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# Inequalities for dual quermassintegrals of the radial *p* th mean bodies

*Journal of Inequalities and Applications*
**volume 2014**, Article number: 252 (2014)

## Abstract

Gardner and Zhang defined the notion of radial *p* th mean body (p>-1) in the Euclidean space {\mathbb{R}}^{n}. In this paper, we obtain inequalities for dual quermassintegrals of the radial *p* th mean bodies. Further, we establish dual quermassintegrals forms of the Zhang projection inequality and the Rogers-Shephard inequality, respectively. Finally, Shephard’s problem concerning the radial *p* th mean bodies is shown when p>0.

**MSC:**52A40, 52A20.

## 1 Introduction

Let {\mathcal{K}}^{n} denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the Euclidean space {\mathbb{R}}^{n} for the set of convex bodies containing the origin in their interiors in {\mathbb{R}}^{n} by {\mathcal{K}}_{o}^{n}. Let {S}^{n-1} denote the unit sphere in {\mathbb{R}}^{n}, denote by V(K) the *n*-dimensional volume of body *K* for the standard unit ball *B* in {\mathbb{R}}^{n}, define {\omega}_{n}=V(B).

If *K* is a compact star-shaped (about the origin) in {\mathbb{R}}^{n}, its radial function, {\rho}_{K}=\rho (K,\cdot ), is defined by (see [1, 2])

for all u\in {S}^{n-1}. If {\rho}_{K} is positive and continuous, *K* will be called a star body (about the origin). Let {\mathcal{S}}_{o}^{n} denote the set of star bodies (about the origin) in {\mathbb{R}}^{n}. Two star bodies *K* and *L* are said to be dilates (of one another) if {\rho}_{K}(u)/{\rho}_{L}(u) is independent of u\in {S}^{n-1}.

The notion of radial *p* th mean body was given by Gardner and Zhang (see [3]). For K\in {\mathcal{K}}^{n}, the radial *p* th mean body {R}_{p}K of *K* is defined for nonzero p>-1 by

for each u\in {S}^{n-1}; define {R}_{0}K by

for each u\in {S}^{n-1}.

In [3], Gardner and Zhang showed the following.

**Theorem 1.A** *If* K\in {\mathcal{K}}^{n}, -1<p<q, *then*

*in each inclusion equality holds if and only if* *K* *is a simplex*. *Here*

*for nonzero* p>-1, {c}_{n,0}={lim}_{p\to 0}{(nB(p+1,n))}^{-\frac{1}{p}}, *and* *DK* *and* {\mathrm{\Pi}}^{\ast}K *denote the difference body and the polar of projection body*, *respectively*.

From Theorem 1.A, Gardner and Zhang [3] again proved the Zhang projection inequality (also see [4]) and the Rogers-Shephard inequality (also see [5]).

**Theorem 1.B** (Zhang projection inequality)

*If* K\in {\mathcal{K}}^{n}, *then*

*with equality if and only if* *K* *is a simplex*.

**Theorem 1.C** (Rogers-Shephard inequality)

*If* K\in {\mathcal{K}}^{n}, *then*

*with equality if and only if* *K* *is a simplex*.

In this paper, we continuously research the radial *p* th mean body. First, we establish inequalities for dual quermassintegrals of the radial *p* th mean body {R}_{p}K as follows.

**Theorem 1.1** *If* K\in {\mathcal{K}}^{n}, p>0, *real* i\ne n, *then there exists* {x}_{0}\in K *such that for* i<n-p *or* i>n,

*for* n-p<i<n,

*In every inequality*, *equality holds if and only if* {R}_{p}K=K-{x}_{0}. *For* i=n-p, (1.7) (*or* (1.8)) *is identic*. *Here*, {\tilde{W}}_{i}(K) *denotes the dual quermassintegrals of* *K* *which are given by* (*see* [6])

*Obviously*, *let* i=0 *in* (1.9), *then*

Let i=0 in Theorem 1.1 and notice that V(K-{x}_{0})=V(K), we easily get the following.

**Corollary 1.1** *If* K\in {\mathcal{K}}^{n}, p>0, *then for* p<n,

*for* p>n,

*All with equality if and only if* {R}_{p}K=K. *For* p=n, *above inequalities are identic*.

Note that Corollary 1.1 can be found in [7].

As an application of Theorem 1.1, we obtain the following dual quermassintegrals form of the Zhang projection inequality.

**Theorem 1.2** *If* K\in {\mathcal{K}}^{n}, p>0, *real* i\ne n, *then there exists* {x}_{0}\in K *such that for* n-p\le i<n,

*with equality for* i=n-p *if and only if* *K* *is a simplex*, *for* n-p<i<n *if and only if* *K* *is a simplex and* {R}_{p}K=K-{x}_{0}.

Note that the case of p=n-i in (1.11) can be found in [8].

If *p* is a positive integer in Theorem 1.2, then by (1.4) we get that

Hence, we have the following.

**Corollary 1.2** *If* K\in {\mathcal{K}}^{n}, *p* *is a positive integer*, *i* *is any real*, *if* n-p\le i<n, *then there exists* {x}_{0}\in K *such that*

*with equality for* i=n-p *if and only if* *K* *is a simplex*, *for* n-p<i<n *if and only if* *K* *is a simplex and* {R}_{p}K=K-{x}_{0}.

Let i=0 in Corollary 1.2, and together with (1.12) and (1.10), we have the following.

**Corollary 1.3** *If* K\in {\mathcal{K}}^{n}, p\ge n *and* *p* *is an integer*, *then*

*with equality for* p=n *if and only if* *K* *is a simplex*, *for* p>n *if and only if* *K* *is a simplex and there exists* {x}_{0}\in K *such that* {R}_{p}K=K-{x}_{0}.

Compared to (1.13) and the Zhang projection inequality (1.5), inequality (1.13) may be regarded as a general Zhang projection inequality.

As another application of Theorem 1.1, we obtain the following dual quermassintegrals form of the Rogers-Shephard inequality.

**Theorem 1.3** *If* K\in {\mathcal{K}}^{n}, p>0 *and real* i\ne n, *if* i\le n-p *or* i>n, *then there exists* {x}_{0}\in K *such that*

*with equality for* i=n-p *if and only if* *K* *is a simplex*, *for* i<n-p *or* i>n *if and only if* *K* *is a simplex and* {R}_{p}K=K-{x}_{0}.

Similarly, if *p* is a positive integer in Theorem 1.3, then by (1.12) we obtain the following.

**Corollary 1.4** *If* K\in {\mathcal{K}}^{n}, *p* *is a positive integer*, *i* *is any real*, *if* i\le n-p *or* i>n, *then there exists* {x}_{0}\in K *such that*

*with equality for* i=n-p *if and only if* *K* *is a simplex*, *for* i<n-p *or* i>n *if and only if* *K* *is a simplex and* {R}_{p}K=K-{x}_{0}.

Taking i=0 in Corollary 1.4, and using (1.12) and (1.10), we get the following.

**Corollary 1.5** *If* K\in {\mathcal{K}}^{n}, *p* *is a positive integer and* p\le n, *then*

*with equality for* p=n *if and only if* *K* *is a simplex*, *for* p<n *if and only if* *K* *is a simplex and there exists* {x}_{0}\in K *such that* {R}_{p}K=K-{x}_{0}.

Compared to (1.15) and the Rogers-Shephard inequality (1.6), inequality (1.15) may be regarded as a general Rogers-Shephard inequality.

In addition, we also give the Shephard-type problem for the radial *p* th mean bodies in Section 4.

## 2 Preliminaries

### 2.1 Support function, difference body and projection body

If K\in {\mathcal{K}}^{n}, then its support function, {h}_{K}=h(K,\cdot ), is defined by (see [1, 2])

where x\cdot y denotes the standard inner product of *x* and *y*.

If K\in {\mathcal{K}}_{o}^{n}, the polar body of *K*, {K}^{\ast}, is defined by (see [1, 2])

If K\in {\mathcal{K}}^{n}, the difference body, DK=K+(-K), of *K* is defined by (see [1])

for all u\in {S}^{n-1}.

For K\in {\mathcal{K}}^{n}, the projection body of *K*, Π*K*, is a centered convex body whose support function is given by (see [1])

for all u\in {S}^{n-1}, where {V}_{n-1} denotes (n-1)-dimensional volume, and K|{u}^{\perp} denotes the image of orthogonal projection of *K* onto the codimensional 1 subspace orthogonal to *u*.

### 2.2 Extended radial function

If *K* is compact star-shaped with respect to x\in {\mathbb{R}}^{n}, its radial function {\rho}_{K}(x,\cdot ) with respect to *x* is defined, for all u\in {S}^{n-1} such that the line through *x* parallel to *u* intersects *K*, by (see [3])

From (1.1) and (2.1), we easily know that

for all u\in {S}^{n-1}. We call {\rho}_{K}(x,\cdot ) the extended radial function of *K* with respect to *x*. If *x* is the origin *o*, then {\rho}_{K}(x,u)={\rho}_{K}(u) for all u\in {S}^{n-1}.

From (2.2) and (1.2), obviously,

### 2.3 {L}_{p}-Dual mixed quermassintegrals

If K,L\in {\mathcal{S}}_{o}^{n}, p>0, \lambda ,\mu \ge 0 (not both zero), the {L}_{p}-radial combination, \lambda \cdot K\phantom{\rule{0.2em}{0ex}}{\tilde{+}}_{p}\phantom{\rule{0.2em}{0ex}}\mu \cdot L\in {\mathcal{S}}_{o}^{n}, of *K* and *L* is defined by (see [9, 10])

Associated with (2.4) and (1.9), we define a class of {L}_{p}-dual mixed quermassintegrals as follows: For K,L\in {\mathcal{S}}_{o}^{n}, p>0, \epsilon >0 and real i\ne n, the {L}_{p}-dual mixed quermassintegrals, {\tilde{W}}_{p,i}(K,L), of *K* and *L* are defined by

Let i=0 in definition (2.5), and together with (1.10), we write that {\tilde{W}}_{p,0}(K,L)={\tilde{V}}_{p}(K,L), then

Here {\tilde{V}}_{p}(K,L) denotes a type of {L}_{p}-dual mixed volume of *K* and *L* which is defined in [9, 11] (for p\ge 1 also see [12]).

From definition (2.5), the integral representation of {L}_{p}-dual mixed quermassintegrals can be established as follows.

**Theorem 2.1** *If* K,L\in {\mathcal{S}}_{o}^{n}, p>0, *and real* i\ne n, *then*

*Proof* From (2.4) and (2.5), for i\ne n, we have that

By Hospital’s rule we see that

thus we get formula (2.6) by definition (2.5). □

From (2.6), we easily know that

The Minkowski inequality for the {L}_{p}-dual mixed quermassintegrals is given as follows.

**Theorem 2.2** *Let* K,L\in {\mathcal{S}}_{o}^{n}, p>0, *and real* i\ne n, *then for* i<n-p,

*for* n-p<i<n *or* i>n,

*In every inequality*, *equality holds if and only if* *K* *and* *L* *are dilates*. *For* i=n-p, (2.9) (*or* (2.10)) *is identic*.

*Proof* For i<n-p, from (2.6) and together with the Hölder inequality (see [13]), we have that

which gives inequality (2.9) when i<n-p. According to the condition that equality holds for the Hölder inequality, we know that the equality holds in inequality (2.9) if and only if *K* and *L* are dilates.

Similarly, we can prove for n-p<i<n or i>n, inequality (2.10) is true.

For i=n-p, by (2.8) and (2.3) then

and

thus (2.9) (or (2.10)) is identic when i=n-p. □

## 3 Proofs of the theorems

The proofs of the theorems require the following lemma.

**Lemma 3.1** *If* K\in {\mathcal{K}}^{n}, p>0, *and real* i\ne n, *then for any* Q\in {\mathcal{S}}_{o}^{n},

*Proof* Using (2.6) and (2.3), then for any Q\in {\mathcal{S}}_{o}^{n}, we have that

□

*Proof of Theorem 1.1* For i<n-p, let Q={R}_{p}K in (3.1), this together with (2.6), (2.7) and (2.9) gives

*i.e.*,

Therefore, according to the integral mean value theorem, there exists {x}_{0}\in K such that

Since p>0 and i<n-p, thus we get inequality (1.7). According to the condition that equality holds in inequality (2.9), we see that with equality in (1.7) if and only if {R}_{p}K and K-{x}_{0} are dilates. This combined with (1.7), we know that equality holds in (1.7) if and only if {R}_{p}K=K-{x}_{0}.

Similarly, for n-p<i<n or i>n, from inequality (2.10) and equality (3.1), then

Hence, we have that for i>n and p>0,

for n-p<i<n and p>0,

From this, we get inequality (1.7) and inequality (1.8), respectively, and equality holds in the above inequalities if and only if {R}_{p}K=K-{x}_{0}.

For i=n-p, by (2.8) and (3.1) we see that (1.7) (or (1.8)) is identic. □

*Proof of Theorem 1.2* From (1.3), we have that {c}_{n,p}{R}_{p}K\subseteq nV(K){\mathrm{\Pi}}^{\ast}K for p>-1, then

with equality if and only if *K* is a simplex. Hence, together with (1.8), then for n-p<i<n and p>0, we obtain that

which is desired (1.11).

Associated with the cases of equality holding in (3.2) and (1.8), we see that equality holds in (1.11) for i=n-p if and only if *K* is a simplex, for n-p<i<n if and only if *K* is a simplex and {R}_{p}K=K-{x}_{0}. □

*Proof of Theorem 1.3* From (1.3), we know that DK\subseteq {c}_{n,p}{R}_{p}K for p>-1, thus

with equality if and only if *K* is a simplex. Hence, together with (1.7), then for p>0, i<n-p or i>n, we get that

this is just (1.14).

Combining with the cases of equality holding in (3.3) and (1.7), we see that equality holds in (1.14) for i=n-p if and only if *K* is a simplex, for i<n-p or i>n if and only if *K* is a simplex and {R}_{p}K=K-{x}_{0}. □

## 4 Shephard-type problem

In this section, we research the Shephard-type problem for the radial *p* th mean bodies. Recall that Zhou and Wang in [7] gave the Shephard-type problem for the radial *p* th mean bodies as follows.

**Theorem 4.A** *Let* K,L\in {\mathcal{K}}^{n}, p>0, *if* {R}_{p}K\subseteq {R}_{p}L, *then*

*with equality if and only if* {R}_{p}K={R}_{p}L *and* *K* *is a translation of* *L*.

Here, we obtain a stronger result for the Shephard-type problem of the radial *p* th mean bodies. Our result is the following theorem.

**Theorem 4.1** *Let* K,L\in {\mathcal{K}}^{n}, p>0, *if* {R}_{p}K\subseteq {R}_{p}L, *then there exist* {x}_{0}\in K *and* {y}_{0}\in L *such that*

*with equality if and only if* {R}_{p}K={R}_{p}L *and* K-{x}_{0}=L-{y}_{0}.

*Proof* Since {R}_{p}K\subseteq {R}_{p}L for p>0, thus {\rho}_{{R}_{p}K}^{p}(u)\le {\rho}_{{R}_{p}L}^{p}(u) for all u\in {S}^{n-1}, *i.e.*,

Therefore, by the integral mean value theorem, there exist {x}_{0}\in K and {y}_{0}\in L such that

thus

for all u\in {S}^{n-1}. This yields (4.1). □

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## Acknowledgements

Research is supported in part by the Natural Science Foundation of China (Grant No. 11371224) and Innovation Foundation of Graduate Student of China Three Gorges University (Grant No. 2014CX097).

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Wang, W., Zhang, T. Inequalities for dual quermassintegrals of the radial *p* th mean bodies.
*J Inequal Appl* **2014**, 252 (2014). https://doi.org/10.1186/1029-242X-2014-252

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DOI: https://doi.org/10.1186/1029-242X-2014-252