Orlicz difference bodies

Wei Shi; Weidong Wang; Tongyi Ma

doi:10.1515/math-2018-0098

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Orlicz difference bodies

Wei Shi , Weidong Wang and Tongyi Ma

Published/Copyright: October 29, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper, by the Orlicz-Minkowski combinations of convex bodies, we define the general Orlicz difference bodies and study their properties. Furthermore, we obtain the extreme values of the general Orlicz difference bodies and their polars.

Keywords: Difference body; Orlicz-Minkowski combination; General Orlicz difference body; Extreme value

MSC 2010: 52A20; 52A40; 52A39

1 Introduction

A convex body in ℝⁿ is a compact convex subset with non-empty interior. Let 𝒦ⁿ denote the set of all convex bodies in ℝⁿ. For the set of convex bodies containing the origin in their interiors, the set of convex bodies whose centroids are at the origin and the set of origin-symmetric convex bodies in ℝⁿ, we write Kon, Kcn and Kosn, respectively. Let B denote the standard Euclidean unit ball in ℝⁿ and write S_n−1 for its surface. In addition, we write V(K) for the n-dimensional volume of a body K. Especially, V(B)=ω_n.

If K∈𝒦ⁿ, then its support function h_K=h(K,⋅) :ℝⁿ→(−∞,+∞) is defined by (see [1,2])

h(K,x)=max{x⋅y:y∈K},x∈Rn,(1.1)

where x⋅y denotes the standard inner product of x and y.

From (1.1), we easily know that: If A∈GL(n), then (see [1])

h(AK,u)=h(K,A′u),forallu∈Sn−1.(1.2)

Here, GL(n) denotes the set of all general (non-singular) affine transformations and A^′ denotes the transpose of A. Further, from (1.2), it is easy to get that h(−K,u)=h(K,−u), for any u∈Sⁿ⁻¹.

For K,L∈𝒦ⁿ, and λ,μ≥0 (not both zero), the Minkowski linear combination, λK+μL∈𝒦ⁿ, of K and L is defined by (see [1,2])

h(λK+μL,⋅)=λh(K,⋅)+μh(L,⋅),(1.3)

where λK={λx:x∈K}.

Taking λ=μ=1/2,L=−K in (1.3), then the difference body, ΔK, of K∈𝒦ⁿ is given by (see [1])

ΔK=12K+12(−K).(1.4)

Obviously, ΔK is an origin-symmetric convex body.

For the difference body, we know that (see [1]): If K∈𝒦ⁿ, then

V(ΔK)≥V(K),

with equality if and only if K is centrally symmetric.

A recent extension of the Brunn-Minkowski theory is the Orlicz-Brunn-Minkowski theory, which was first launched by Lutwak, Yang and Zhang ([4, 3]) with affine isoperimetric inequalities for Orlicz centroid and projection bodies. In addition, Gardner, Hug and Weil ([5]) built the foundation and provided a general framework for the Orlicz-Brunn-Minkowski theory. In [6], Xi, Jin and Leng obtained the beautiful Orlicz-Brunn-Minkowski inequality, which can be viewed as a generalization of the classical Brunn-Minkowski inequality. Corresponding to Orlicz-Brunn-Minkowski theory, Zhu, Zhou and Xu ([7]) also established the dual Orlicz-Brunn-Minkowski theory. For the recent development of the Orlicz theory, see also [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29, 21, 22, 23].

Let Φ denote the set of convex and strictly increasing functions φ:[0,∞)→[0,+∞) and φ(0)=0. In 2014, Gardner, Hug and Weil (see [5], also see [6]) defined the Orlicz-Minkowski combination: Let φ∈Φ satisfy φ(1)=1, for K, K,L∈Kon, α,β≥0 (not both zero), define the Orlicz-Minkowski combination +φ(K,L,α,β) by

h+φ(K,L,α,β)(u)=inf{λ>0:αφ(hK(u)λ)+βφ(hL(u)λ)≤φ(1)}.(1.5)

Notice that since the function z→αφ(hK(u)z)+βφ(hL(u)z) is strictly decreasing, we have

h+φ(K,L,α,β)(u)=λu,

if and only if

αφ(hK(u)λu)+βφ(hL(u)λu)=φ(1).(1.6)

From definition (1.5), we first give the concept of Orlicz difference body as follows:

For K∈Kon, define the Orlicz difference body ΔφK of K by

hΔφK(u)=inf{λ>0:12φ(hK(u)λ)+12φ(h−K(u)λ)≤φ(1)}.(1.7)

From (1.7), it is easy to conclude that ΔφK∈Kosn, and K∈Kosn implies ΔφK=K. If φ(t)=t in (1.7), then by (1.4), we have ΔφK=ΔK.

By (1.7), for K∈Kon and real τ∈[−1,1], we also define the general Orlicz difference body, ΔφτK, of K by

hΔφτK(u)=inf{λ>0:f1(τ)φ(hK(u)λ)+f2(τ)φ(h−K(u)λ)≤φ(1)}.(1.8)

Here,

f1(τ)=(1+τ)22(1+τ2),f2(τ)=(1−τ)22(1+τ2)(1.9)

for τ∈[−1,1]. Obviously, by (1.9), functions f₁(τ) and f₂(τ) satisfy

f1(τ)+f2(τ)=1,(1.10)

f1(−τ)=f2(τ).(1.11)

From (1.7), (1.8) and (1.9), we easily know that if τ=0, then Δφ0K=ΔφK, and if τ=±1, then Δφ+1K=K,Δφ−1K=−K.

Remark 1.1

For p≥1, let φ(t)=t^p in (1.8), then ΔφτK implies the general L_p-difference body ΔpτK which is given by Wang and Ma (see [30]). In particular, for p=1, we write ΔpτK=ΔτK and call Δ^τK the general difference body of K. □

The aim of this paper is to study the general Orlicz difference bodies and their polars. First, we obtain the following inclusion relationship between general difference bodies Δ^τK and general Orlicz difference bodies ΔφτK.

Theorem 1.2

If K∈Kon, then for any τ∈[−1,1], we have

ΔτK⊆ΔφτK,(1.12)

with equality either if and only if K∈Kosn or φ is a linear function.

Obviously, by (1.12) we immediately get the following corollary.

Corollary 1.3

If K∈Kon, τ∈[−1,1], and φ∈Φ, then

V(ΔφτK)≥V(ΔτK),

with equality either if and only if K∈Kosn or φ is a linear function.

Next, the extreme value of the general Orlicz difference bodies is also obtained:

Theorem 1.4

If K∉Kosn, τ∈[−1,1], and φ∈Φ, then

V(ΔφτK)≥V(K).(1.13)

If K∉Kosn, equality holds in (1.13) if and only if τ=±1; if K∈Kosn, then (1.13) becomes an equality.

Let Δφτ,cK=(ΔφτK)c denote the polar body of general Orlicz difference body ΔφτK, the following result gives the extreme value of the volume, V(Δφτ,∗K), of Δφτ,∗K.

Theorem 1.5

If K∉Kosn, τ∈[−1,1], and φ∈Φ, then

V(Δφτ,∗K)≤V(K∗).(1.14)

If K\hspace*{1.6ex}{\not}\hspace*{-1.3ex}{\in}\,\mathcal{K}_{os}^n, equality holds in (1.14) if and only if τ=±1; When K∈Kosn, inequality (1.14) becomes an identity.

According to the fact that the Orlicz-Minkowski combination +φ(K,L,α,β)∈K_no (see [6]), we easily deduce that ΔφτK∈Kon. Using inequality (1.13) and the Blaschke-Santalo´ inequality for convex bodies, we have

Theorem 1.6

If K∈Kosn, φ∈Φ, then for any τ∈[−1,1], we have

V(K)V(Δφτ,cK)≤ωn2,(1.15)

with equality if and only if K is an ellipsoid centered at the origin. Here Δφτ,cK=(ΔφτK)c, and for Q∈Kon, Q_c=(Q−centQ)_∗ and centQ denotes the centroid of Q.

Obviously, when φ(t)=t^p (p≥1), it is easily checked that Theorems 1.2-1.6 reduce to the results of general L_p-difference body Δ_τpK (see [30]).

This paper is organized as follows. In Section 2, we list some basic notions that will be indispensable to the proofs of our results. In Section 3, several elementary properties of the general Orlicz difference bodies are listed. In Section 4, the proofs of Theorems 1.2-1.6 are completed.

2 Basic notions

2.1 Radial functions and the polar of convex bodies

If K is a compact star-shaped (with respect to the origin) in ℝⁿ, then its radial function, ρ_K=ρ(K,⋅):ℝⁿ∖{0}→[0,+∞), is defined by (see [1])

ρ(K,x)=max{λ≥0:λx∈K},x∈Rn∖{0}.

If ρ_K is positive and continuous, then K will be called a star body (with respect to the origin). Denote by Son the set of star bodies (about the origin) in ℝⁿ. When ρ_K(u)/ρ_L(u) is independent of u∈Sⁿ⁻¹, we say that two star bodies K and L are dilates (of each other).

For a non-empty subset E⊆ℝⁿ, the polar set E^∗ of E is defined by (see [1,2])

E∗={x∈Rn:x⋅y≤1,y∈E}.(2.1)

From definition (2.1), we easily get that for K∉Kosn,

h(K,u)=1ρ(K∗,u), u∈Sn−1.(2.2)

The well-known Blaschke-Santalo´ inequality for convex bodies has the following representation ([31]):

Theorem 2.1

If K∈Kcn, then

V(K)V(K∗)≤ωn2,(2.3)

with equality if and only if K is an ellipsoid centered at the origin.

Remark 2.2

For Q∈𝒦ⁿ, let centQ∈intQ denote the centroid of Q. Associated with each Q∈𝒦ⁿ is a point s=San(Q)∈intK, called the Santalo´ point of Q, defined as the unique point s∈intQ, such that Cent((−s+Q)_∗)=0. LetKsn denote the set of convex bodies having their Santalo´ point at the origin. Thus, Q∈Ksn if and only if Q∗∈Kcn

2.2 The Orlicz mixed volume

Using the Orlicz-Minkowski combination +φ(α,β,K,L), Gardner, Hug and Weil ([5], also see [6]) defined the following Orlicz mixed volume: For K,K,L∈Kon, φ∈Φ, the Orlicz mixed volume of K and L is defined by

Vφ(K,L)=1n∫Sn−1φ(hL(u)hK(u))hK(u)dSK(u).

Associated with the definition of Orlicz mixed volume, Gardner, Hug and Weil ([5], also see [6]) presented the following Orlicz-Minkowski inequality.

Theorem 2.3

If K, K,L∈Kon, φ∈Φ, then

Vφ(K,L)≥V(K)φ(V(L)1nV(K)1n).(2.4)

If φ is strictly convex, then equality holds either if and only if K and L are dilates or L={o}.

2.3 Dual Orlicz mixed volume

In [17], Ma and Wang made a further study on Orlicz theory. They introduced the notion of Orlicz radial combination as follows: For K, K,L∈Son, φ∈Φ, the Orlicz radial combination, α∘K+~−φβ∘L, of K and L is defined by (see [17])

ρ(α∘K+~−φβ∘L,u)=sup{λ>0:αφ(λρ(K,u))+βφ(λρ(L,u))≤1}

for all u∈Sⁿ⁻¹. Further, Ma and Wang in [17] gave the corresponding definition of dual Orlicz mixed volume V˜_−φ(K,L):

−nφr′(1)V~−φ(K,L)=limε→0+V(K+~−φε∘L)−V(K)ε.

Here φ^′_r (1) denote the right derivative of φ at 1.

Based on the above definition, an important integral representation of the dual Orlicz mixed volume was obtained immediately in [17].

Theorem 2.4

Suppose φ∈Φ and φ(1)=1. If K, K,L∈Son, then

V~−φ(K,L)=1n∫Sn−1φ(ρK(u)ρL(u))ρKn(u)dS(u).(2.5)

Obviously, by (2.5) we get

V~−φ(K,K)=1n∫Sn−1φ(ρK(u)ρK(u))ρKn(u)dS(u)

=1n∫Sn−1ρKn(u)dS(u)=V(K).(2.6)

=1n∫_Sn−1ρ_nK(u)dS(u)=V(K). (2.6)

For dual Orlicz mixed volume V˜_−φ(K,L), using the similar argument of Orlicz-Minkowski inequality (2.4), the corresponding dual Orlicz-Minkowski inequality can be stated as follows (see [17]):

Theorem 2.5

If φ∈Φ, K, K,L∈Son, then

V~−φ(K,L)≥V(K)φ(V(K)1nV(L)1n).(2.7)

Equality holds if and only if K and L are dilates.

3 Basic properties of general Orlicz difference bodies

In this section, we will list some properties of general Orlicz difference bodies, which are essential for the proofs of our Theorems.

Lemma 3.1

Let K∈Kon, φ∈Φ, if A∈GL(n), then for any τ∈[−1,1], we have

ΔφτAK=AΔφτK.

Proof

By (1.2) and (1.8), this gives the desired result. □

Lemma 3.2

If K∉Kosn, τ∈[−1,1], and φ∈Φ, then

Δφτ(−K)=Δφ−τK=−ΔφτK.

Proof

For all u∈Sⁿ⁻¹, by (1.8) and (1.11), and together with h(−K,u)=h(K,−u), Lemma 3.2 can be easily proved. □

Lemma 3.3

If K∈Kon, τ∈[−1,1], φ ∈ Φ,0 and φ∈Φ, then ΔφτK=Δφ−τK if and only if K∈Kosn.

Proof

By (1.8) and (1.11), and notice that (1.6), it is easy to get the desired result. Contrarily, if K∈Kosn, together with Lemma 3.2, we immediately get Δ_τφK=Δ_−τφK for all u∈S_n−1. □

From Lemma 3.3, we obtain immediately the following result.

Corollary 3.4

Let K∈Kosn, φ∈Φ. If K∉K_nos, then for any τ∈[−1,1], we have ΔφτK=Δφ−τK if and only if τ=0.

By Lemma 3.2, we see that if ΔφτK=Δφ−τK, then ΔφτK=−ΔφτK. This means ΔφτK∈Kosn. This and Lemma 3.3 yield

Corollary 3.5

Let K∉Kosn, τ∈[−1,1] and φ∈Φ. If τ ≠ 0, then ΔφτK∈Kosn if and only if K∈Kosn.

Lemma 3.6

Suppose K∈Kosn, τ∈[−1,1] and φ∈Φ, then

ΔφτK=K.(3.1)

Proof

Since K∈Kosn, from (1.8), (1.10) and (1.6), we easily can get

φ(hK(u)hΔφτK(u))=φ(1).

The strictly increasing property of convex function φ shows that

hΔφτK(u)=hK(u)

for all u∈Sⁿ⁻¹. This gives (3.1). □

Lemma 3.4 immediately infers

Corollary 3.7

Suppose K, L∈Kosn, τ∈[−1,1] and φ∈Φ, then

ΔφτK=ΔφτL⟺K=L.

4 Results and proofs

In this section, we will give the proofs of Theorems 1.2-1.6. First, the following Orlicz-Brunn-Minkowski inequality (see [5,6]) is useful and indispensable for the proofs of our results.

Lemma 4.1

If L∈Kon, φ∈Φ and α,β>0, then we have

αφ(V(K)1nV(+φ(α,β,K,L))1n)+βφ(V(L)1nV(+φ(α,β,K,L))1n)≤φ(1).(4.1)

with equality if K and L are dilates. If φ is strictly convex, equality holds if and only if Kand L are dilates of each other.

Proof of Theorem 1.2

Notice that the function φ is a strictly increasing convex function on [0,+∞), and together with (1.3) and (1.6) we get for all u∈Sⁿ⁻¹,

φ(1)=f1(τ)φ(hK(u)hΔφτK(u))+f2(τ)φ(h−K(u)hΔφτK(u))≥φ(f1(τ)hK(u)hΔφτK(u)+f2(τ)h−K(u)hΔφτK(u))=φ(hΔτK(u)hΔφτK(u)).(4.2)

Therefore, we have for all u∈Sⁿ⁻¹,

hΔτK(u)≤hΔφτK(u).

i.e.,

ΔτK⊆ΔφτK.

So we obtain (1.12).

Now, we discuss the case of equality about (1.12). Using the increasing property of convex function φ, we have hK(u)hΔφτK(u)=h−K(u)hΔφτK(u), namely, K=−K. That is to say, K∈Kosn. On the contrary, if K∈Kosn, from Lemma 3.6 and definition (1.3), it is easy to see that Δ^τK=Δ_τφK.

When φ is not strictly convex, the equality holds in (4.2) means φ must be a linear function. If φ is linear, we will show that

φ(1)=f1(τ)φ(hK(u)hΔφτK(u))+f2(τ)φ(h−K(u)hΔφτK(u))=φ(f1(τ)hK(u)hΔφτK(u))+φ(f2(τ)h−K(u)hΔφτK(u))=φ(f1(τ)hK(u)hΔφτK(u)+f2(τ)h−K(u)hΔφτK(u))=φ(hΔτK(u)hΔφτK(u)).

Thus, h_Δτ_K(u)=h_Δτφ_K(u). It can concluded that the equality holds in (1.12).

By the above discussions, this gives the proof of Theorem 1.2. □

Lemma 4.2

If K, L∈Kon, φ∈Φ. Then for any 0<α<1, one gets

V(+φ(α,1−α,K,L))≥V(K)αV(L)1−α.(4.3)

Equality holds if and only if K=L.

Proof

For the sake of brevity, let Kφ=+φ(α,1−α,K,L). Notice that φ∈Φ, so, we know that φ is convex and strictly increasing, from the Orlicz-Brunn-Minkowski inequality (4.1) and the harmonic-geometric-arithmetic mean (HG-AM) inequality (see [32], p.515), we have

φ(1)≥αφ(V(K)1nV(Kφ)1n)+(1−α)φ(V(L)1nV(Kφ)1n)

≥φ(αV(K)1nV(Kφ)1n+(1−α)V(L)1nV(Kφ)1n)

≥φ(V(K)αnV(L)1−αnV(Kφ)1n).

Therefore, we get

1≥V(K)αnV(L)1−αnV(Kφ)1n,

i.e.

V(+φ(α,1−α,K,L))≥V(K)αV(L)1−α.

From (4.1), and the characteristic of convex function φ, together with the equality condition of HG-AM inequality, we know that equality holds in (4.3) if and only if K=L. □

Proof of Theorem 1.4

Since τ∈(−1,1), thus 0<f₁(τ)<1. Let α=f₁(τ) in (4.3), we have

V(ΔφτK)≥V(K)f1(τ)V(−K)1−f1(τ)=V(K).

This gives inequality (1.13).

By Lemma 4.1, if τ∈(−1,1) (i.e. τ ≠ ±1), then equality holds in (1.13) if and only if K=−K, thus, K∈Kosn. So we see that if K∉K_nos, then equality holds in (1.13) if and only if τ=±1. □

Proof of Theorem 1.5

According to the equality (1.6), we know

f1(τ)φ(hK(u)hΔφτK(u))+f2(τ)φ(h−K(u)hΔφτK(u))=φ(1).

This and (2.2) yield

f1(τ)φ(ρΔφτ,∗K(u)ρK∗(u))+f2(τ)φ(ρΔφτ,∗K(u)ρ(−K)∗(u))=φ(1).(4.4)

Observing (−K)^∗=−K^∗, then (4.4) can be written as

f1(τ)φ(ρΔφτ,∗K(u)ρK∗(u))+f2(τ)φ(ρΔφτ,∗K(u)ρ−K∗(u))=φ(1).(4.5)

Using (2.6), (2.5), (4.5) and notice that φ(1)=1, then

V(Δφτ,∗K)=1n∫Sn−1ρΔφτ,∗Kn(u)dS(u)=1n∫Sn−1φ(1)ρΔφτ,∗Kn(u)dS(u)=1n∫Sn−1[f1(τ)φ(ρΔφτ,∗K(u)ρK∗(u))+f2(τ)φ(ρΔφτ,∗K(u)ρ−K∗(u))]ρΔφτ,∗Kn(u)dS(u)=f1(τ)1n∫Sn−1φ(ρΔφτ,∗K(u)ρK∗(u))ρΔφτ,∗Kn(u)dS(u)+f2(τ)1n∫Sn−1φ(ρΔφτ,∗K(u)ρ−K∗(u))ρΔφτ,∗Kn(u)dS(u)=f1(τ)V~−φ(Δφτ,∗K,K∗)+f2(τ)V~−φ(Δφτ,∗K,−K∗).

This and the dual Orlicz-Minkowski inequality (2.7) yield

V(Δφτ,∗K)≥f1(τ)V(Δφτ,∗K)φ(V(Δφτ,∗K)1nV(K∗)1n)+f2(τ)V(Δφτ,∗K)φ(V(Δφτ,∗K)1nV(−K∗)1n).

Therefore, by (1.10) we get that

1=φ(1)≥(f1(τ)+f2(τ))φ(V(Δφτ,∗K)1nV(K∗)1n)=φ(V(Δφτ,∗K)1nV(K∗)1n).

For the function φ, it is strictly increasing on [0,+∞). So, we have

V(Δφτ,∗K)≤V(K∗).

This is just inequality (1.14).

About the condition of equality in (1.14), if K∈Kosn, from Lemma 3.6, we know that (1.14) is evidently true. If K∉Kⁿ_os, associated with the condition of equality in (2.7), we know that equality holds in (1.14) if and only if Δ^τ,^∗_φK and K^∗, Δ_τ,∗φK and −K^∗ both are dilates. These together with V(Δ^τ,_∗_φK)=V(K^∗) give that Δ^τ_φK=±K, i.e., τ=±1. □

Proof of Theorem 1.6

Since Δ_τφK∉Kosn, thus Δ_τφK−c∈K_nc, where c is the centroid of Δ_τφK. Because of Δ_τ,cφK=(Δ_τφK)_c=(Δ_τφK−c)^∗, hence, together with inequality (1.13) and the Blaschke-Santalo´ inequality (2.3), we obtain

V(K)V(Δφτ,cK)≤V(ΔφτK)V(Δφτ,cK)=V(ΔφτK−c)V((ΔφτK−c)∗)≤ωn2.

This gives inequality (1.15).

The equality conditions of inequalities (2.3) and (1.13) show that equality holds in inequality (1.15) if and only if K is an ellipsoid centered at the origin. □

Acknowledgement

Research is supported in part by the Natural Science Foundation of China (Grant No.11371224), and the National Natural Science Foundation of China (Grant No.11561020) and Excellent Foundation of Graduate Student of China Three Gorges University (Grant No.2017YPY077).

References

[1] Gardner R.J., Geometric Tomography, second ed., Encyclopedia Math. Appl., vol. 58, Cambridge University Press, Cambridge, 2006.Search in Google Scholar

[2] Schneider R., Convex Bodies: The Brunn-Minkowski theory, 2nd edn, Cambridge University Press, Cambridge, 2014.Search in Google Scholar

[3] Lutwak E., Yang D., Zhang G.Y., Orlicz centroid bodies, J. Differ. Geom., 2010, 84, 365-387.10.4310/jdg/1274707317Search in Google Scholar

[4] Lutwak E., Yang D., Zhang G.Y., Orlicz projection bodies, Adv. Math., 2010, 223, 220-242.10.1016/j.aim.2009.08.002Search in Google Scholar

[5] Gardner R.J., Hug D., Weil W., The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities, J. Differ. Geom., 2014, 97, 427-476.10.4310/jdg/1406033976Search in Google Scholar

[6] Xi D.M., Jin H.L., Leng G.S., The Orlicz Brunn-Minkowski inequality, Adv. Math., 2014, 260, 350-374.10.1016/j.aim.2014.02.036Search in Google Scholar

[7] Zhu B.C., Zhou J.Z., Xu W.X., Dual Orlicz-Brunn-Minkowski theory, Adv. Math., 2014, 264, 700-725.10.1016/j.aim.2014.07.019Search in Google Scholar

[8] Böröczky K.J., Stronger versions of the Orlicz-Petty projection inequality, J. Differ. Geom., 2013, 95, 215-247.10.4310/jdg/1376053446Search in Google Scholar

[9] Chen F.W., Yang C.L., Luo M., Successive radii and Orlicz Minkowski sum, Monatsh Math., 2016, 179, 201-219.10.1007/s00605-015-0754-3Search in Google Scholar

[10] Chen F.W., Zhou J.Z., Yang C.L., On the reverse Orlicz Busemann-Petty centroid inequality, Adv. Appl. Math., 2011, 47, 820-828.10.1016/j.aam.2011.04.002Search in Google Scholar

[11] Du C.M., Guo L.J., Leng G.S., Volume inequalities for Orlicz mean bodies, Proc. Indian Acad. Sci., 2015, 125, 57-70.10.1007/s12044-015-0214-ySearch in Google Scholar

[12] Gardner R.J., Hug D., Weil W., Ye D.P., The dual Orlicz-Brunn-Minkowski theory, J. Math. Anal. Appl., 2015, 430, 810-829.10.1016/j.jmaa.2015.05.016Search in Google Scholar

[13] Guo L.J., Leng G.S., Du C.M., The Orlicz mean zonoid operator, J. Math. Anal. Appl., 2015, 424, 1261-1271.10.1016/j.jmaa.2014.12.002Search in Google Scholar

[14] Haberl C., Lutwak E., Yang D., Zhang G.Y., The even Orlicz Minkowski problem, Adv. Math., 2010, 224, 2485-2510.10.1016/j.aim.2010.02.006Search in Google Scholar

[15] Huang Q.Z., He B.W., On the Orlicz Minkowski problem for polytopes, Discrete. Comput. Geom., 2012, 48, 281-297.10.1007/s00454-012-9434-4Search in Google Scholar

[16] Li A.J., Leng G.S., A new proof of the Orlicz Busemann-Petty centroid inequality, Proc. Amer. Math. Soc., 2011, 139, 1473-1481.10.1090/S0002-9939-2010-10651-2Search in Google Scholar

[17] Ma T.Y., Wang W.D., Dual Orlicz geominimal surface area, J. Inequal. Appl., 2016, 2016, 1-13.10.1186/s13660-016-1005-4Search in Google Scholar

[18] Wang G.T., Leng G.S., Huang Q.Z., Volume inequalities for Orlicz zonotopes, J. Math. Anal. Appl., 2012, 391, 183-189.10.1016/j.jmaa.2012.02.018Search in Google Scholar

[19] Wang W.D., Shi W., Ye S., Dual mixed Orlicz-Brunn-Minkowski inequality and dual Orlicz mixed quermassintegrals, Indag. Math., 2017, 28, 721-735.10.1016/j.indag.2017.04.001Search in Google Scholar

[20] Xiong G., Zou D., Orlicz mixed quermassintegrals, Sci. China Math., 2014, 57, 2549-2562.10.1007/s11425-014-4812-4Search in Google Scholar

[21] Ye D.P., Dual Orlicz-Brunn-Minkowski theory: Orlicz φ-radial addition, Orlicz L_φ-dual mixed volume and related inequalities, arXiv: 1404.6991v1., 2014.Search in Google Scholar

[22] Ye D.P., Dual Orlicz-Brunn-Minkowski theory: Dual Orlicz L_φaffine and geominimal surface areas, J. Math. Anal. Appl., 2016, 443, 352-371.10.1016/j.jmaa.2016.05.027Search in Google Scholar

[23] Ye D.P., New Orlicz affine isoperimetric inequalities, J. Math. Anal. Appl., 2015, 427, 905-929.10.1016/j.jmaa.2015.02.084Search in Google Scholar

[24] Yuan S.F., Jin H.L., Leng G.S., Orlicz geominimal surface areas, Math. Inequal. Appl., 2015, 18, 353-362.10.7153/mia-18-25Search in Google Scholar

[25] Zhu B.C., Hong H., Ye D.P., The Orlicz-Petty bodies, arXiv: 1611.04436v1., 2016.Search in Google Scholar

[26] Zhu G.X., The Orlicz centroid inequality for star bodies, Adv. Appl. Math., 2012, 48, 432-445.10.1016/j.aam.2011.11.001Search in Google Scholar

[27] Zou D., Xiong G., The minimal Orlicz surface area, Adv. Appl. Math., 2014, 61, 25-45.10.1016/j.aam.2014.08.006Search in Google Scholar

[28] Zou D., Xiong G., Orlicz-John ellipsoids, Adv. Math., 2014, 265, 132-168.10.1016/j.aim.2014.07.034Search in Google Scholar

[29] Zou D., Xiong G., Orlicz-Legendre ellipsoids, J. Geom. Anal., 2016, 26, 2474-2502.10.1007/s12220-015-9636-0Search in Google Scholar

[30] Wang W.D., Ma T.Y., Asymmetric L_p-difference bodies, Proc. Amer. Math. Soc., 2014, 142, 2517-2527.10.1090/S0002-9939-2014-11919-8Search in Google Scholar

[31] M. Meyer and A. Pajor, On Santalós inequality, In: Lindenstrauss, J, Milman, VD (eds.) Geometric Aspects of Functional Analysis. Springer Lecture Notes in Math., 1989, 1376, 261-263.10.1007/BFb0090059Search in Google Scholar

[32] Gardner R.J., Zhang G.Y., Affine inequalities and radial mean bodies, Amer. J. Math., 1998, 120, 505-528.10.1353/ajm.1998.0021Search in Google Scholar

Received: 2018-01-22

Accepted: 2018-09-05

Published Online: 2018-10-29

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0098

Keywords for this article

Difference body; Orlicz-Minkowski combination; General Orlicz difference body; Extreme value

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