Dual Orlicz–Brunn–Minkowski theory Advances in Mathematics

Contents lists available atScienceDirect

Advances in Mathematics

www.elsevier.com/locate/aim

Dual Orlicz–Brunn–Minkowski theory

Baocheng Zhu^a,b,1, Jiazu Zhou^a,c,∗,2,Wenxue Xu^a

a SchoolofMathematicsandStatistics,SouthwestUniversity,Chongqing400715, People’sRepublicofChina

b DepartmentofMathematics,HubeiUniversityforNationalities,Enshi,Hubei 445000,People’sRepublicofChina

cCollegeofMathematicalScience,KailiUniversity,Kaili,Guizhou556000, People’sRepublicofChina

a r t i c l e i n f o a bs t r a c t

Article history:

Received19September2013 Accepted14July2014 Availableonlinexxxx

CommunicatedbyErwinLutwak

MSC:

52A20 52A40 53A15

Keywords:

Starbody Orliczradialsum DualOrliczmixedvolume DualOrlicz–Brunn–Minkowski theory

DualOrlicz–Minkowskiinequality DualOrlicz–Brunn–Minkowski inequality

In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumesareintroduced.ThedualOrlicz–Minkowskiinequality andthedual Orlicz–Brunn–Minkowskiinequality areestab- lished.Thevariationalformulaforthevolumewithrespectto theOrliczradialsumisproved.Theequivalencebetweenthe dualOrlicz–MinkowskiinequalityandthedualOrlicz–Brunn–

Minkowski inequality is demonstrated. Orlicz intersection bodies aredefinedandtheOrlicz–Busemann–Pettyproblem isposed.

© 2014ElsevierInc.All rights reserved.

* Corresponding author at: School of Mathematics and Statistics, Southwest University, Chongqing 400715,People’sRepublicofChina.

E-mailaddresses:zhubaocheng814@163.com(B. Zhu),zhoujz@swu.edu.cn(J. Zhou),xwxjk@163.com (W. Xu).

1 Supported in part by the Fundamental Research Funds for the Central Universities (No.

XDJK2013D022).

2 Supported inpartbyNSFC(No.11271302)andthePh.D.ProgramofHigherEducationResearchFund (No.2012182110020).

http://dx.doi.org/10.1016/j.aim.2014.07.019 0001-8708/© 2014ElsevierInc.All rights reserved.

1. Introduction

TheclassicalBrunn–Minkowskitheory,alsoknownasthetheoryofmixedvolumes,is thecoreofconvex geometric analysis.Itoriginatedwith Minkowskiwhenhe combined hisconceptofmixedvolumewiththeBrunn–Minkowskiinequality.OneofMinkowski’s majorcontributions tothetheory wasto showhow histheorycouldbe developedfrom afewbasicconcepts,suchassupportfunction,vectoraddition,andvolume.

Duringthelasttwo decades,theBrunn–Minkowskitheoryhasbeen extendedtothe Lp Brunn–Minkowskitheory,whichcombinesvolumeand ageneralizedvector addition ofcompactconvexsetsintroducedbyFirey intheearly1960s (see[10])andnowcalled L_paddition.ThirtyyearsafterFieryintroducedhisnewL_paddition,thenewL_pBrunn–

Minkowskitheorywasborn inLutwak’spapers [37,38]and hassincewitnessedarapid growth.See[4,5,7,22,39,41,45,51,52]foradditionalreferences.

Recently, progress towards an Orlicz–Brunn–Minkowski theory was made by Lut- wak, Yang and Zhang [43,44] and Ludwig [32]. This theory is far more general than theL_pBrunn–Minkowskitheory,wereferthereaderto[6,16,24,26,29,32,40,43,44,46,53, 57].TheOrliczextensionwasmotivatedbyconceptswithintheasymmetricL_p Brunn–

Minkowskitheorydevelopedby,e.g.,Ludwig[30],HaberlandSchuster[23],andLudwig and Reitzner [33]. As part of the new Orlicz–Brunn–Minkowski theory, Lutwak, Yang andZhangestablished twofundamentalaffine inequalities,theOrliczBusemann–Petty centroid inequality [44] and the Orlicz Petty projection inequality [43]. The concepts of the Orlicz–Brunn–Minkowski theory are much more general than those of the Lp

Brunn–Minkowskitheory.

Duringthelastthree decades,convexgeometric analysishasachieved importantde- velopments. Lutwak’s dual Brunn–Minkowski theory, introduced in the 1970s, helped achievingamajorbreakthrough inthe solutionoftheBusemann–Petty probleminthe 1990s. In contrast to the Brunn–Minkowski theory, in the dual theory, convex bodies arereplaced bystar-shaped sets, andprojections ontosubspacesarereplaced byinter- sectionswithsubspaces.Themachineryofthedualtheoryincludesdualmixedvolumes andimportantauxiliarybodiesknownas intersectionbodies(see[11–13,18,27,28,35,36, 54–56]).

LpharmonicradialcombinationofconvexbodieswereinvestigatedbyFirey(see[8,9]).

Then,LutwakextendedtheLp harmonicradialcombinationtostarbodiesin[38].IfK andLarestarbodies(seeSection2forprecisedefinitionandunexplainedterminology), and a,b ≥ 0 (not both zero), then for p ≥ 1, the L_p harmonic radial combination, a·K+_pb·L isastarbodyanddefinedby(see[38])

ρ(a·K+_pb·L, u)^−p=aρ(K, u)^−p+bρ(L, u)^−p, u∈Sⁿ⁻¹, (1.1) where thefunction ρis theradial functionof theset involved. In[38],Lutwak showed thatassociated withL_p harmonicradialcombinationthere wereintegralrepresentation andinequalitiesfortheLpdualmixedvolume.SincethenanLpdualBrunn–Minkowski

theory hasbeen developed(see [20,31,33,38,42,56]).Motivatedbythemanner inwhich Lutwak, Yang and Zhang extended the Lp Brunn–Minkowski theory to the Orlicz–

Brunn–Minkowski theory, we extend the dual Lp Brunn–Minkowski theory to a dual Orlicz–Brunn–Minkowski theory.

Todoso,wedefineanOrliczradialsumanddualOrliczmixedvolumesasextensions of dual L_p mixed volumes. Considerconvex function φ : (−∞,0)∪(0,+∞) → (0,∞) such that lim_t→∞φ(t) = 0, lim_t→0φ(t) = ∞, and set φ(0) = ∞. This means thatφ mustbeincreasingon(−∞,0) anddecreasingon(0,∞).Wewillassumethatφiseither strictlyincreasingon(−∞,0) orstrictlydecreasingon(0,∞) throughoutthispaper.Let Cdenotetheclassofallsuchconvexfunctionsφ.LetC⁺betheclassofconvexandstrictly decreasing functionsφ: (0,∞)→ (0,∞) such thatlim_t→∞φ(t)= 0, lim_t→0φ(t)=∞, and φ(0)=∞.

For φ ∈ C⁺, we define the Orlicz radial sum K+φL of two star bodies about the originK andLinRⁿ,by

ρ(K+φL, u) = sup

t >0 :φ

ρ(K, u) t

+φ

ρ(L, u) t

≤φ(1)

, u∈Sⁿ⁻¹. (1.2) We will show thatthe Orlicz radialsum reduces to the Lp harmonicradial sumwhen φ(t)=t^−p inSection3.

Wealso definethedual OrliczmixedvolumeV_φ(K,L),and provethe followingdual Orlicz–Minkowski inequality:Forφ∈ C⁺,ifK,L∈ Soⁿ (see Section2forprecise defini- tion),then

V_φ(K, L)≥V(K)φ

V(L) V(K)

_n¹ ,

where V(·) isthevolumefunctionofthesetinvolved.

InSection5,wewillestablishOrliczradialsumversionsofthedualBrunn–Minkowski inequality forstar bodiesincluding theequality conditions. Forexample, thefollowing dual Orlicz–Brunn–Minkowski inequality:

φ(1)≥φ

V(K) V(K+_φL)

_n¹ +φ

V(L) V(K+_φL)

_n¹

, (1.3)

withequalityifandonlyifKandLaredilatesofeachother.Hereφ∈ C⁺andV denotes volume.

When φ(t) = t^−p, with p≥ 1. Theabove dual Orlicz–Brunn–Minkowski inequality reduces toLutwak’sLpdual Brunn–Minkowskiinequality(see[38]):

V(K+_φL)^−np ≥V(K)^−np +V(L)^−pn, (1.4) with equalityifand onlyifK andL aredilatesofeachother.

This paperis organizedas follows. In Section2, wecollect somebasicconcepts and various facts that will be used in the proofs of our results. In Section 3, the Orlicz radialsumis introduced, and someofits basicpropertiesare shown.The definitionof thedual Orlicz mixedvolumeisgiven inSection4. InSection5,we establishthedual Orlicz–Brunn–MinkowskiinequalityandthedualOrlicz–Minkowskiinequality.Weview theseinequalitiesascentraltothedualOrlicz–Brunn–MinkowskiTheory.InSection6,we introduceOrliczintersectionbodiesandproposetheOrlicz–Busemann–Pettyproblem.

2. Preliminaries

The unit ball in Rⁿ and its surface are denoted by B and Sⁿ⁻¹, respectively. The volume of the unit ball B is denoted by ωn = π^n/2/Γ(1+n/2), where Γ(·) is the Gammafunction.Wewrite V(K) forthevolumeofthecompactsetK inRⁿ.

ForA∈GL(n) writeA^tforthetransposeofAandA^−tfortheinverseofthetranspose ofA.Write|A| fortheabsolute valueof thedeterminantofA.

Wesaythatthesequence{φi},ofφi∈ C⁺,issuchthatφi→φ0∈ C⁺ provided

|φi−φ0|I := max

s∈Iφi(s)−φ0(s)→0, foreverycompactinterval I⊂R.

Theradialfunctionρ_K=ρ(K,·):Rⁿ\{0}→[0,∞),ofacompactstar-shaped(about theorigin)K⊂Rⁿ,isdefinedby(see[14,50])

ρ(K, x) = max{λ≥0 :λx∈K}.

If ρ_K is positive and continuous, then K is called a star body (about the origin).

WriteSoⁿ forthesetofstarbodiesabouttheorigininRⁿ.TwostarbodiesKandLare dilates(ofoneanother)ifρK(u)/ρL(u) isindependentofu∈Sⁿ⁻¹.Ifs>0,wehave

ρ(sK, x) =sρ(K, x), for allx∈Rⁿ\ {0}. (2.1) Moregenerally,from thedefinitionoftheradialfunctionitfollowsimmediatelythat for A∈GL(n) the radialfunctionof the image AK ={Ay :y ∈K}of K is given by (see[14,50])

ρ(AK, x) =ρ

K, A⁻¹x , for allx∈Rⁿ. (2.2) ForK∈ Soⁿ,define therealnumbersRK andrK by

R_K = max

u∈Sⁿ⁻¹ρ_K(u) and r_K= min

u∈Sⁿ⁻¹ρ_K(u). (2.3) Notethat0< rK < RK <∞, forallK∈ Soⁿ.

Obviously, forK,L∈ Soⁿ,

K⊂L if and only if ρK ≤ρL. (2.4) TheradialHausdorffmetricbetweenthestarbodiesKand Lis

δ(K, L) = max

u∈Sⁿ⁻¹

ρK(u)−ρL(u). A sequence{K_i}ofstarbodiesissaidtobeconvergentto Kif

δ(K _i, K)→0, asi→ ∞.

Therefore, asequenceofstarbodiesKiconvergesto Kifandonlyifthesequenceof radialfunctionsρ(Ki,·) convergesuniformlyto ρ(K,·) (see[49,Theorem 7.9]).

The radialMinkowskiaddition and scalarproduct of sets Qand T inRⁿ is defined by(see[14,35])

aQ+bT ={ax+by:x∈Q, y∈T}, for alla, b∈R. IfK,L∈ Soⁿ anda,b≥0,aK+bL canbe definedasastar bodysuchthat

ρ_aK+bL(u) =aρ_K(u) +bρ_L(u), for allu∈Sⁿ⁻¹. (2.5) Thepolarcoordinateformulaforthevolumeof acompactsetK is

V(K) = 1 n

Sⁿ⁻¹

ρⁿ_K(u)dS(u), (2.6)

where S istheLebesguemeasureonSⁿ⁻¹(i.e.,the(n−1)-dimensionalHausdorffmea- sure).

Thefirstdual mixedvolume,V₁(K,L),ofK,L∈ Soⁿ,isdefinedby nV1(K, L) = lim

ε→0

V(K+εL)−V(K)

ε (2.7)

In [34] Lutwak proved the following integral representation for the first dual mixed volume:IfK,L∈ Soⁿ,then

V1(K, L) = 1 n

Sⁿ⁻¹

ρⁿ_K⁻¹(u)ρL(u)dS(u). (2.8)

This integral representation, together with the Hölderinequality (see [25]), and the polarcoordinateformula,immediatelyleads tothefollowingdualMinkowski inequality

forthefirstdualmixedvolume(see[35]):IfK,L∈ Soⁿ,then

V1(K, L)ⁿ ≤V(K)ⁿ⁻¹V(L), (2.9) withequalityifandonlyifKand Laredilates.

From the dual Minkowski inequality, we can obtain the dual Brunn–Minkowski in- equality(see[35]):IfK,L∈ Soⁿ anda,b≥0 then

V(aK+bL)ⁿ¹ ≤aV(K)¹ⁿ+bV(L)ⁿ¹, (2.10) withequalityifandonlyifKand Laredilates.

ForK,L∈ S0ⁿ,p≥1,theLp dualmixedvolumeV₋p(K,L) ofKandL isdefinedby

−n

pV_−p(K, L) = lim

ε→0

V(K+_pε·L)−V(K)

ε . (2.11)

In[38]LutwakalsoprovedthefollowingintegralrepresentationfortheLpdualmixed volume:ForK,L∈ S0ⁿ,andp≥1,

V_−p(K, L) = 1 n

Sⁿ⁻¹

ρ^n+p_K (u)ρ⁻_L^p(u)dS(u). (2.12)

Suppose that μ is a probability measure on a space X and g : X → I ⊂ R is a μ-integrable function,where I is apossibly infiniteinterval. Jensen’sinequality states thatifφ:I→Ris aconvexfunction,then

X

φ

g(x) dμ(x)≥φ

X

g(x)dμ(x)

. (2.13)

Ifφ is strictly convex, equality holds ifand only ifg(x) isconstantfor μ-almost all x∈X (see[25]).

3. Orliczradialsum

WefirstdefinetheOrliczradialsum.

Definition3.1.LetK,L∈ Soⁿ withradialfunctionsρK,ρL,respectively.Fora,b≥0 (not bothzero)andφ∈ C⁺,definetheOrlicz radialsuma·K+_φb·Lof Kand Lby

ρ_a·K+

φb·L(u) = sup

t >0 :aφ

ρ_K(u) t

+bφ

ρ_L(u) t

≤φ(1)

,

u∈Sⁿ⁻¹. (3.1)

From (2.2)andthedefinitionof Orliczradialsum,wehave:

Proposition 3.1. SupposeK,L∈ Soⁿ,anda,b≥0.If φ∈ C⁺,thenforA∈GL(n), A(a·K+φb·L) =a·AK+φb·AL. (3.2) Proof. Foru∈Sⁿ⁻¹,by(2.2)

ρ_a·AK+

φb·AL(u) = sup

t >0 :aφ

ρ_AK(u) t

+bφ

ρ_AL(u) t

≤φ(1)

= sup

t >0 :aφ

ρ_K(A⁻¹u) t

+bφ

ρ_L(A⁻¹u) t

≤φ(1)

=ρ_a·K+

φb·L A⁻¹u

=ρ_A(a·K+

φb·L)(u). 2

Since K,L ∈ Soⁿ, 0 < ρ_K < ∞ and 0 < ρ_L < ∞, hence ^ρK_t → 0 and ^ρ_t^L → 0 as t → ∞. By this and the assumption that φ is strictly decreasing in (0,∞), the function

t →aφ ρ_K

t

+bφ ρ_L

t

is strictlyincreasingin(0,∞).Itisalsocontinuous.Thus,we have:

Lemma 3.1.Suppose K,L∈ Soⁿ andu∈Sⁿ⁻¹.If φ∈ C⁺,then

aφ

ρ_K(u) t

+bφ

ρ_L(u) t

=φ(1) if andonly if

ρ_a·K+

φb·L(u) =t.

Remark3.1.Whenφ(t)=t^−p,withp≥1,itiseasytoshowthattheOrliczradialsum reduces toLutwak’sLpharmonicradialcombination(see[38]):

ρ(a·K+φb·L, u)^−p=aρ(K, u)^−p+bρ(L, u)^−p, u∈Sⁿ⁻¹.

If K,L ∈ Soⁿ, let R = max{R_K,R_L} and r = min{r_K,r_L}. For a,b ≥ 0, let C = max{a,b}and c =a+b. Since φ iscontinuous and strictly deceasing in(0,∞), hence theinverseφ⁻¹ isalsocontinuousand deceasingin(0,∞).

Lemma3.2. SupposeK,L∈ Soⁿ.If φ∈ C⁺,then r

φ⁻¹(^φ(1)_2C ) ≤ρ_a·K+

φb·L(u)≤ R φ⁻¹(^φ(1)_c ), forallu∈Sⁿ⁻¹.

Proof. Supposeu∈Sⁿ⁻¹ andletρ_a·K+

φb·L(u)=t.ByLemma 3.1andthefactthatφ isstrictlydeceasingon(0,∞),wehave

φ(1) =aφ

ρK(u) t

+bφ

ρL(u) t

≤Cφ rK

t

+Cφ rL

t

≤2Cφ r

t

. (3.3)

Since the inverseφ⁻¹ of φ is strictly deceasing on(0,∞), we have the lower bound forρ_a·K+

φb·L(u):

t≥ r φ⁻¹(^φ(1)_2C ).

Onthe other hand,from Eq. (3.3), together with theconvexityand the strictly de- creasingon(0,∞) ofφ,wehave

φ(1) a+b = a

a+bφ

ρK(u) t

+ b

a+bφ ρL(u)

t

≥ a a+bφ

RK

t

+ b

a+bφ RL

t

≥φ a

a+bRK+_a+b^b RL

t

≥φ R

t

. Thenweobtaintheupperestimate:

t≤ R

φ⁻¹(^φ(1)_c ). 2

WenowprovethattheOrliczradialsumoftwo starbodiesisagainastarbody.

Lemma 3.3. Supposeφ∈ C⁺ anda,b≥0(not both zero).If K,L∈ Soⁿ,then a·K+φ

b·L∈ Soⁿ.

Proof. Letu0∈Sⁿ⁻¹,for anysubsequence {ui}⊂Sⁿ⁻¹ such thatui →u0 asi→ ∞, we needtoshow

ρ_a·K+

φb·L(ui)→ρ_a·K+

φb·L(u0), asi→ ∞. Let

ρ_a·K+

φb·L(u_i) =t_i. Then Lemma 3.2gives

r

φ⁻¹(^φ(1)_2C) ≤t_i ≤ R φ⁻¹(^φ(1)_c ).

Since K,L ∈ Soⁿ, we have 0 < r_K ≤ R_K < ∞, 0 < r_L ≤ R_L < ∞. Thus, there exist λ,μsuchthat0< λ≤ti ≤μ<∞, foralli. Toshow thatthe boundedsequence {ti} converges to ρ_a·K+

φb·L(u0), we show that every convergent subsequence of {ti} convergestoρ_a·K+

φb·L(u₀).Denoteanarbitraryconvergentsubsequenceof{t_i}by{t_i} as well,and supposethatforthissubsequence

ti→t0.

It isclearthatλ≤t0≤μ.Lemma 3.1andthefactρ_a·K+

φb·L(ui)=ti showthat aφ

ρK(ui) ti

+bφ

ρL(ui) ti

=φ(1).

SinceρKandρLarecontinuousonSⁿ⁻¹,togetherwiththecontinuityofφandti→t0, itfollows that

aφ

ρK(u0) t₀

+bφ

ρL(u0) t₀

=φ(1).

ByLemma 3.1,wehave

t0=ρ_a·K+

φb·L(u0).

This shows

ρ_a·K+

φb·L(u_i)→ρ_a·K+

φb·L(u₀).

Therefore, thecontinuityofρ_a·K+φb·L isprovedanda·K+φb·L∈ Soⁿ. 2

FromthedefinitionoftheOrliczradialsum(3.1),forσ >0,wehave ρ_a·(σK)+

φb·(σL)(u) = sup

t >0 :aφ

ρ_σK(u) t

+bφ

ρ_σL(u) t

≤φ(1)

= sup

σt >0 :aφ

ρ_K(u) t

+bφ

ρ_L(u) t

≤φ(1)

=σρ_a·K+

φb·L(u).

Thisgivesthat

ρ_{a·(σK)+φb·(σL)}(u) =σρ_a·K+φb·L(u). (3.4) Next,weshowthattheOrliczradialsum+_φ:Soⁿ→ Soⁿ iscontinuous.

Lemma3.4.Supposeφ∈ C⁺.IfKi,Li ∈ SoⁿandKi →K∈ Soⁿ,Li→L∈ Soⁿ,asi→ ∞, then

a·K_i+_φb·L_i→a·K+_φb·L, asi→ ∞ forallaandb.

Proof. Supposeu∈Sⁿ⁻¹. Wewill showthat ρ_a·Ki+

φb·Li(u)→ρ_a·K+

φb·L(u), as i→ ∞. (3.5) Let

ρ_{a·Ki+φb·Li}(u) =ti.

WriteR_i= max{R_Ki,R_Li}andr_i= min{r_Ki,r_Li}. ThenLemma 3.2gives r_i

φ⁻¹(^φ(1)_2C )≤ti≤ R_i φ⁻¹(^φ(1)_c ).

SinceK_i →K ∈ Soⁿ and L_i →L∈ Soⁿ, wehaveR_Ki →R_K <∞, R_Li →R_L <∞, and r_Ki →r_K >0, r_Li →r_L >0.By thefact thatthe functionsR = max{R_K,R_L} and r = min{rK,rL}are continuous, we have Ri → R <∞, ri → r > 0.Thus, there existλ,μsuchthat

0< λ≤t_i≤μ <∞, for alli. (3.6) Toshowthattheboundedsequence{t_i}convergestoρ_a·K+

φb·L(u),weshowthatevery convergent subsequence of {ti} converges to ρ_a·K+φb·L(u). Denote an arbitrary con-

vergent subsequence of {ti}by {ti}as well, and suppose that for this subsequencewe have

ti→t0.

It isclearthatλ≤t₀≤μ.LetK_i=t⁻_i ¹K_i andL_i=t⁻_i ¹L_i. SinceK_i→K,L_i →L, and t⁻¹_i →t⁻¹₀ ,wehavet⁻¹_i Ki→t⁻¹₀ K andt⁻¹_i Li→t⁻¹₀ L.

Now (3.4), andthefactρ_a·Ki+

φb·Li(u)=ti,show thatρ_a·K

i+φb·Li(u)= 1.Thatis, aφ

ρ_K

i(u) +bφ ρ_L

i(u) =φ(1), for alli.

Sincet⁻¹_i K_i→t⁻¹₀ Kandt⁻¹_i L_i→t⁻¹₀ L,togetherwiththecontinuityofφ,and(2.1),it follows that

aφ

ρK(u) t₀

+bφ

ρL(u) t₀

=φ(1).

ByLemma 3.1,wehave

t0=ρ_a·K+φb·L(u).

This shows

ρ_a·Ki+

φb·Li(u)→ρ_a·K+

φb·L(u).

Now thepointwiseconvergence(3.5)hasbeen proved.

Wewill showthattheconvergence(3.5)isuniformforany u₀∈Sⁿ⁻¹.

Assumethatρ_{a·Ki+φb·Li}doesnotconvergeuniformlytoρ_a·K+φb·L.Then,thereexists aδ0>0 such that,fori≥N0∈N,

ρ_{a·Ki+φb·Li}(ui)−ρ_a·K+φb·L(ui)≥δ0. (3.7) Since Sⁿ⁻¹ is compact, for some u₀ ∈ Sⁿ⁻¹, there exists asubsequence {u_i} ⊂ Sⁿ⁻¹ suchthatui →u0 asi→ ∞.

From Lemma 3.2, there exist an N0 ∈ Nand positive λ,μ such that(3.6)holds for i≥N₀. Then,thereexists apositives₀suchthat

ρ_a·Ki+

φb·Li(ui)→s0. From (3.7),wehave

s0−ρ_a·K+φb·L(u0)≥δ0.

Thisimplies

s₀=ρ_a·K+

φb·L(u₀). (3.8)

Lets_i=ρ_a·Ki+

φb·Li(u_i).ByLemma 3.1,wehave aφ

ρ_Ki(u_i) si

+bφ

ρ_Li(u_i) si

=φ(1).

This,togetherwith thefactsthatK_i→K,L_i →Lands_i→s₀,gives aφ

ρK(u0) s₀

+bφ

ρL(u0) s₀

=φ(1).

ByLemma 3.1again,wehave

s₀=ρ_a·K+

φb·L(u₀).

Thiscontradictsto (3.8). Therefore, ρ_a·Ki+

φb·Li→ρ_a·K+

φb·L

uniformlyonSⁿ⁻¹ andhence

a·Ki+φb·Li→a·K+φb·L. 2

WewillseethattheOrlicz radialsum+_φ iscontinuousinaandb.

Lemma3.5. Supposeφ∈ C⁺.If ai,bi≥0andai→a,bi→b,asi→ ∞,then ai·K+φbi·L→a·K+φb·L, asi→ ∞,

forallK,L∈ Soⁿ.

Proof. Supposeu∈Sⁿ⁻¹and K,L∈ Soⁿ.Wewillshow that

ρ_{ai·K+φbi·L}(u)→ρ_a·K+φb·L(u), asi→ ∞. (3.9) Let

ρ_ai·K+

φbi·L(u) =t_i. ThenLemma 3.2gives

r φ⁻¹(^φ(1)_2C

i) ≤ti≤ R φ⁻¹(^φ(1)_c

i ).

Since ai → a,bi → b and the facts that the functions Ci = max{ai,bi} and ci = ai+bi arecontinuous, we haveCi →C and ci →c. Since theinverse φ⁻¹ of φ isalso continuous anddeceasing in(0,∞), thereexist λ,μsuchthat0< λ≤ti≤μ<∞,for all i. To show thatthe bound sequence {ti}converges to ρ_a·K+

φb·L(u), we show that every convergent subsequence of {t_i} converges to ρ_a·K+

φb·L(u). Denote an arbitrary convergent subsequenceof {t_i} by{t_i} as well, and suppose that forthis subsequence we have

t_i→t₀. It isclearthat0< λ≤t₀≤μ.Sinceρ_ai·K+

φbi·L(u)=t_i,thatis, a_iφ

ρ_K(u) ti

+b_iφ

ρ_L(u) ti

=φ(1) for alli.

Since a_i→aandb_i →b,together withthecontinuityof φ, andt_i→t₀ itfollowsthat aφ

ρK(u) t₀

+bφ

ρL(u) t₀

=φ(1).

ByLemma 3.1,wehave

t₀=ρ_a·K+

φb·L(u).

This shows

ρ_ai·K+

φbi·L(u)→ρ_a·K+

φb·L(u).

Now thepointwiseconvergence(3.9)hasbeen proved.

To showtheconvergence(3.9)isuniformonSⁿ⁻¹,weassumethatρ_ai·K+

φbi·L does not converge uniformly to ρ_a·K+

φb·L. Then, there exist a positive δ₀ and an N₀ ∈ N suchthat,fori≥N₀,

ρ_ai·K+

φbi·L(u_i)−ρ_a·K+

φb·L(u_i)≥δ₀. (3.10) SinceSⁿ⁻¹iscompact,foru0∈Sⁿ⁻¹,thereexistsasubsequence{ui}⊂Sⁿ⁻¹suchthat u_i→u₀ asi→ ∞.

From Lemma 3.2,thereexist anN₀∈Nand positiveλ,μsuchthat,fori≥N₀, 0< λ≤ρ_ai·K+

φbi·L(ui)≤μ <∞. Then, thereexists apositives0suchthat,fori≥N0,

ρ_{ai·K+φbi·L}(ui)→s0.

From(3.10),wehave

s0−ρ_a·K+φb·L(u0)≥δ0. Thisimplies

s₀=ρ_a·K+

φb·L(u₀). (3.11)

Letsi=ρ_ai·K+

φbi·L(ui).ByLemma 3.1,wehave aiφ

ρK(ui) s_i

+biφ

ρL(ui) s_i

=φ(1).

This,togetherwith thefactsthata_i →a,b_i→bands_i→s₀,gives aφ

ρ_K(u₀) s0

+bφ

ρ_L(u₀) s0

=φ(1).

ByLemma 3.1again,wehave

s0=ρ_a·K+

φb·L(u0).

Thiscontradictsto (3.11).Therefore,

ρ_{ai·K+φbi·L}→ρ_a·K+φb·L uniformlyonSⁿ⁻¹ and

a_i·K+_φb_i·L→a·K+_φb·L. 2

The following lemma shows that the Orlicz radial sum and the radial Minkowski additionarecloselyrelated.

Lemma3.6. LetK,L∈ Soⁿ.For0< λ<1,if φ∈ C⁺,then

(1−λ)·K+φλ·L⊆(1−λ)K+λL. (3.12) If φisstrictlyconvex,equalityholds ifandonly ifK andLare dilatesof eachother.

Proof. LetK_λ= (1−λ)·K+_φλ·L.ByLemma 3.1and convexityofφ,we have φ(1) = (1−λ)φ

ρ_K(u) ρKλ(u)

+λφ

ρ_L(u) ρKλ(u)

≥φ

(1−λ)ρ_K(u) +λρ_L(u) ρKλ(u)

. (3.13)

Since φisstrictlydecreasingon(0,∞),thenwehave (1−λ)ρ_K(u) +λρ_L(u)≥ρ_Kλ(u).

This is,

ρ_{(1−λ)K+λL}(u)≥ρ_Kλ(u).

By (2.4), we obtain the desired inclusion. From the equality condition in Jensen’s inequality (2.13), ifφ is strictly convex, thenequation holds in(3.13) if and onlyif K and Laredilatesofeachother. 2

4. DualOrliczmixedvolume

Letus introducethedualOrlicz mixedvolume.

Definition 4.1.Forφ∈ C⁺,wedefine thedual OrliczmixedvolumeVφ(K,L) by Vφ(K, L) = 1

n

Sⁿ⁻¹

φ

ρL(u) ρK(u)

ρⁿ_K(u)dS(u), (4.1)

forallK,L∈ Soⁿ.

Remark 4.1. When φ(t) =t^−p, with p≥ 1.The dual Orlicz mixedvolume reduces to Lutwak’s L_p dualmixedvolume(see[38]):

V₋p(K, L) = 1 n

Sⁿ⁻¹

ρ^n+p_K (u)ρ^−p_L (u)dS(u),

forallK,L∈ Soⁿ.

Wedenotetherightderivativeofareal-valuedfunctionf byf_r.Forφ∈ C⁺,thereis φ_r(1)<0 becauseφisconvexand strictlydecreasing.

Lemma 4.1.Letφ∈ C⁺ andK,L∈ Soⁿ.Then

ε→0lim⁺

ρ_K+φε·L(u)−ρK(u)

ε =ρ_K(u)

φ_r(1)φ

ρ_L(u) ρK(u)

, (4.2)

uniformly forallu∈Sⁿ⁻¹.

Proof. Supposeε>0,K,L∈ Soⁿ,andu∈Sⁿ⁻¹.Let t(ε) =ρ_K+φε·L(u).

Then,byLemma 3.5,wehave

t(ε)→ρK(u) as ε→0. (4.3)

ByLemma 3.1,wehave φ

ρK(u) t(ε)

+εφ

ρL(u) t(ε)

=φ(1).

Then

ρK(u) t(ε) =φ⁻¹

φ(1)−εφ ρL(u)

t(ε)

.

Let

s=φ⁻¹

φ(1)−εφ ρ_L(u)

t(ε)

(4.4) andnotethats→1⁺ asε→0⁺.Thus

t(ε)−ρK(u)

t(ε) = 1−ρK(u)

t(ε) = 1−s. (4.5)

Combining(4.5)andLemma 3.5,weobtain

ε→0lim⁺ ρ_K+

φε·L(u)−ρ_K(u)

ε = lim

ε→0⁺

t(ε)

ε ·t(ε)−ρK(u) t(ε)

= lim

ε→0⁺t(ε)·φ ρL(u)

t(ε)

·

t(ε)−ρK(u) t(ε)

φ(1)−(φ(1)−εφ(^ρ_t(ε)^L(u)))

=ρ_K(u)·φ

ρ_L(u) ρK(u)

· lim

s→1⁺

1−s φ(1)−φ(s)

= ρ_K(u) φ_r(1)φ

ρ_L(u) ρK(u)

. (4.6)

Thenthepointwiselimit(4.2)hasbeenproved.

Moreover,theconvergenceisuniformforanyu∈Sⁿ⁻¹.Indeed,by(4.4)and(4.6),it sufficestorecallthatbyLemma 3.5,

εlim→0⁺ρ_K+φε·L(u) =ρK(u), uniformlyforu∈Sⁿ⁻¹. 2