On the Dual Orlicz Mixed Volumes?

Hailin JINShufeng YUANGangsong LENG

1 Introduction

The dual mixed volumes,as a core concept in the dual Brunn-Minkowskitheory,were firstly introduced by Lutwak[1],and played an important role in convex geometry.They are closely related to such important bodies as:Intersection bodies(see[2]),centroid bodies(see[3]),and projection bodies(see[4]).In[5],Gardner gave some stability results of these inequalities about dual mixed volumes.In[6],Klain presented a classification theorem for homogeneous valuations on star-sharped sets by dual mixed volumes.

Quite recently,Gardner,Hug and Weil[7]constructed a general framework for the Orlicz-Brunn-Minkowskitheory,which was introduced by Lutwak,Yang and Zhang(see[8–11]),and they made clear for the first time its relation to Orlicz spaces and norms.In[7],Gardner,Hug and Weil gave a reasonable definition of Orlicz addition,then obtained the Orlicz-Brunn-Minkowksi inequality,and in the end gave the Orlicz mixed volume of convex bodies which contain the origin in their interiors and get the Orlicz mixed volume inequality.In[12],Xi,Jin and Leng also obtained the Orlicz-Brunn-Minkowski inequality by Steiner symmetry and the Orlicz Minkowksi mixed volume inequality.

denotes the usualn-dimensional Euclidean space.A setA?is said to be starshaped,if 0∈A,and for each linelpassing through the origin inthe setA∩lis a closed interval.Denote bythe set of all star bodies ini.e.,the set of all star-shaped sets with a positive and continuous radial function.

In this paper,we define the harmonic Orlicz sumof star bodiesKandLinimplicitly by

forx∈Here?∈Φ2,and we have the set of convex functions?:[0,∞)2→[0,∞)that are strictly increasing in each variable and satisfy?(0,0)=0 and?(1,0)=?(0,1)=1.

In Section 2,we introduce a new notion of the Orlicz harmonic combination(K,L,α,β)by means of an appropriate modification of(1.1).The particular instance of interest corresponds to using(1.1)with?(x1,x2)=?1(x1)+2(x2)for>0 and?1,?2∈Φ,the set of strictly increasing convex functions?:[0,∞)→[0,∞)that satisfy?(0)=0 and?(1)=1,in which case we writeLinstead ofL,and we obtain the following equation.

Theorem 1.1Suppose ?∈Φ2.For all K,L∈we have

where↓0means thatis decreasing and tends to0.

The integral on the right-hand side of(1.2)with?2replaced by?,a new dual Orlicz mixed volumeV?(K,L)is introduced,and we see that either side of the equation(1.2)is equal toV?2(K,L)and establish the following dual Orlicz-Minkowski inequality and the harmonic Orlicz addition version of the Brunn-Minkowski inequality.

Theorem 1.2Suppose ?∈Φ,K,L∈and then

with equality if and only if K and L are dilates.

Theorem 1.3Suppose ?∈Φ2such that ?(x1,x2)=?1(x1)+?2(x2)and ?i∈Φ,i=1,2,xi∈If K,L∈then

with equality if and only if K and L are dilates.

2 Preliminaries

We shall denote the(n?1)-dimensional unit sphere inbySn?1.A star bodyKis determined uniquely by its radial functionρK=ρ(K,·):Sn?1→R,defined foru∈Sn?1by

Suppose thatK,L∈and the radial Hausdorf fmetricis defined by

IfK∈then the polar coordinate formula for volumeV(K)is

where dS(u)is the spherical Lebesgue measure ofSn?1.

Throughout the paper,Φm,m∈and denote the set of convex functions?:[0,∞)m→[0,∞)that are strictly increasing in each variable and satisfy?(0)=0 and?(ej)=1>0,j=1,···,m.Whenm=1,we shall write Φ instead of Φ1.

Letm≥2 andKj∈The harmonic Orlicz sum ofK1,···,Km,denoted by(K1,···,Km)is defined by

for allx∈

Equivalently,the harmonic Orlicz sum(K1,···,Km)can be defined implicitly(and uniquely)by

An important special case is obtained when

for some fixed?j∈Φ,j=1,···,msuch that?1(1)=···=?m(1)=1.We then write(K1,···,Km)=Km.This means thatis defined either by

for allx∈or by the corresponding special case of(2.3).

Letm=2 and?(x1,x2)=p≥1,and we get the harmonicLpsum(see[13]).

Suppose thatαj≥0 and?j∈Φ,j=1,···,m.IfKj∈j=1,···,m,we define the Orlicz linear combination(K1,···,Km,α1,···,αm)by

for allx∈Unlike theL?pcase,it is not generally possible to isolate a harmonic Orlicz scalar multiplication,since there is a dependence not just on one coefficientαj,but on allK1,···,Kmandα1,···,αm.

For our purposes,it suffices to focus on the casem=2.The harmonic Orlicz combination(K,L,α,β)forK,L∈andα,β≥0(not both zero),is defined equivalently via the implicit equation for allx∈

It is easy to verify that when?1(t)=?2(t)=tp,p≥1,we get that the harmonic Orlicz linear combination(K,L,α,β)equals the harmonicLpcombination(see[13]).In[14],He and Leng got a strong law of large numbers on the harmonicLpcombination.

Henceforth we shall writeLinstead of(K,L,1,and assume throughout that this is de fined by(2.6),whereα=1,β=and?j∈Φ,j=1,2.

The left derivative of a real-valued functionfis denoted by

Suppose thatμis a probability measure on a spaceXandg:X→I?is aμ-integrable function,whereIis a possibly in finite interval.Jensen’s inequality states that if?:I→is a convex function,then

If?is a strictly convex,the equality holds if and only ifg(x)is constant forμ-almost allx∈X.

3 Dual Orlicz Mixed Volume

We firstly give some properties of the harmonic Orlicz addition.

Lemma 3.1Let ?∈Φ2.If K,Ki,L,Li∈then the harmonic Orlicz additionhas the following properties:

(1)(Continuity)KiLi→KL,i.e.,for all x∈as Ki→K and Li→L in the radial Hausdor ffmetric.

(2)(Monotonicity)

(3)(GL(n)Covariance)

Proof(1)Since=1,by the continuity of?,we have

Hence,

(2)By the monotonicity of?,(2)is easy to get.

(3)Since=1,we have

SetA?1x=y.Then

SoA?1

Proof of Theorem 1.1Set=It is easy to see thatρ(u)→ρK(u)for eachu∈Sn?1.Then

From the definition ofwe haveThen

Letand note thatz→1?as↓0.Consequently,

Sinceρ(u)is monotonic with respect toby Lemma 3.1(2)and Theorem 7.11 of[15],we haveρ(u)=ρK(u)uniformly onSn?1.Hence,

Therefore,by the polar coordinate formula for volume(2.1),we obtain

For?∈Φ,the dual Orlicz mixed volume(K,L)is defined by

for allK,L∈If?(t)=tp,p≥1,we get the dual mixed volume(see[13,Proposition 1.9]).The following theorem gives a connection between the dual Orlicz mixed volume and the harmonic Orlicz combination.

Theorem 3.1Suppose ?i∈Φ,i=1,2and ?(x1,x2)=?1(x1)+?2(x2).For all K,L∈we have

The following theorems show that the dual Orlicz volume is SL(n)invariant,continuous.

Theorem 3.2Suppose ?∈Φ.If K,L∈and A∈SL(n),then

ProofBy the same method of the proof of Lemma 3.1(3),we get for∈Φ2.By Theorem 3.2,we get(AK,AL)=(K,L).

By the continuity of?,we have the following.

Theorem3.3Suppose?∈Φ.If K,L∈and Ki→K,Li→L in the radial Hausdor ff metric,then(Ki,Li)→(K,L).

Theorem 3.4Suppose K,L∈If ?i,?∈Φand ?i→?,i.e.,|?i(t)??(t)|→ 0for every compact interval I?then(K,L)→(K,L).

ProofSinceK,L∈and?i→?,we haveuniformly onSn?1.By(3.1),we obtain thati(K,L)→(K,L).

4 Inequality of the Dual Orlicz Mixed Volume

In[13],Lutwak proved the dualL?p-mixed volume inequality:Ifp≥1,andK,L∈then

with equality if and only ifKandLare dilates.

For the dual Orlicz mixed volumes,we also establish the dual Orlicz mixed volume inequalities.

The following lemma will be needed in the proof of Theorem 1.2,which is easy to be obtained by the H?lder inequality and the polar coordinate formula for volumes.

Lemma 4.1(see[13,Proposition 1.10])If K,L∈then

with equality if and only if K and L are dilates.

Proof of Theorem 1.2By Jensen’s inequality and Lemma 4.1,we have

If the equality holds,by the equality conditions of Jensen’s inequality and Lemma 4.1,we have thatKandLare dilates.Conversely,ifKandLare dilates,it is easy to see that the equality holds.

In Theorem 1.2,if we set?(t)=tp,p≥1,it leads to the Lutwak’s result of the dualL?p-mixed volume.

By Theorem 1.2,we have the following uniqueness result which is the Orlicz version of Proposition 1.11 of[13].

Proposition 4.1Suppose ?∈Φ,and M?such that K,L∈M.If

then K=L.

ProofTakingQ=Lgives=1.Now Theorem 1.2 givesV(L)≥V(K),with equality if and only ifKandLare dilates.TakingQ=K,we getV(K)≥V(L).Hence,V(K)=V(L),andKandLmust be dilates.ThusK=L.

In the following,we give the dual Orlicz-Brunn-Minkowski inequality for the harmonic Orlicz addition.

Proof of Theorem 1.3By Theorem 1.2,we have

If the equality holds,by the equality condition of Theorem 1.2,we have thatKandLare dilates.Conversely,ifKandLare dilates,it is easy to check that the equality holds.

Let?(x1,x2)=and we have the following result.

Corollary 4.1(see[13,Proposition 1.12])Suppose K,L∈If p≥1,then

with equality if and only if K and L are dilates.

AcknowledgementThe authors are grateful to the referees for their valuable suggestions and comments.

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