LET K n denote the class of convex bodies (compact,

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1 IAENG Iteratioal Joural of Applied Matheatics 46:3 IJAM_46_3_16 Orlicz Mixed Geoiial Surface Area Yuayua Guo Togyi Ma Li Gao Abstract I this paper we deal with the Orlicz geoiial surface area ad give a itegral represetatio by the Orlicz- Petty body. The otio of Orlicz ixed geoiial surface area will be itroduced as a extesio of the Orlicz geoiial surface area. Furtherore soe related iequalities are established icludig Alexadrov-Fechel type iequality aalogous cyclic iequality Blaschke-Sataló type iequality ad affie isoperietric iequality. Idex Ters Covex bodies Orlicz geoiial surface area Orlicz ixed geoiial surface area. I. INTRODUCTION LET K deote the class of covex bodies copact covex subsets with oepty iteriors i Euclidea -space R. For the class of covex bodies cotaiig the origi i their iteriors ad the class of origi-syetric covex bodies i R we write Ko ad Kc respectively. So deotes the class of star bodies about the origi i R. Write S 1 ad B for the uit sphere ad the stadard Euclidea uit ball i R respectively. Besides we use V K to deote the -diesioal volue of a body K ad write ω V B for the -diesioal volue of B. The classical geoiial surface area was firstly itroduced by Petty [1] which serves as a bridge coectig ay areas of geoetry: affie differetial geoetry relative differetial geoetry ad Mikowskia geoetry. For K Ko the geoiial surface area GK of K is defied by see [1] ω 1/ GK if{v 1 K QV Q 1/ : Q K o } 1 where Q deotes the polar of covex body Q ad V 1 K Q is the ixed volue of K Q K o see [2]. The developet of L p -space appeared i the early 1960s see [3] ad started to ake rapidly progress fro the iitial Lutwak s cotributios [4] [5] i the id 1990s. I his seial paper [5] Lutwak exteded the classical geoiial surface area to L p -versio ad obtaied related iequalities. Ma et al. also studied this topic of L p -space. For istace Ma et al. [6] defied the cocept of ith L p -ixed affie surface areas ad established related ootoic iequality. I [7] Ma et al. obtaied soe Bru-Mikowski type iequalities of L p -geoiial surface area. There are ay papers o L p -space see e.g. [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]. Mauscript received Jauary ; revised March This work was supported by the Natioal Natural Sciece Foudatio of Chia uder Grat No ad was supported by the Sciece ad Techology Pla of the Gasu Provice uder Grat No.145RJZG227. T. Y. Ma is with the College of Matheatics ad Statistics Hexi Uiversity Zhagye Chia. Ad is with the College of Matheatics ad Statistics Northwest Noral Uiversity Lazhou Chia. e-ail: atogyi@126.co. Y. Y. Guo is with the College of Matheatics ad Statistics Northwest Noral Uiversity Lazhou Chia. L. Gao is with the College of Matheatics ad Statistics Northwest Noral Uiversity Lazhou Chia. Lutwak [5] defied the L p -geoiial surface area G p K of K K o as follows: For p 1 ω p/ G p K if{v p K QV Q p/ : Q K o } 2 where V p K Q deotes the L p -ixed volue of K Q K o see [5]. Whe p 1 G 1 K is just classical geoiial surface area GK. Based o the hoogeeous of volue ad L p -ixed volue the L p -geoiial surface area ca be defied by G p K if{v p K Q : Q K o ad V Q ω }. 3 I recet years the Orlicz Bru-Mikowski theory has aroused icreasig attetio which plays such a sigificat role that it is udeiably applied to a large uber of areas of geoetry. The beautiful Orlicz Bru-Mikowski theory a ew extesio of L p -Bru-Mikowski theory origiated fro Lutwak Yag ad Zhag see [18] [19]. I these papers the affie isoperietric iequalities for L p - projectio bodies ad L p -cetroid bodies were expaded to Orlicz space. However because of lackig hoogeeity for ohoogeeous fuctio t the way of defiig Orlicz additio is otrivial extreely to be foud appropriately. Fortuately i the groudbreakig paper [20] Garder Hug ad Weil have gotte over the difficulty. They itroduced the defiitio of Orlicz additio ad Orlicz ixed volue. O the basis of the liear Orlicz additio for covex bodies they also established the ew Orlicz Bru-Mikowski iequality ad the Orlicz Mikowski ixed volue iequality. Their classical couterparts are the Bru-Mikowski iequality ad Mikowski iequality which have bee applied i ay fields. The ore developet of the Orlicz Bru- Mikowski theory see for exaple [21] [22] [23] [24] [25] [26] [27] [28] aog others. More recetly Yua et al. itroduced the Orlicz geoiial surface area G K of K K o see [29]. Let Φ deote the set of covex fuctios : [0 [0 such that 0 0 ad 1 1. For K K o ad Φ G K if{v K Q : Q K o ad V Q ω } 4 where V K Q deotes Orlicz ixed volue of K Q see [25]. Yua et al. [29] have proved the existece property of Orlicz geoiial surface area. I this paper the first goal is to establish the uiqueess property of Orlicz geoiial surface area. The we ca give the itegral represetatio of Orlicz geoiial surface area by the Orlicz-Petty body see Sectio 3.1. Motivated by the work of Zhu et al. [30] we itroduce Orlicz geoiial surface area ad exted the related theory to Orlicz versio. We defie the Orlicz ixed geoiial surface area see Sectio 3.2 which exteds the cocept of Orlicz geoiial surface area. I additio we devote to the geeral ith Orlicz ixed geoiial surface area. Soe Advace olie publicatio: 26 August 2016

2 IAENG Iteratioal Joural of Applied Matheatics 46:3 IJAM_46_3_16 related iequalities for Orlicz ixed geoiial surface are established icludig Alexadrov-Fechel type iequality aalogous cyclic iequality Blaschke-Sataló type iequality ad isoperietric iequality see Sectio 3.3. These iequalities are atural extesios of iequalities for L p - geoiial surface area. Our paper is orgaized as follows. Firstly i Sectio II we provide soe preliiaries icludig defiitios ad kow results we will use. The i Sectio III we give our ai results ad proofs. II. PRELIMINARIES For a copact covex set K K o its support fuctio h K hk : S 1 R is defied by h K u ax{u x : x K} u S 1 where u x deotes the stadard ier product of u ad x i R. Note that the copact covex set K is uiquely deteried by its support fuctio h K. Let GL deote the group of liear trasforatios. If A GL the h AK u h K A t u where A t deotes the traspose of A see [31]. For K L K o the Hausdorff etric is defied by δk L sup hk u hl u. u S 1 A set K R is said to be a star body about the origi if the lie seget fro the origi to ay poit x K is cotaied i K ad K has cotiuous ad positive radial fuctio ρ K. Here the radial fuctio of K ρ K ρk : S 1 [0 is defied by ρ K u ax{λ : λu K} ad it uiquely deteries the copact covex set K. Two star bodies K L are said to be dilates of oe aother if ρ K u/ρ L u is idepedet of u S 1. For K K o the K the polar body of K is defied by see [31] [32] K {x R : x y 1 y K}. Whe K K o it ca be easily proved that K K. Fro the defiitios it follows obviously that for each covex body K K o we easily get h K u 1 ρ K u ad ρ K u 1 h K u for all u S 1. For K K o the Blaschke-Sataló iequality see [33] [34] states as follows: If K K c the V KV K ω 2 5 with equality if ad oly if K is a ellipsoid. For K L K ad λ µ 0 ot both zero the Mikowski liear cobiatio λk + µl K is defied by see [5] hλk + µl λhk + µhl. The classical Bru-Mikowski iequality states that for K L K ad λ µ 0 ot both zero the volue of the bodies ad of their Mikowski liear cobiatio λk + µl K are related by see [35] V λk + µl 1 λv K 1 + µv L 1 with equality if ad oly if K ad L are hoothetic. For real p 1 K L K o ad λ µ 0 ot both zero the Firey liear cobiatio λ K + p µ L is defied by see [3] hλ K + p µ L λhk p + µhl p where i λ K deotes the Firey scalar ultiplicatio. After that Firey [3] established the L p -Bru-Mikowski iequality. If p > 1 λ µ 0 ot both zero ad K L K o the V λ K + p µ L p λv K p + µv L p with equality if ad oly if K ad L are dilates. I [5] Lutwak defied the haroic L p -cobiatio λ K + p µ L as follows: For K L So ad λ µ 0 ot both zero ρλ K + p µ L p λρk p + µρl p. If K L K o rather tha beig i S o the λ K + p µ L λ K + p µ L. Further Lutwak established the L p Bru-Mikowski iequality. If p 1 λ µ 0 ot both zero ad K L K o the V λ K + p µ L p λv K p + µv L p 6 with equality if ad oly if K ad L are dilates see [5]. I [20] Garder Hug ad Weil itroduced the defiitio of Orlicz ixed volue. For Φ K L Ko the Orlicz ixed volue V K L of K L is defied by V K L 1 hl u h K uds K u 7 h K u S 1 where S K is the surface area easure of K. Apparetly we have V K K V K. 8 For t t p with p 1 the Orlicz ixed volue V K L reduces to L p -ixed volue V p K L of K L see [5]: V p K L 1 hl u p ds p K u S 1 where S p K is the L p -surface area easure of K. Xi Ji ad Leg [25] established the Orlicz Mikowski iequality: Let Φ if K L Ko the 1 V L V K L V K 9 V K with equality if K ad L are dilates. Whe is strictly covex equality i 9 holds if ad oly if K ad L are dilates. Whe t t p p 1 the correspodig results of above reduces to the L p -Bru-Mikowski iequality. The followig result provides a Orlicz geoiial surface area iequality by Yua et al. [29]. If K Kc ad Φ the 1 ω G K V K 10 V K with equality oly if K is a ellipsoid. Advace olie publicatio: 26 August 2016

3 IAENG Iteratioal Joural of Applied Matheatics 46:3 IJAM_46_3_16 III. MAIN RESULTS AND PROOFS A. Orlicz geoiial surface area At first we prove the uiqueess of Orlicz geoiial surface area. The cobiig with the existece of Orlicz geoiial surface area proved by Yua et al. [29] we give the itegral represetatio of Orlicz geoiial surface area. Theore 3.1. If K Ko ad Φ the there exists a uique body K Ko such that G K V K K ad V K ω. Proof. I [29] the existece property of Theore 3.1 has bee prove ow we just prove the uiqueess. Suppose L 1 L 2 K o with L 1 L 2 such that V L 1 ω V L 2 ad Defied L K o by Sice obviously V K L 1 V K L 2. L 1 2 L L 2. L 1 2 L L 2 ad V L 1 ω V L 2 it follows fro the 6 that V L ω with equality if ad oly if L 1 L 2. By the defiitio 7 of Orlicz ixed volue together with the covexity of we have V K L 1 hl u h K uds K u S h 1 K u 1 h 1 2 L L u 2 h K uds K u S h 1 K u 1 1 h L1 u S 2 h 1 K u + 1 h L2 u h K uds K u 2 h K u 1 hl1 u h K uds K u 2 S h 1 K u + 1 hl2 u h K uds K u 2 h K u S V K L V K L 2 V K L 1 V K L 2 with equality if ad oly if L 1 L 2. Thus V K L < V K L 1 V K L 2 is the cotradictio that would arise if it were the case that L 1 L 2. The uique covex body whose existece is guarateed by Theore 3.1 ca be deoted by T K Orlicz-Petty body of K. We use T K to deote the polar body of T K. Thus for K K o the body T K is defied by Let G K V K T K ad V T K ω. T { K K o : G K V K K ad V K ω }. Lea 3.1. ad Defiitio For K K o ad Φ there exists a uique covex body T K T with G K V K T K. By Lea 3.1 ad 7 we get the followig itegral represetatio of G K. Theore 3.2. For K Ko ad Φ there exists a uique covex body T K T with ht Ku G K h K udsu. h K u S 1 B. Orlicz ixed geoiial surface area We ow defie a ew cocept: the Orlicz ixed geoiial surface area G K 1 K of K 1 K Ko as follow: Defiitio 3.1. For each K i Ko there exists a uique covex body Orlicz-Petty body of K i T K i T i 1 with G K 1 K [ ht K 1 u S h 1 K1 u ht K u h K u Let g K i u h K1 u h K u ] 1 forula ca be expressed as follows: G K 1 K ht K 2 u dsu. h K2 u h K2 u h T K i u h Ki u h Ki u the the above S 1 [g K 1 u g K u] 1 dsu. 11 Lea 3.2. Hölder s itegral iequality see [36] [37] Let f 0 f 1 f k be Borel easurable fuctios o X. Suppose that p 0 p 1 p k are ozero real ubers with k i1 1 p i 1. The X f 0 uf 1 u f k udu k i1 f 0 uf i u p i du X 1 p i with equality if ad oly if either a there are costats b 1 b 2 b k ot all zero such that b 1 f 1 u p1 b 2 f 2 u p 2 b k f k u p k or b oe of the fuctios is ull. The classical Alexadrov-Fechel iequality for ixed volue see [32] [38] is oe of iportat iequalities i covex geoetry. It states that 1 i0 V K 1 K K i K i }{{} V K 1 K. We the prove the followig Alexadrov-Fechel type iequality for Orlicz ixed geoiial surface area. Theore 3.3. If K 1 K K o the for 1 < G K 1 K 1 i0 G K 1 K K i K i }{{} with equality if the K j are dilates of each other for j + 1. If 1 equality holds trivially. Advace olie publicatio: 26 August 2016

4 IAENG Iteratioal Joural of Applied Matheatics 46:3 IJAM_46_3_16 I particular if the G K 1 K G K 1 G K 12 equality holds if the K i are dilates of each other. Proof. Let H 0 u [g K 1 u g K u] 1 ad H i+1 u [g K i u] 1 for i Fro 11 ad Lea 3.2 we get G K 1 K [g K 1 u g K u] 1 dsu S 1 H 0 uh 1 u H udsu S 1 1 i0 1 S 1 H 0 uh i+1 u dsu 1 G 1 K 1... K K i... K i. }{{} i0 As ca be see fro Lea 3.2 the equality of above iequality holds if ad oly if H 0 uhi+1 u c ij H 0uHj+1 u for soe c ij > 0 ad all 0 i j 1. This is equivalet to ht K i u h K i u h K i u c ij ht K j u h K j u h K j u which ca observe the equality holds if K i ad K j are dilates. For Φ by the defiitio of Orlicz ixed volue V K L of K L Ko takig L B the we iediately get V K B 1 hb u h K uds K u. h K u S 1 Now we defie the Orlicz geoiial surface area S K aely S K V K B. The special case of t t p with p 1 is L p -geoiial surface area. The we ca prove the aalogous isoperietric iequality for Orlicz ixed geoiial surface area. Theore 3.4. Let K i Ko 1 i the G K 1 K G B B S K 1 S K V B V B. 13 Whe is strictly covex equality holds if ad oly if the K i are ellipsoids with dilates of each other. Proof. By iequality 10 we have G B V B ω the G B B G B ω. By iequalities ad 9 we get G K 1 K G B B G K 1 G K G B G B V K 1 ω V K 1 1 V K ω V K V B V B V K 1 B V K B V B V B S K 1 S K V B V B. 1 With the existece ad uiqueess of T K equality holds i 10 if ad oly if K is a ellipsoid. Cobiig with the equality coditio of 9 we see that whe is strictly covex equality holds i 13 if ad oly if the K i are ellipsoids with dilates of each other. C. The ith Orlicz ixed geoiial surface area I this sectio we itroduce the cocept of ith Orlicz ixed geoiial surface area. For K L Ko ad i R the ith Orlicz ixed geoiial surface area G i K L of K L is defied by G i K L By the Lea 3.1 we get sice S 1 g K u i g L u i dsu. 14 G B V B T B 15 G B ω V B B 16 cobiig ad the uiqueess of Lea 3.1 we have T B B. Let L B ad write G i K B G i K. 17 Cobiig ad h T B h B 1 we have G i K g K u i dsu. S 1 By ad 17 we easily get G 0 K B G K G i K K G K 18 G 0 K L G K G K L G L. 19 The followig Theore deals with the cyclic iequality for the ith Orlicz ixed geoiial surface. Theore 3.5. Let K L K o i j k R ad i < j < k the G i K L k j G k K L j i G j K L k i 20 equality holds if K ad L are dilates. Proof. Fro defiitio 14 ad Hölder s itegral iequality see [36] we get G i K L k j k i Gk K L j i k i [ ] k j g K u i g L u i k i dsu S 1 [ ] j i g K u k g L u k k i dsu S 1 { ] k i } k j k j k i [g K u α1 g L u α2 dsu S 1 { [ ] k i } j i g K u β 1 g L u β j i k i 2 dsu S 1 g K u j g L u j dsu S 1 G j K L Advace olie publicatio: 26 August 2016

5 IAENG Iteratioal Joural of Applied Matheatics 46:3 IJAM_46_3_16 where α 1 ik j k i α 2 ik j k i β 1 kj i k i ad β 2 kj i k i. We prove iequality 20. Accordig to the equality coditio of Hölder s itegral iequality the equality i 20 holds if ad oly if for ay u S 1 g K u i g L u i g K u k g L u k is a costat. Naely for ay u S 1 g K u/g L u is a costat. By the siilar to the proof of Theore 3.3 we coclude that equality i 20 holds if K ad L are dilates of oe aother. Takig L B i Theore 3.5 ad usig 17 we iediately obtai: Corollary 3.1. If K L K o i j k R ad i < j < k the G i K k j G k K j i G j K k i equality holds if K is a ball with cetroid at the origi. The followig iequalities are the Mikowski iequalities for the ith Orlicz ixed geoiial surface area. Theore 3.6. For K L K o i R the for i < 0 or i > for 0 < i < G i K L G K L i G L i 21 G i K L G K L i G L i. 22 Equality of each iequality holds if K ad L are dilates. For i 0 or i above iequalities are idetical. Proof. i For i < 0 let i j k i 0 i Theore 3.5 we get G i K L G K L i G 0 K L i equality holds if K ad L are dilates. Fro 19 we have i.e. G i K L G L i G K i G i K L G K i G L i equality holds if K ad L are dilates. ii For i > let i j k 0 i i Theore 3.5 we get G 0 K L i G i K L G K L i equality holds if K ad L are dilates. Fro 19 we have i.e. G K i G i K L G L i G i K L G K i G L i equality holds if K ad L are dilates. iii For 0 < i < let i j k 0 i i Theore 3.5 we get G 0 K L i G K L i G i K L equality holds if K ad L are dilates. Fro 19 we have iequality 22. iv For i 0 or i by 19 iequality 21 or 22 is idetical. Takig L B i Theore 3.6 usig 17 ad G B ω we iediately obtai: Corollary 3.2. For K L K o i R the for i < 0 or i > for 0 < i < G i K ω i G K i 23 G i K ω i G K i. 24 Equality of each iequality holds if K is a ball with cetroid at the origi. For i 0 or i above iequalities are idetical. Zhu Li ad Zhou [39] established the Blaschke-Sataló type iequality for L p -geoiial surface area. It states that if K K o p 1 the G p KG p K ω 2 25 with equality if ad oly if K is a ellipsoid. The we ca prove the followig Blaschke-Sataló type iequality for Orlicz geoiial surface area. Theore 3.7. Let K Kc ad Φ the G KG K ω 2. Whe is strictly covex equality holds if ad oly if K is a elliposoid. Proof. Fro the defiitio of Orlicz geoiial surface area 4 ad Orlicz Mikowski iequality 9 we have V K 1 ω G K V Q V K V K Q V Q V K QV Q K. 1 V Q Sice K K o takig Q K ad together with the equatio 8 ad the Blaschke-Sataló iequality 5 we get i.e. Siilarly ω G K V KV K ω 2 Cobiig 26 ad 27 we get G K ω. 26 G K ω. 27 G KG K ω Suppose that is strictly covex. By the equality coditios of 9 ad 5 equality holds i 28 if ad oly if K is a ellipsoid. Whe t t p with p 1 the above Blaschke- Sataló type iequality 28 for Orlicz geoiial surface area reduces to the Blaschke-Sataló type iequality 25 for L p -geoiial surface area. The the followig siilar results of the ith Orlicz ixed geoiial surface area ca be established. Theore 3.8. If K L Kc ad 0 i the G i K LG i K L ω 2. Advace olie publicatio: 26 August 2016

6 IAENG Iteratioal Joural of Applied Matheatics 46:3 IJAM_46_3_16 For 0 < i < equality holds if K ad L are ellipsoids with dilates of each other. For i 0 or i the equality of above iequality holds if K or L is a ellipsoid. Proof. For 0 < i < by 22 ad Theore 3.7 we get That is G i K L G i K L [G KG K ] i [G LG L ] i ω 2. G i K LG i K L ω 2 equality holds if K ad L are ellipsoids with dilates of each other. Based o 19 ad 22 we ca see Theore 3.8 is obviously for i 0 or i ad equality holds if K or L is a ellipsoid. Theore 3.9. If K L K c the i 0 i G i KG i K ω 2 ii i G i KG i K ω 2 Proof. i By Theore 3.8 lettig L B we get By 17 we get G i K BG i K B ω 2. G i KG i K ω 2 ii For all i by iequality 23 we have i Gi K G K. G i B G B Fro Theore 3.7 ad G B ω we obtai Gi KG i K G KG K i G i B 2 G B 2 1. That is G i KG i K ω 2 Now we establish the geeralized isoperietric iequalities for G i K. Theore If K L Kc the i 0 i 1 G i K i V K G i B V B ω V K ii i 1 G i K i V K G i B V B ω V K Proof. i For i 0 by 18 ad 10 we have 1 V K G K G B V B ω V K ω V B 1 V K 1 V B ω V K For i by the equality holds trivially. For 0 < i < by 24 we get i G K Gi K G i B G B V K V B ω V K 1 i ii For i by the equality holds trivially. For i > by 23 we get i G K Gi K G i B We coplete the proof. G B V K V B ω ACKNOWLEDGMENT V K 1 i. The referee of this paper proposed ay very valuable coets ad suggestios to iprove the accuracy ad readability of the origial auscript. We would like to express our ost sicere thaks to the aoyous referee. REFERENCES [1] C. M. Petty Geoiial surface area Geoetry Dedicata vol. 3 o. 1 pp [2] E. Lutwak Volue of ixed bodies Trasactios of the Aerica Matheatical Society Vol. 294 o. 2 pp [3] W. J. Firey p-eas of covex bodies Matheatica Scadiavica vol. 10 o. 1 pp [4] E. Lutwak The Bru-Mikowski-Firey theory I: Mixed volues ad the Mikowski proble Joural Difieretial Geoetry vol. 38 o. 1 pp [5] E. Lutwak The Bru-Mikowski-Firey theory II: Affie ad geoiial surface areas Advaces i Matheatics vol. 118 o. 2 pp [6] T. Y. Ma ad W. D. Wag Soe Iequalities for Geeralized - ixed Affie Surface Areas IAENG Iteratioal Joural of Applied Matheatics vol. 45 o. 4 pp [7] T. Y. Ma ad Y. B. Feg Soe Iequalities for p-geoiial Surface Area ad Related Results IAENG Iteratioal Joural of Applied Matheatics vol. 46 o. 1 pp [8] E. Lutwak D. Yag ad G. Zhag L p affie isoperietric iequalities Joural Difieretial Geoetry vol. 56 o. 1 pp [9] W. D. Wag ad Y. B. Feg A geeral L p-versio of Prttys affie projectio iequality Taiwaese Joural of Matheatics vol. 17 o. 2 pp [10] W. D. Wag ad G. S. Leg L p-ixed affie surface areas Joural of Matheatical Aalysis ad Applicatios vol. 335 o. 1 pp [11] E. Werer O L p-affie surface areas Idiaa Uiversity Matheatics Joural vol. 56 o. 5 pp [12] E. Werer ad D. Ye New L p-affie isoperietric iequalities Advaces i Matheatics vol. 218 o. 6 pp [13] E. Werer ad D. Ye Iequalities for ixed p-affie surface area Matheatische Aale vol. 347 o.3 pp Advace olie publicatio: 26 August 2016

7 IAENG Iteratioal Joural of Applied Matheatics 46:3 IJAM_46_3_16 [14] J. Yua S. J. Lv ad G. S. Leg The p-affie surface area Matheatical Iequalities Applicatios vol. 10 o. 3 pp [15] D. Ye L p -geoiial surface area ad related iequalities Iteratioal Matheatics Research Notices vol o. 9 pp [16] D. Ye B. C. Zhu ad J. Z. Zhou The ixed L p -geoiial surface areas for ultiple covex bodies to be published. [17] B. C. Zhu J. Z. Zhou ad W. X. Xu Affie isoperietric iequalities for L p -geoiial surface area i Proceedigs of ICMSC 2014 Spriger [18] E. Lutwak D. Yag ad G. Zhag Orlicz projectio bodies Advaces i Matheatics vol. 223 o. 1 pp [19] E. Lutwak D. Yag ad G. Zhag Orlicz cetroid bodies Joural Difieretial Geoetry vol. 84 o. 2 pp [20] R. J. Garder D. Hug ad W. Weil The Orlicz-Bru-Mikowski theory: a geeral fraework additios ad iequalities Joural of Differetial Geoetry vol. 97 o. 3 pp [21] F. W. Che J. Z. Zhou ad C. L. Yag O the reverse Orlicz Busea-Petty cetroid iequality Advaces i Applied Matheatics vol. 47 o. 4 pp [22] C. Haberl E. Lutwak D. Yag ad G. Zhag The eve Orlicz Mikowski proble Advaces i Matheatics Vol. 224 o. 6 pp [23] Q. Z. Huag ad B. W. He O the Orlicz Mikowski proble for polytopes Discrete ad Coputatioal Geoetry Vol. 48 o. 2 pp [24] A. J. Li ad G. S. Leg A ew proof of the Orlicz Busea- Petty cetroid iequality Proceedigs of the Aerica Matheatical Society Vol. 139 o. 4 pp [25] D. M. Xi H. L. Ji ad G. S. Leg The Orlicz Bru-Mikowski Theory Advaces i Matheatics vol. 260 o. 1 pp [26] D. Ye New Orlicz Affie Isoperietric Iequalities Joural of Matheatical Aalysis ad Applicatios vol. 427 o. 2 pp [27] G. X. Zhu The Orlicz cetroid iequality for star bodies Advaces i Applied Matheatics vol. 48 o. 2 pp [28] D. Zou G. Xiog Orlicz-Legedre ellipsoids The Joural of Geoetric Aalysis pp [29] S. F. Yua H. L. Ji ad G. S. Leg Orlicz geoiial surface areas Matheatical Iequalities ad Applicatios vol. 18 o. 1 pp [30] B. C. Zhu J. Z. Zhou ad W. X Xu L p -ixed geoiial surface area Joural of Matheatical Aalysis ad Applicatios vol. 422 o. 2 pp [31] R. J. Garder Geoetric toography secod editio Cabridge Uiversity Press New York [32] R. Scheider Covex bodies: The Bru-Mikowski theory Cabridge Uiversity Press Cabridge [33] M. Meyer ad A. Pajor O Satalós iequality i Geetric aspects of fuctioal aalysis Jora Lidestrauss ad Vitali D. Mila Eds. Spriger Lecture Notes i Matheatics vol pp Spriger-Verlag New York/Berli [34] M. Meyer ad A. Pajor O the Blaschke-Satalós iequality Archiv Der Matheatik vol. 55 o. 1 pp [35] R. J. Garder The Bru-Mikowski iequality Bulleti of the Aerica Matheatical Society vol. 39 o. 3 pp [36] H. Federer Geoetric Measure Theory Spriger-Verlag New York [37] G. H. Hardy J. E. Littlewood ad G. Pólya iequalities Cabridge Uiversity Press Lodo [38] M. Ludwig C. Schütt ad E. Werer Approxiatio of the Euclidea ball by polytopes Studia Matheatica Vol. 173 o. 1 pp [39] B. C. Zhu N. Li ad J. Z. Zhou Isoperietric iequalities for L p - geoiial surface area Glasgow Matheatical Joural vol. 53 o. 3 pp Advace olie publicatio: 26 August 2016