽ φ K, L=Ṽ φ K, L

7 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 7 of 13 Proof For x R \{0,let x = x/ x By26ad22, we have Ṽ φ K, L= 1 = 1 = 1 S 1 φ S 1 φ S 1 φ = Ṽ φ K, 1 L, ρ K u ρ L u ρk 1 u ρ L 1 u ρk 1 u ρ 1 L 1 u ρ K udsu ρk 1 u ds 1 u ρk 1 u det ds 1 u where 1 deotes the iverse of It is easy to check that Ṽ φ λk, λl =λ Ṽ φ K, L, for λ > 0 Therefore, we have the followig Propositio 22 Suppose K, L So If φ ad GL, the Ṽ φ K, L= det Ṽ φ K, L Lemma 23 Suppose f i, f are strictly positive ad cotiuous fuctios o S 1 ; φ j, φ ; μ k, μ are Borel probability measures o S 1 ; i, j, k N If f i f poitwise, φ j φ uiformly, ad μ k μ weakly, the φ j f i dμ k φf dμ, as i, j, k 29 S 1 S 1 Proof The cotiuity of f i ad f,adf i f poitwise guaratee that f i f uiformly Thus, there exists a N 0 N,suchthat Let ad 1 2 mi u S 1 f u f iu 2 max u S 1 f u, for i > N 0 { 1 c m = mi 2 mi f u, mi f iu, with i N 0 u S 1 u S 1 { c M = mi 2 max f u, max f iu, with i N 0 u S 1 u S 1 The strict positivity ad the cotiuity of f i ad f imply that 0 < c m c M < Thus, c m f u c M ad c m f i u c M, foru S 1 ad i N 210 Sice φ j φ uiformly o [c m, c M ], by 210 adf i f uiformly, it follows that as i, j, φ j f i φf, uiformly o S 1 Combiedwithμ k μ weakly, oe cocludes

8 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 8 of 13 that, as i, j, k, φ j f i dμ k φf dμ, S 1 S 1 as for 29asdesired Usig Lemma 23, we immediately obtai the followig result Lemma 24 Suppose K, K i, L, L j S o ad φ, φ k If K i K, L j Ladφ k φ, the lim i,j,k Ṽ φk K i, L j =Ṽ φ K, L 3 Dual Orlicz geomiimal surface areas I this sectio we first propose the followig problem Problem 31 For K So,fidacovexbodyQ, amogst all covex bodies cotaiig the origi i their iteriors, which solves the costraied ifimum problem G φ K:=if Ṽ φ K, Q subject to V Q = ω I order to demostrate the existece ad uiqueess of Problem 31, the followigsimple fact will be eeded Lemma 32 [6] Let C deote the set of compact covex subsets of Euclidea -space R, ad suppose K i K o such that K i L C If the sequece VK i is bouded, the L K o Now we aswer Problem 31, amely, the proofof Theorem 11 is give Proof of Theorem 11 Problem 31 ca be equivaletly restated as: For a fixed star body K So,thereexistsasequeceM i Ko such that VM i =ω,withṽ φ K, B Ṽ φ K, M i, for all i,adṽ φ K, M i G φ K To see that the M i Ko are uiformly bouded, let R i = RM i =ρm i, u i =max { ρm i, u:u S 1, where u i is ay of the poits i S 1 at which this maximum is attaied Let r K = mi S 1 ρ K Ther K B K From the defiitio 26 of the dual Orlicz mixed volume ad the Jese iequality, it follows that Ṽ φ K, B VK Ṽ φk, M i VK ρk u = φ S 1 ρ Mi u φ S 1 dṽ K ρ K u ρ Mi u dṽ K ρ K u φ dṽ S 1 K R i 1 = φ ρ K u +1 dsu VKR i S 1

9 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 9 of 13 Namely, r K φ ρ K u dsu VKR i S 1 rk = φ R i ω rk φ rk Ṽ φ K, M i Ṽ φ K, B< R i Sice the M i are uiformly bouded, the Blaschke selectio theorem guaratees the existece of a subsequece of the M i,whichwillalsobedeotedbym i,adacompact covex L C,suchthatM i LSiceVMi =ω, Lemma 32 gives L Ko Now,M i L implies that Mi L, ad sice VMi =ω, it follows that VL =ω Lemma 24 ca ow be used to coclude that L will serve as the desired body K The uiqueess of the miimizig body is easily demostrated as follows Suppose L 1, L 2 Ko ad L 1 L 2,suchthatVL 1 =ω = VL 2, ad Ṽ φ K, L 1 =Ṽ φ K, L 2 Defie L K o,by L = 1 2 L L 2 Sice obviously L = 1 2 L L 2, ad VL 1 =ω = VL 2, it follows from the Bru-Mikowski iequality 21that V L ω, with equality if ad oly if L 1 = L 2 By the defiitio 26 ofthe dualorliczmixedvolume, together with the covexity of φ, we have Ṽ φ K, L = 1 = 1 = S 1 φ S 1 φ S 1 φ S 1 φ ρ K u ρ 1 2 L L 2 u ρ K u 1 2ρ L1 u + 1 2ρ L2 u 1 ρk u 2ρ L1 u + ρ Ku 2ρ L2 u ρk u ρ L1 u = 1 2Ṽ φk, L Ṽ φk, L 2 ρ K u dsu ρ K u dsu ρ K u dsu ρ K u dsu+ 1 2 S 1 φ ρk u ρ K u dsu ρ L2 u

10 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 10 of 13 = Ṽ φ K, L 1 = Ṽ φ K, L 2, with equality if ad oly if L 1 = L 2 Thus, Ṽ φ K, L<Ṽ φ K, L 1 =Ṽ φ K, L 2, is the cotradictio that would arise if it were the case that L 1 L 2 Thiscompletesthe proof I view of Theorem 11, aturally, we itroduce the followig defiitio Defiitio 33 For K So,adφ, we defie the dual Orlicz geomiimal surface area, G φ K, of K by G φ K=if { Ṽ φ K, Q:Q K o ad V Q = ω Whe φt=t p,withp 1, the dual Orlicz geomiimal surface area reduces to Wag s p-geomiimal surface area We will show that the dual Orlicz geomiimal surface area of a stat body is ivariat uder uimodular cetro-trasformatios of the body Propositio 34 Suppose K So If φ ad SL, the G φ K= G φ K Proof From Defiitio 33 ofthe dual Orlicz geomiimal surface area, ad Lemma 21,we have for SL G φ K=if { Ṽ φ K, Q:Q K o ad V Q = ω = if { Ṽ φ K, 1 Q : 1 Q K o ad V 1 Q = V t Q = ω = G φ K The uique body whose existece is guarateed by Theorem 11 will be deoted by T φ K, ad it is called the dual Orlicz-Petty body of KThepolarbodyof T φ K will be deoted by T φ K, rather tha T φ K Thus,forK So,thebody T φ K is defied by G φ K=Ṽ φ K, T φ K ad V T φ K = ω is a uimodular cetro- The ext propositio shows that the mappig T φ : So K o affie ivariat mappig Propositio 35 If K So, the for SL, T φ K = T φ K

11 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 11 of 13 Proof From the defiitio of T φ ad Propositio 34, Ṽ φ K, T φ K= G φ K= G φ K=Ṽ φ K, T φ K By Lemma 21, Ṽ φ K, T φ K=Ṽ φ K, T φ K=Ṽ φ K, 1 T φ K The uiqueess part of Theorem 11 shows that T φ K = 1 T φ K, whichisthedesired result A immediate cosequece of the defiitio of G φ ad Lemma 24 is the followig Propositio 36 If φ, the the fuctioal G φ : So 0,, is cotiuous For Q K,letCeQ it Q deote the cetroid of Q AssociatedwitheachQ K is a poit s = SaQ it K, called the Sataló poit of Q, the Sataló poit also may be defied as the uique s Q,suchthat V s + Q = mi { V x + Q : x it Q, or equivaletly, as the uique s Q,suchthat uh s + Q, u +1 dsu=0 S 1 We recall that Ks deotes the set of covex bodies havig their Sataló poit at the origi Thus see [6], Q Ks if ad oly if Q Kc Now we give the proof of Theorem 12 Proof of Theorem 12 Suppose Q Ko,ads is the Sataló poit of Q LetQ 0 = s + Q Ks SiceVQ 0 VQ see[6], p263, the by the defiitio of the dual Orlicz geomiimal surface area, the dual Orlicz-Mikowski iequality 28adtheBlaschke-Sataló iequality 23, it follows that G φ K =if { Ṽ φ K, Q:Q Ko ad V Q = ω { if VKφ { = if VKφ { = if VKφ VK VKφ VK VQ VK Vs + Q 0 VK ω VQ 0 : Q K o ad V Q = ω : Q 0 Ks ad V Q 0 ω : Q 0 Ks ad V Q 0 ω

12 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 12 of 13 By the equality coditios of the dual Orlicz-Mikowski iequality 28adtheBlaschke- Sataló iequality 23, we see that equality holds i 15 ifadolyifk is a ellipsoid cetered at the origi Proof of Theorem 13 By the dual Orlicz-Mikowski iequality 28, we get VQ ω G φ Kφ VK { VQ = if Ṽ φ K, Qφ V Q : Q K o VK ad V Q = ω if { Ṽ φ K, QṼ φ Q, K : Q K o ad V Q = ω 31 Sice K Kc,takigQ = K i 31, ad combiig equatio 27 with iequality 23, we get ω G φ K VKV K ω 2, ie, G φ K ω 32 Similarly, G φ K ω 33 Combiig 32ad33, we get G φ K G φ K ω 2 34 By the equality coditios of the dual Orlicz-Mikowski iequality 28adtheBlaschke- Sataló iequality 23, we see that equality i 34 holds if ad oly if K is a ellipsoid cetered at the origi Proof of Theorem 14 By the defiitio of the dual Orlicz geomiimal surface area ad φ i i = 1, 2 beig strictly icreasig o 0,, we have VK φ 1 G φ2 K φ 2 VK 1 VK = φ 1 if{ Ṽ φ 2 K, Q : Q Ko ad VQ =ω φ 2 VK 1 VK φ 1 if{ Ṽ φ 1 K, Q : Q Ko ad VQ =ω φ 1 VK 1 VK φ 2 if{ Ṽ φ 1 K, Q : Q Ko ad VQ =ω φ 1 VK 1 VK = φ 2 G φ1 K φ 1 VK 1

13 Ma ad Wag Joural of Iequalities ad Applicatios :56 Page 13 of 13 Accordig to the iverse φ i 1 i =1,2ofφ i i = 1, 2 beig strictly icreasig ad havig cotiuity o 0,, it followsthat φ 2 1 VK G φ2 K φ 2 VK 1 φ 1 1 VK G φ1 K φ 1 VK 1 This completes the proof Competig iterests The authors declare that they have o competig iterests Authors cotributios The authors completed the paper, ad read ad approved the fial mauscript Author details 1 College of Mathematics ad Statistics, Hexi Uiversity, Zhagye, , PR Chia 2 Departmet of Mathematics, Chia Three Gorges Uiversity, Yichag, , PR Chia Ackowledgemets This work was supported by the Natioal Natural Sciece Foudatio of Chia Grat No , was supported by the Sciece ad Techology Pla of the Gasu Provice Grat No 145RJZG227, ad was partly supported by the Natioal Natural Sciece Foudatio of Chia Grat No Received: 14 October 2015 Accepted: 2 February 2016 Refereces 1 Petty, CM: Geomiimal surface area Geom Dedic 31, Lutwak, E: Volume of mixed bodies Tras Am Math Soc 2942, Ludwig, M, Reitzer, M: A characterizatio of affie surface area Adv Math 147, Werer, E: O L p -affie surface areas Idiaa Uiv Math J 56, Werer, E, Ye, D: New L p -affie isoperimetric iequalities Adv Math 218, Lutwak, E: The Bru-Mikowski-Firey theory II: affie ad geomiimal surface areas Adv Math 1182, Wag, WD, Feg, YB: A geeral L p -versio of Petty s affie projectio iequality Taiwa J Math 172, Zhu, BC, Li, N, Zhou, JZ: Isoperimetric iequalities for L p -geomiimal surface area Glasg Math J 53, Lutwak, E, Yag, D, Zhag, G: Orlicz projectio bodies Adv Math 223, Lutwak,E,Yag,D,Zhag,G:OrliczcetroidbodiesJDifferGeom84, Che, F, Zhou, J, Yag, C: O the reverse Orlicz Busema-Petty cetroid iequality Adv Appl Math 47, Garder, R, Hu, D, Weil, W: The Orlicz-Bru-Mikowski theory: a geeral framework, additios, ad iequalities J Differ Geom 97, Haberl, C, Lutwak, E, Yag, D, Zhag, G: The eve Orlicz Mikowski problem Adv Math 224, Zhu, G: The Orlicz cetroid iequality for star bodies Adv Appl Math 48, Yua, SF, Ji, HL, Leg, GS: Orlicz geomiimal surface areas Math Iequal Appl 181, Xi, DM, Ji, HL, Leg, GS: The Orlicz Bru-Mikowski iequality Adv Math 260, Lutwak, E: Cetroid bodies ad dual mixed volumes Proc Lod Math Soc 60, Schuster, FE: Volume iequalities ad additive maps of covex bodies Mathematika 53, Wag, WD, Qi, C: L p -Dual geomiimal surface area J Iequal Appl 2011, Garder, RJ, Hug, D, Weil, W, Ye, D: The dual Orlicz-Bru-Mikowski theory J Math Aal Appl 4302, Ye, D: Dual Orlicz-Bru-Mikowski theory: Orlicz ϕ-radial additio, Orlicz L φ -dual mixed volume ad related iequalities arxiv preprit 2014 arxiv: Ye, D: New Orlicz affie isoperimetric iequalities J Math Aal Appl 427, Zhu, BC, Zhou, JZ, Xu, WX: Dual Orlicz-Bru-Mikowski theory Adv Math 264, Garder, RJ: The Bru-Mikowski iequality Bull Am Math Soc 39, Firey,WJ: p-meas of covex bodies Math Scad 10, Scheider, R: Covex Bodies: The Bru-Mikowski Theory Cambridge Uiversity Press, Cambridge Meyer, M, Pajor, A: O Satalós iequality I: Lidestrauss, J, Milma, VD eds Geometric Aspects of Fuctioal Aalysis Spriger Lecture Notes i Math, vol 1376, pp Spriger, Berli 1989