Inequalities for Dual Orlicz Mixed Quermassintegrals.

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Advaces Pue Mathematcs 206 6 894-902 http://wwwscpog/joual/apm IN Ole: 260-0384 IN Pt: 260-0368 Iequaltes fo Dual Olcz Mxed Quemasstegals jua u chool of Mathematcs ad Computatoal cece Hua Uvesty of

1 Advaces Pue Mathematcs IN Ole: IN Pt: Iequaltes fo Dual Olcz Mxed Quemasstegals jua u chool of Mathematcs ad Computatoal cece Hua Uvesty of cece ad Techology Xagta Cha How to cte ths pape: u J (206 Iequaltes fo Dual Olcz Mxed Quemasstegals Advaces Pue Mathematcs Receved: Octobe Accepted: Novembe Publshed: Novembe Abstact I ths pape we establsh the dual Olcz-Mkowsk equalty ad the dual Olcz- Bu-Mkowsk equalty fo dual Olcz mxed quemasstegals eywods ta Body Olcz Radal um Dual Olcz Mxed Volume Copyght 206 by autho ad cetfc Reseach Publshg Ic Ths wok s lcesed ude the Ceatve Commos Attbuto Iteatoal cese (CC BY 40 Ope Access Itoducto Recetly Covex Geomety Aalyss has made geat achevemet Olcz space (see []-[4] Zhu Zhou ad Xu [2] defed the Olcz adal sum ad dual Olcz mxed volumes et be the set of covex ad stctly deceasg fuctos : ( 0 ( 0 such that lmt ( t = 0 lm t 0 ( t = ad ( 0 = et ad be two sta bodes about the og ad ab 0; the Olcz adal sum a b was defed by [3] ρ ρ ρa b sup t 0 : a b ( u = > ( t t = of the Olcz adal sum s the p hamoc adal sum whch was defed by utwak (see [5] et f deote the ght devatve of a eal-valued fucto f Fo thee s < 0 because s covex ad stctly deceasg The dual Olcz mxed volume V ( s defed by V ( V ( V ( = lm (2 0 p The case ( t t ( p I ths pape we wll defe the dual Olcz mxed quemasstegal DOI: 04236/apm Novembe 7 206

2 J u = 0 by ( ( ( = lm (3 0 The ma pupose of ths pape s to establsh the dual Olcz-Mkowsk equalty ad the dual Olcz-Bu-Mkowsk equalty fo dual Olcz mxed quemasstegals Theoem et ad be two sta bodes about the og ad If 0 < the ( ( ( ( wth equalty f ad oly f ad ae dlates of each othe Theoem 2 et ad be two sta bodes about the og 0 < the ( ( ( ( ( (4 ad If (5 wth equalty f ad oly f ad ae dlates of each othe Ths pape s ogazed as follows: I ecto 2 we toduce above teelated otatos ad the backgoud mateals ecto 3 cotas the poofs of ou ma esults 2 Notato ad Backgoud Mateal The adal fucto : [ 0 ρ u of a compact sta-shaped about the og s defed fo u by If ρ { } ρ u = max λ 0 : λu (2 s postve ad cotuous the s called a sta body about the og The set of sta bodes about the og s deoted by 0 Obvously fo If ρ ρ 0 s depedet of ρ ρ u u u (22 u the we say sta bodes ad ae dlates of each othe If 0 ( = 2 m ad λ ( = 2 m ae oegatve eal umbes the the volume of λ λmm s a homogeeous polyomal of degee λ gve by V λ λmm = V λ λ whee the sum s take ove all -tuples ( of postve teges ot exceedg m 895

3 J u depeds oly o the bodes ad s uquely detemed by the above detty t s called the dual mxed volume of Moe explctly the dual mxed volume V ( has the followg tegal epesetato [6]: The coeffcet V ( ( ρ ρ V = u u d u (23 whee s the ebesgue measue o The coeffcets V ( ae oegatve symmetc ad mootoe (wth espect to set cluso They ae also multlea wth espect to the adal sum ad V ( = V ( et = = = ad = = = the the dual mxed volume V ( s usually wtte as V ( If = B the V ( B Fo 0 the dual mxed quemasste- s the dual quemasstegal deotes the dual mxed volume gal ( the ( = ( The dual mxed quemasstegal ( has the followg tegal epesetato: whee s the ebesgue measue o V B B Fo = ( = ρ ρ d (24 By usg the Mkowsk s tegal equalty we ca obta the dual Mkowsk equalty fo dual mxed quemasstegals: If 0 ad 0 < the (25 equalty holds f ad oly f ad ae dlates of each othe uppose that µ s a pobablty measue o a space X ad g: X I s a µ- tegable fucto whee I s a possbly fte teval Jesse s equalty states that f : X I s a covex fucto the ( g( x d µ ( x g( x d µ ( x (26 X If s stctly covex equalty holds f ad oly f g( x s a costat fo µ-almost all x X (see [7] 3 Ma Results et 0 ad Fo = 0 the dual Olcz mxed quemasste- s defed by gal X ρ ( = ρ d ρ (3 Fo = the ( = ( The case 0 mxed quemasstegal ( s the dual Olcz mxed volume V ( was defed by Zhu Zhou ad Xu [2] = of the dual Olcz whch 896

4 J u s mootoe wth espect to set cluso Poof et 2 0 ad 2 By (3 (22 ad the fact that s stctly Coollay 3 The dual Olcz mxed quemasstegal ( deceasg o ( 0 we have emma 3 [2] et 0 ad ρ ρ ρ 2 ρ ( ( = ρ u d u ρ = a 2 u ρ ρ b = t t d If the f ad oly f ρ u t a b = emma 32 [2] et 0 ad The 0 ρ u u ρ ρ u ρ u lm = ρ (32 ufomly fo all u Theoem 3 et 0 ad Fo = 0 the 0 ( ρ lm = ρ d ( ρ Poof uppose > 0 0 ad u Note that as 0 (see [2] By emma 32 t follows that ufomly o Hece ( ρ ρ u ρ u ρ u ρ u lm = ρ lm 0 = 0 0 u ρ u = ρ ( ( u ( u ρ ρ = lm lm d 0 0 ρ = lm 0 ρ = ρ ρ ρ d u d 897

5 J u e complete the poof of Theoem 3 Fom (3 ad Theoem 3 we have Fo 0 ( ( = lm (33 0 sce ρ d = ( ty measue o the d ( ρ Poof of Theoem By (3 (26 (25 ad the fact that s deceasg o ( 0 we obta ( ρ u ( ( = ρ ρ d ρ ρ d( u ( ρ ( = ( ( = s a pobabl- Ths gves the desed equalty ce s stctly deceasg fom the equalty codto of the dual Mkowsk equalty (25 we have that ad ae dlates of each othe Covesely whe = λ by (3 we have ( ( = ( ( λ = ( ( The followg uqueess s a dect cosequece of the dual Olcz-Mkowsk equalty (4 Coollay 32 uppose ad 0 such that Fo 0 < f M = M fo all M (34 o M M = fo all M (35 the = 898

6 J u Poof uppose (34 holds If we take fo M the fom (3 we obta ( ( ( = = Hece fom the dual Olcz-Mkowsk equalty (4 we have ( ( ( wth equalty f ad oly f ad ae dlates of each othe ce s stctly deceasg o ( 0 we have wth equalty f ad oly f ad ae dlates of each othe If we take fo M we smlaly have ( ( Hece ( = ( ad fom the equalty codto we ca coclude that ad ae dlates of each othe Howeve sce they have the same volume they must be equal Next suppose (35 holds If we take fo M the fom (3 we obta ( = = The fom the dual Olcz-Mkowsk equalty (4 we have ( ( ( wth equalty f ad oly f ad ae dlates of each othe ce s stctly deceasg o ( 0 we have wth equalty f ad oly f ad ae dlates of each othe If we take fo M we smlaly have ( ( Hece ( = ( ad fom the equalty codto we ca coclude that ad ae dlates of each othe Howeve sce they have the same volume they must be equal Fom the dual Olcz-Mkowsk equalty we wll pove the followg dual Olcz-Bu-Mkowsk equalty whch s moe geeal tha Theoem 2 Theoem 32 et 0 ab> 0 ad If 0 < the ( ( ( a b ( a b ( a b wth equalty f ad oly f ad ae dlates of each othe Poof et = a b Fom (23 emma 3 ad (4 t follows that 899

7 J u = ρ d ( ρ ρ = a b ρ u u ϕ ( ρ ρ ( u ( ( d a ρ b ρ = ρ d ρ ( ( a b = d u u u u ρ ρ ( a b ( ( By the equalty codto of the dual Olcz-Mkowsk equalty (4 equalty (36 holds f ad oly f ad ae dlates of each othe Ideed we also ca pove the dual Olcz-Mkowsk equalty by the dual Olcz- Bu-Mkowsk equalty Poof Fo 0 let = Note that as 0 By the dual Olcz-Bu-Mkowsk equalty the followg fucto G ( ( ( = ( ( s o-postve Obvously G ( 0 = 0 Thus O the othe had we have 0 ( G( 0 G lm 0 (37 ( ( ( ( G( G( 0 lm = lm ( ( ( ( = lm 0 ( 0 0 ( ( ( = lm ( lm 0 0 ( ( ( (38 900

8 J u et s ( = ( By (33 we have ad ote that s as 0 Cosequetly ( ( ( ( s lm = lm = ( s ( ( 0 s lm ( ( 0 = lm lm 0 0 lm 0 ( ( ( ( = ( = Fom (38 (39 ad (30 t follows that 0 G( G( 0 ( ( lm = ( ( Combg (37 ad (3 we have ( ( 0 ( ( Theefoe the equalty (32 holds f ad oly f G G (39 (30 (3 (32 = 0 = 0 ths mples that ad ae dlates of each othe Remak 3 The case = 0 of Theoem ad Theoem 2 wee establshed by Zhu Zhou ad Xu [2] The dual foms of Theoem ad Theoem 2 wee establshed by Xog ad Zou [] Refeeces [] Che F Zhou J ad Yag C (20 O the Revese Olcz Busema-Petty Cetod Iequalty Advaces Appled Mathematcs [2] Gade RJ Hu D ad el (204 The Olcz-Bu-Mkowsk Theoy: A Geeal Famewok Addtos ad Iequaltes Joual of Dffeetal Geomety [3] Gade RJ Hug D el ad Ye D (205 The Dual Olcz-Bu-Mkowsk 90

9 J u Theoy Joual of Mathematcal Aalyss ad Applcatos [4] Habel C utwak E Yag D ad Zhag G (200 The Eve Olcz Mkowsk Poblem Advaces Mathematcs [5] Huag Q ad He B (202 O the Olcz Mkowsk Poblem fo Polytopes Dscete & Computatoal Geomety [6] A ad eg G (20 A New Poof of the Olcz Busema-Petty Cetod Iequalty Poceedgs of the Ameca Mathematcal ocety [7] udwg M (200 Geeal Affe uface Aeas Advaces Mathematcs [8] utwak E Yag D ad Zhag G (200 Olcz Pojecto Bodes Advaces Mathematcs [9] utwak E Yag D ad Zhag G (200 Olcz Cetod Bodes Joual of Dffeetal Geomety [0] X D J H ad eg G (204 The Olcz-Bu-Mkowsk Iequalty Advaces Mathematcs [] Xog G ad Zou D (204 Olcz Mxed Quemasstegals cece Cha Mathematcs [2] Zhu B Zhou J ad Xu (204 Dual Olcz-Bu-Mkowsk Theoy Advaces Mathematcs [3] Zhu G (202 The Olcz Cetod Iequalty fo ta Bodes Advaces Appled Mathematcs [4] Zou D ad Xog G (204 Olcz-Joh Ellpsods Advaces Mathematcs [5] utwak E (996 The Bu-Mkowsk-Fey Theoy II: Affe ad Geommal uface Aeas Advaces Mathematcs [6] utwak E (975 Dual Mxed Volumes Pacfc Joual of Mathematcs [7] Hady GH ttlewood JE ad PO lya G (988 Iequaltes 2d Edto Cambdge Uvesty Pess Cambdge 902

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ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE

O The Covegece Theoems... (Muslm Aso) ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE Muslm Aso, Yosephus D. Sumato, Nov Rustaa Dew 3 ) Mathematcs