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

- Weidong Wang
^{1}Email author and - Ting Zhang
^{1}

**2014**:252

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

© Wang and Zhang; licensee Springer. 2014

**Received: **8 January 2014

**Accepted: **26 June 2014

**Published: **22 July 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.

## Keywords

*p*th mean bodydual quermassintegralsZhang projection inequalityRogers-Shephard inequality

## 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 ${\omega}_{n}=V(B)$.

*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 $u\in {S}^{n-1}$.

*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 $u\in {S}^{n-1}$.

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].

*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

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

*K*, ${K}^{\ast}$, is defined by (see [1, 2])

*K*is defined by (see [1])

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

*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

*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])

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 $u\in {S}^{n-1}$.

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

*K*and

*L*is defined by (see [9, 10])

*K*and

*L*are defined by

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

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

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.

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.*,

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 ${R}_{p}K=K-{x}_{0}$.

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

*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

*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.*,

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

## Declarations

### 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).

## Authors’ Affiliations

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