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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

Orlicz difference bodies

Wei Shi / Weidong Wang
  • Corresponding author
  • Department of Mathematics, China Three Gorges University, Yichang, 443002, China
  • Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, 443002, China
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/ Tongyi Ma
Published Online: 2018-10-29 | DOI: https://doi.org/10.1515/math-2018-0098

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 Sn−1 for its surface. In addition, we write V(K) for the n-dimensional volume of a body K. Especially, V(B)=ωn.

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

h(K,x)=max{xy:yK},xRn,(1.1)

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

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

h(AK,u)=h(K,Au),foralluSn1.(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 uSn−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:xK}.

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,LKon, 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 zαφ(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 KKon, define the Orlicz difference body ΔφK of K by

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

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

By (1.7), for KKon 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(hK(u)λ)φ(1)}.(1.8)

Here,

f1(τ)=(1+τ)22(1+τ2),f2(τ)=(1τ)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,Δφ1K=K.

Remark 1.1

For p≥1, let φ(t)=tp in (1.8), then ΔφτK implies the general Lp-difference body ΔpτK which is given by Wang and Ma (see [30]). In particular, for p=1, we write ΔpτK=ΔτK and call ΔτK 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 ΔτK and general Orlicz difference bodies ΔφτK.

Theorem 1.2

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

ΔτKΔφτK,(1.12)

with equality either if and only if KKosn or φ is a linear function.

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

Corollary 1.3

If KKon, τ∈[−1,1], and φ∈Φ, then

V(ΔφτK)V(ΔτK),

with equality either if and only if KKosn or φ is a linear function.

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

Theorem 1.4

If KKosn, τ∈[−1,1], and φ∈Φ, then

V(ΔφτK)V(K).(1.13)

If KKosn, equality holds in (1.13) if and only if τ=±1; if KKosn, 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 KKosn, 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 KKosn, inequality (1.14) becomes an identity.

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

Theorem 1.6

If KKosn, 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 QKon, Qc=(Q−centQ) and centQ denotes the centroid of Q.

Obviously, when φ(t)=tp (p≥1), it is easily checked that Theorems 1.2-1.6 reduce to the results of general Lp-difference body ΔτpK (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, ρK=ρ(K,⋅):ℝn∖{0}→[0,+∞), is defined by (see [1])

ρ(K,x)=max{λ0:λxK},xRn{0}.

If ρK 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 ρK(u)/ρL(u) is independent of uSn−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={xRn:xy1,yE}.(2.1)

From definition (2.1), we easily get that for KKosn,

h(K,u)=1ρ(K,u),uSn1.(2.2)

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

Theorem 2.1

If KKcn, 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 centQintQ 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 sintQ, such that Cent((−s+Q))=0. Let Ksn denote the set of convex bodies having their Santalo´ point at the origin. Thus, QKsn if and only if QKcn

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,LKon, φ∈Φ, the Orlicz mixed volume of K and L is defined by

Vφ(K,L)=1nSn1φ(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,LKon, 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,LSon, φ∈Φ, the Orlicz radial combination, αK+~φβL, of K and L is defined by (see [17])

ρ(αK+~φβL,u)=sup{λ>0:αφ(λρ(K,u))+βφ(λρ(L,u))1}

for all uSn−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)=limε0+V(K+~φεL)V(K)ε.

Here φr (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,LSon, then

V~φ(K,L)=1nSn1φ(ρK(u)ρL(u))ρKn(u)dS(u).(2.5)

Obviously, by (2.5) we get

V~φ(K,K)=1nSn1φ(ρK(u)ρK(u))ρKn(u)dS(u) =1nSn1ρKn(u)dS(u)=V(K).(2.6)

=1nSn−1ρnK(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,LSon, 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 KKon, φ∈Φ, if AGL(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 KKosn, τ∈[−1,1], and φ∈Φ, then

Δφτ(K)=ΔφτK=ΔφτK.

Proof

For all uSn−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 KKon, τ∈[−1,1], φ ∈ Φ,0 and φ∈Φ, then ΔφτK=ΔφτK if and only if KKosn.

Proof

By (1.8) and (1.11), and notice that (1.6), it is easy to get the desired result. Contrarily, if KKosn, together with Lemma 3.2, we immediately get ΔτφK=ΔτφK for all uSn−1. □

From Lemma 3.3, we obtain immediately the following result.

Corollary 3.4

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

By Lemma 3.2, we see that if ΔφτK=ΔφτK, then ΔφτK=ΔφτK. This means ΔφτKKosn. This and Lemma 3.3 yield

Corollary 3.5

Let KKosn, τ∈[−1,1] and φ∈Φ. If τ ≠ 0, then ΔφτKKosn if and only if KKosn.

Lemma 3.6

Suppose KKosn, τ∈[−1,1] and φ∈Φ, then

ΔφτK=K.(3.1)

Proof

Since KKosn, 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 uSn−1. This gives (3.1). □

Lemma 3.4 immediately infers

Corollary 3.7

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

ΔφτK=ΔφτLK=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 LKon, 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 uSn−1,

φ(1)=f1(τ)φ(hK(u)hΔφτK(u))+f2(τ)φ(hK(u)hΔφτK(u))φ(f1(τ)hK(u)hΔφτK(u)+f2(τ)hK(u)hΔφτK(u))=φ(hΔτK(u)hΔφτK(u)).(4.2)

Therefore, we have for all uSn−1,

hΔτK(u)hΔφτK(u).

i.e.,

ΔτKΔφτK.

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)=hK(u)hΔφτK(u), namely, K=−K. That is to say, KKosn. On the contrary, if KKosn, from Lemma 3.6 and definition (1.3), it is easy to see that ΔτK=ΔτφK.

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(hK(u)hΔφτK(u))=φ(f1(τ)hK(u)hΔφτK(u))+φ(f2(τ)hK(u)hΔφτK(u))=φ(f1(τ)hK(u)hΔφτK(u)+f2(τ)hK(u)hΔφτK(u))=φ(hΔτK(u)hΔφτK(u)).

Thus, hΔτK(u)=hΔτφK(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, LKon, φ∈Φ. Then for any 0<α<1, one gets

V(+φ(α,1α,K,L))V(K)αV(L)1α.(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)+(1α)φ(V(L)1nV(Kφ)1n) φ(αV(K)1nV(Kφ)1n+(1α)V(L)1nV(Kφ)1n) φ(V(K)αnV(L)1αnV(Kφ)1n).

Therefore, we get

1V(K)αnV(L)1αnV(Kφ)1n,

i.e.

V(+φ(α,1α,K,L))V(K)αV(L)1α.

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)1f1(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, KKosn. So we see that if K∉Knos, 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(hK(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)ρK(u))=φ(1).(4.5)

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

V(Δφτ,K)=1nSn1ρΔφτ,Kn(u)dS(u)=1nSn1φ(1)ρΔφτ,Kn(u)dS(u)=1nSn1[f1(τ)φ(ρΔφτ,K(u)ρK(u))+f2(τ)φ(ρΔφτ,K(u)ρK(u))]ρΔφτ,Kn(u)dS(u)=f1(τ)1nSn1φ(ρΔφτ,K(u)ρK(u))ρΔφτ,Kn(u)dS(u)+f2(τ)1nSn1φ(ρΔφτ,K(u)ρK(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 KKosn, from Lemma 3.6, we know that (1.14) is evidently true. If K∉Knos, associated with the condition of equality in (2.7), we know that equality holds in (1.14) if and only if Δτ,φK and K, Δτ,∗φK and −K both are dilates. These together with V(Δτ,φK)=V(K) give that ΔτφKK, i.e., τ=±1. □

Proof of Theorem 1.6

Since ΔτφKKosn, thus ΔτφKc∈Knc, where c is the centroid of ΔτφK. Because of Δτ,K=(ΔτφK)c=(ΔτφKc), 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(ΔφτKc)V((ΔφτKc))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).

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About the article

Received: 2018-01-22

Accepted: 2018-09-05

Published Online: 2018-10-29


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1187–1195, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0098.

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© 2018 Shi et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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