# On dual mixed quermassintegral quotient functions

## Abstract

We introduce the notion of dual mixed quermassintegral quotient functions and establish the Brunn-Minkowski inequalities for them in this paper.

## Introduction and main results

The setting for this paper is Euclidean n-space $$\mathbb{R}^{n}$$. Let $$\mathcal{S}^{n}_{o}$$ denote the set of star bodies containing the origin in their interiors in $$\mathbb{R}^{n}$$. Let $$S^{n-1}$$ denote the unit sphere in $$\mathbb{R}^{n}$$, and let $$V(K)$$ denote the n-dimensional volume of body K. For the standard unit ball B in $$\mathbb{R}^{n}$$, we use $$\omega_{n}=V(B)$$ to denote its volume.

In 1975, Lutwak (see ) gave the notion of dual mixed volumes as follows: For $$K_{1}, K_{2}, \ldots, K_{n}\in\mathcal{S}^{n}_{o}$$, the dual mixed volume, $$\widetilde{V}(K_{1}, K_{2}, \ldots, K_{n})$$, of $$K_{1}, K_{2}, \ldots, K_{n}$$ is defined by

$$\widetilde{V}(K_{1},\ldots,K_{n})=\frac{1}{n}\int _{S^{n-1}}\rho (K_{1},u)\cdots\rho(K_{n},u) \,dS(u).$$
(1.1)

Taking $$K_{1}=\cdots=K_{n-i}=K$$, $$K_{n-i+1}=\cdots=K_{n}=L$$ in (1.1), we write $$\widetilde{V}_{i}(K, L)=\widetilde{V}(K, n-i; L, i )$$, where K appears $$n-i$$ times and L appears i times. Then

$$\widetilde{V}_{i}(K, L)=\frac{1}{n}\int_{S^{n-1}} \rho(K,u)^{n-i}\rho (L,u)^{i}\, dS(u).$$
(1.2)

Let $$L=B$$ in (1.2) and notice $$\rho(B, \cdot)=1$$, and allow i is any real, then the dual quermassintegrals can be defined as follows: For $$K\in\mathcal{S}^{n}_{o}$$ and i is any real, the dual quermassintegrals, $$\widetilde{W}_{i}(K)$$, of K are given by (see )

$$\widetilde{W}_{i}(K)=\frac{1}{n}\int_{S^{n-1}} \rho(K,u)^{n-i}\,dS(u).$$
(1.3)

Associated with dual quermassintegrals, Zhao (see ) defined the dual quermassintegral quotient functions of a star body K by

$$Q_{\widetilde{W}_{i,j}(K)}=\frac{\widetilde{W}_{i}(K)}{\widetilde {W}_{j}(K)}\quad (i,j\in\mathbb{R}).$$
(1.4)

Further, in  the Brunn-Minkowski type inequalities for the dual quermassintegral quotient functions of star bodies were established as follows.

### Theorem A

If $$K, L\in\mathcal{S}^{n}_{o}$$ and reals i, j satisfy $$i\leq n-1\leq j\leq n$$, then

$$Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(K\, \,\tilde{+}\,\, L)} \leq Q^{\frac {1}{j-i}}_{\widetilde{W}_{i,j}(K)}+Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(L)}.$$

Here $$\tilde{+}$$ is the radial Minkowski sum.

### Theorem B

If $$K, L\in\mathcal{S}^{n}_{o}$$ and reals i, j satisfy $$i\leq1\leq j\leq n$$, then

$$Q^{\frac{n-1}{j-i}}_{\widetilde{W}_{i,j}(K\,\breve{+}\,L)} \leq Q^{\frac {n-1}{j-i}}_{\widetilde{W}_{i,j}(K)}+Q^{\frac{n-1}{j-i}}_{\widetilde {W}_{i,j}(L)}.$$

Here $$\breve{+}$$ is the radial Blaschke sum.

### Theorem C

If $$K, L\in\mathcal{S}^{n}_{o}$$ and reals i, j satisfy $$i\leq-1\leq j\leq n$$, then

$$\frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(K\,\hat{+}\,L)}}{V(K\hat {+}L)} \leq\frac{Q^{\frac{n+1}{j-i}}_{\widetilde {W}_{i,j}(K)}}{V(K)}+\frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(L)}}{V(L)}.$$

Here $$\hat{+}$$ is the harmonic Blaschke sum.

Motivated by the work of Zhao, we give the following definition of dual mixed quermassintegral quotient function.

Let $$K_{1}=\cdots=K_{n-i-1}=K$$, $$K_{n-i}=\cdots=K_{n-1}=B$$, $$K_{n}=L$$ in (1.1), then we write $$\widetilde{W}_{i}(K, L)=\widetilde{V}(K, n-i-1; B, i; L, 1)$$, where K appears $$n-i-1$$ times, B appears i times and L appears 1 time. Here, we allow i to be any real and define as follows: For $$K, L\in\mathcal{S}^{n}_{o}$$ and i any real, the dual mixed quermassintegrals, $$\widetilde{W}_{i}(K, L)$$, of K and L are given by

$$\widetilde{W}_{i}(K, L)=\frac{1}{n}\int_{S^{n-1}} \rho(K,u)^{n-i-1}\rho (L,u)\,dS(u).$$
(1.5)

Obviously, from (1.3) and (1.5), we have $$\widetilde{W}_{i}(K, K)=\widetilde{W}_{i}(K)$$. According to (1.5), we define the following.

### Definition 1.1

Let $$K, L\in\mathcal{S}^{n}_{o}$$ and $$i,j\in\mathbb{R}$$, the dual mixed quermassintegral quotient function, $$Q_{\widetilde{W}_{i,j}(K,L)}$$, of K and L can be defined by

$$Q_{\widetilde{W}_{i,j}(K,L)}=\frac{\widetilde{W}_{i}(K,L)}{\widetilde {W}_{j}(K, L)}.$$
(1.6)

Obviously, if $$L=K$$, then (1.6) is just (1.4).

The aim of this paper is to establish the following Brunn-Minkowski type inequalities for dual mixed quermassintegral quotient functions of star bodies.

### Theorem 1.1

For $$K, K', L\in\mathcal {S}^{n}_{o}$$, if $$i\leq n-2\leq j< n-1$$, then

$$Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(K\,\tilde{+}\,K',L)} \leq Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(K,L)}+Q^{\frac {1}{j-i}}_{\widetilde{W}_{i,j}(K',L)};$$
(1.7)

if $$n-2\leq i< n-1<j$$, then

$$Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(K\,\tilde{+}\,K',L)} \geq Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(K,L)}+Q^{\frac {1}{j-i}}_{\widetilde{W}_{i,j}(K',L)}.$$
(1.8)

In each case, equality holds if and only if K and $$K'$$ are dilates. Here $$\tilde{+}$$ is the radial Minkowski sum.

### Theorem 1.2

For $$K, K', L\in\mathcal {S}^{n}_{o}$$, if $$i\leq0\leq j< n-1$$, then

$$Q^{\frac{n-1}{j-i}}_{\widetilde{W}_{i,j}(K\,\breve{+}\,K',L)} \leq Q^{\frac{n-1}{j-i}}_{\widetilde{W}_{i,j}(K,L)}+Q^{\frac {n-1}{j-i}}_{\widetilde{W}_{i,j}(K',L)};$$
(1.9)

if $$0\leq i< n-1< j$$, then

$$Q^{\frac{n-1}{j-i}}_{\widetilde{W}_{i,j}(K\,\breve{+}\,K',L)} \geq Q^{\frac{n-1}{j-i}}_{\widetilde{W}_{i,j}(K,L)}+Q^{\frac {n-1}{j-i}}_{\widetilde{W}_{i,j}(K',L)}.$$
(1.10)

In each case, equality holds if and only if K and $$K'$$ are dilates. Here $$\breve{+}$$ is the radial Blaschke sum.

### Theorem 1.3

For $$K, K', L\in\mathcal {S}^{n}_{o}$$, if $$i\leq-2\leq j< n-1$$, then

$$\frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(K\,\hat{+}\,K',L)} }{V(K\,\hat{+}\,K')}\leq\frac{Q^{\frac{n+1}{j-i}}_{\widetilde {W}_{i,j}(K,L)}}{V(K)} +\frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(K',L)}}{V(K')};$$
(1.11)

if $$-2\leq i< n-1< j$$, then

$$\frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(K\,\hat{+}\,K',L)} }{V(K\,\hat{+}\,K')}\geq\frac{Q^{\frac{n+1}{j-i}}_{\widetilde {W}_{i,j}(K,L)}}{V(K)} +\frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(K',L)}}{V(K')}.$$
(1.12)

In each case, equality holds if and only if K and $$K'$$ are dilates. Here $$\hat{+}$$ is the harmonic Blaschke sum.

## Preliminaries

For a compact set K in $$\mathbb{R}^{n}$$ which is star shaped with respect to the origin, we define the radial function $$\rho_{K}(u)=\rho(K,u)$$ of K by

$$\rho(K,u)=\max\{\lambda\geq0: \lambda u\in K\}, \quad u\in S^{n-1}.$$

If $$\rho_{K}$$ is positive and continuous, K will be called a star body (about the origin). Two star bodies K and L are said to be dilates (of one another) if $$\rho_{K}(u)/\rho_{L}(u)$$ is independent of $$u\in S^{n-1}$$.

For $$K_{1}, K_{2}\in\mathcal{S}^{n}_{o}$$, and $$\lambda_{1},\lambda_{2}\geq 0$$ (not both 0), the radial function of the radial Minkowski linear combination $$\lambda_{1}K_{1}\,\tilde{+}\,\lambda_{2}K_{2}$$ is given by Zhang (see ):

$$\rho(\lambda_{1}K_{1}\,\tilde{+}\,\lambda_{2}K_{2},u)= \lambda_{1}\rho (K_{1},u)+\lambda_{2} \rho(K_{2},u).$$
(2.1)

For $$K_{1}, K_{2}\in\mathcal{S}^{n}_{o}$$, and $$\lambda_{1},\lambda_{2}\geq 0$$ (not both 0), the radial Blaschke linear combination $$\lambda _{1}\cdot K_{1} \,\check{+}\,\lambda_{2}\cdot K_{2}$$ is a star body whose radial function is given by Lutwak (see ):

$$\rho(\lambda_{1}\cdot K_{1}\,\check{+}\, \lambda_{2}\cdot K_{2},u)^{n-1}= \lambda_{1}\rho(K_{1},u)^{n-1}+\lambda_{2} \rho (K_{2},u)^{n-1}.$$
(2.2)

For $$K_{1}, K_{2}\in\mathcal{S}^{n}_{o}$$, and $$\lambda_{1},\lambda_{2}\geq 0$$ (not both 0), the harmonic Blaschke linear combination $$\lambda _{1} \circ K_{1} \,\hat{+}\, \lambda_{2}\circ K_{2}$$ is a star body whose radial function is given by Lutwak (see ):

$$\frac{\rho(\lambda_{1}\circ K_{1} \,\hat{+}\, \lambda_{2}\circ K_{2},u)^{n+1}}{V(\lambda_{1}\circ K_{1} \,\hat{+}\, \lambda_{2}\circ K_{2})}=\lambda_{1}\frac{\rho(K_{1},u)^{n+1}}{V( K_{1})}+\lambda _{2}\frac{\rho(K_{2},u)^{n+1}}{V( K_{2})}.$$
(2.3)

## Proofs of theorems

According to a generalization of the Dresher inequality (see ), we get the reverse Dresher inequality.

### Lemma 3.1

(Dresher’s inequality)

Let functions $$f_{1}, f_{2}, g_{1}, g_{2} \geq0$$, E is a bounded measurable subset in $$\mathbb{R}^{n}$$. If $$p \geq1 \geq r \geq0$$, then

$$\biggl(\frac{\int_{E}(f_{1}+f_{2})^{p}\,dx}{\int_{E}(g_{1}+g_{2})^{r}\,dx} \biggr)^{\frac{1}{p-r}} \leq \biggl(\frac{\int_{E}f_{1}^{p}\,dx}{\int_{E}g_{1}^{r}\,dx} \biggr)^{\frac{1}{p-r}} + \biggl(\frac{\int_{E}f_{2}^{p}\,dx}{\int_{E}g_{2}^{r}\,dx} \biggr)^{\frac{1}{p-r}},$$
(3.1)

equality holds if and only if $$f_{1}/ f_{2}= g_{1}/g_{2}$$.

### Lemma 3.2

(Reverse Dresher’s inequality)

Let functions $$f_{1}, f_{2}, g_{1}, g_{2} \geq0$$, E is a bounded measurable subset in $$\mathbb{R}^{n}$$. If $$1\geq p>0>r$$, then

$$\biggl(\frac{\int_{E}(f_{1}+f_{2})^{p}\,dx}{\int_{E}(g_{1}+g_{2})^{r}\,dx} \biggr)^{\frac{1}{p-r}} \geq \biggl(\frac{\int_{E}f_{1}^{p}\,dx}{\int_{E}g_{1}^{r}\,dx} \biggr)^{\frac{1}{p-r}} + \biggl(\frac{\int_{E}f_{2}^{p}\,dx}{\int_{E}g_{2}^{r}\,dx} \biggr)^{\frac{1}{p-r}},$$
(3.2)

equality holds if and only if $$f_{1}/ f_{2}= g_{1}/g_{2}$$.

### Proof of Lemma 3.2

If $$f_{1}, f_{2}, g_{1}, g_{2} \geq0$$, and $$1\geq p>0>r$$, according to the Minkowski inequality,

\begin{aligned}& \biggl(\int_{E}(f_{1}+f_{2})^{p} \,dx \biggr)^{\frac{1}{p}}\geq \biggl(\int_{E}f_{1}^{p} \,dx \biggr)^{\frac{1}{p}}+ \biggl(\int_{E} f_{2}^{p}\,dx \biggr)^{\frac{1}{p}}, \\& \biggl(\int_{E}(g_{1}+g_{2})^{r} \,dx \biggr)^{\frac{1}{r}}\geq \biggl(\int_{E}g_{1}^{r} \,dx \biggr)^{\frac{1}{r}}+ \biggl(\int_{E} g_{2}^{r}\,dx \biggr)^{\frac{1}{r}}. \end{aligned}

For $$1\geq p>0>r$$, we have

\begin{aligned}& \int_{E}(f_{1}+f_{2})^{p} \,dx \geq \biggl(\biggl(\int_{E}f_{1}^{p} \,dx\biggr)^{\frac{1}{p}}+\biggl(\int_{E} f_{2}^{p}\,dx\biggr)^{\frac{1}{p}} \biggr)^{p}, \end{aligned}
(3.3)
\begin{aligned}& \int_{E}(g_{1}+g_{2})^{r} \,dx \leq \biggl(\biggl(\int_{E}g_{1}^{r} \,dx\biggr)^{\frac{1}{r}}+\biggl(\int_{E} g_{2}^{r}\,dx\biggr)^{\frac{1}{r}} \biggr)^{r}. \end{aligned}
(3.4)

According to the Hölder inequality, $$\frac{p-r}{p}>1$$, and (3.3), (3.4),

\begin{aligned} \biggl(\frac{\int_{E}(f_{1}+f_{2})^{p}\,dx}{\int_{E}(g_{1}+g_{2})^{r}\,dx} \biggr)^{\frac{1}{p-r}} \geq& \biggl[\frac{((\int_{E}f_{1}^{p}\,dx)^{\frac{1}{p}}+(\int_{E} f_{2}^{p}\,dx)^{\frac{1}{p}})^{p}}{((\int_{E}g_{1}^{r}\,dx)^{\frac{1}{r}}+(\int_{E} g_{2}^{r}\,dx)^{\frac{1}{r}})^{r}} \biggr]^{\frac{1}{p-r}} \\ =&\frac{ [(\int_{E}f_{1}^{p}\,dx)^{\frac{1}{p}}+(\int_{E} f_{2}^{p}\,dx)^{\frac {1}{p}} ]^{\frac{p}{p-r}}}{ [(\int_{E}g_{1}^{r}\,dx)^{\frac {1}{r}}+(\int_{E} g_{2}^{r}\,dx)^{\frac{1}{r}} ]^{\frac{r}{p-r}}} \\ =& \biggl[ \biggl(\biggl(\int_{E}f_{1}^{p} \,dx\biggr)^{\frac{1}{p-r}} \biggr)^{\frac {p-r}{p}}+ \biggl(\biggl(\int _{E} f_{2}^{p}\,dx\biggr)^{\frac{1}{p-r}} \biggr)^{\frac {p-r}{p}} \biggr]^{\frac{p}{p-r}} \\ &{}\times \biggl[ \biggl(\biggl(\int_{E}g_{1}^{r} \,dx\biggr)^{\frac{-1}{p-r}} \biggr)^{\frac {-(p-r)}{r}}+ \biggl(\biggl(\int _{E} g_{2}^{r}\,dx\biggr)^{\frac{-1}{p-r}} \biggr)^{\frac {-(p-r)}{r}} \biggr]^{\frac{-r}{p-r}} \\ \geq& \biggl(\int_{E}f_{1}^{p}\,dx \biggr)^{\frac{1}{p-r}} \biggl(\int_{E}g_{1}^{r} \,dx \biggr)^{\frac{-1}{p-r}}+ \biggl(\int_{E}f_{2}^{p} \,dx \biggr)^{\frac {1}{p-r}} \biggl(\int_{E}g_{2}^{r} \,dx \biggr)^{\frac{-1}{p-r}} \\ =& \biggl(\frac{\int_{E}f_{1}^{p}\,dx}{\int_{E}g_{1}^{r}\,dx} \biggr)^{\frac {1}{p-r}}+ \biggl(\frac{\int_{E}f_{2}^{p}\,dx}{\int_{E}g_{2}^{r}\,dx} \biggr)^{\frac{1}{p-r}}. \end{aligned}

According to the equality condition of the Minkowski inequality and the Hölder inequality, equality holds in (3.2) if and only if $$f_{1}/ f_{2}= g_{1}/g_{2}$$. □

### Proof of Theorem 1.1

From (2.1), for $$K, K', L\in\mathcal {S}^{n}_{o}$$,

\begin{aligned} \widetilde{W}_{n-p-1}\bigl(K\,\tilde{+}\,K',L\bigr) =& \frac{1}{n}\int_{S^{n-1}}\rho_{K\,\tilde{+}\,K'}^{p}(u) \rho_{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}}\bigl(\rho_{K}(u)+ \rho_{K'}(u)\bigr)^{p}\rho_{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}}\bigl(\rho_{K}(u) \rho^{\frac {1}{p}}_{L}(u)+\rho_{K'}(u)\rho^{\frac{1}{p}} _{L}(u)\bigr)^{p}\,dS(u) \end{aligned}
(3.5)

and

$$\widetilde{W}_{n-r-1}\bigl(K\,\tilde{+}\,K',L\bigr)= \frac{1}{n}\int_{S^{n-1}}\bigl(\rho_{K}(u) \rho^{\frac{1}{r}} _{L}(u)+\rho_{K'}(u)\rho ^{\frac{1}{r}} _{L}(u)\bigr)^{r}\,dS(u).$$
(3.6)

From (3.1), (3.5), and (3.6), for $$p\geq1\geq r>0$$, we have

\begin{aligned} \biggl(\frac{\widetilde{W}_{n-p-1}(K\, \tilde {+}\, K',L)}{\widetilde{W}_{n-r-1}(K\,\tilde{+}\,K',L)} \biggr) ^{\frac{1}{p-r}} =& \biggl(\frac{\int_{S^{n-1}}(\rho_{K}(u)\rho^{\frac {1}{p}}_{L}(u)+\rho_{K'}(u)\rho^{\frac{1}{p}} _{L}(u))^{p}\,dS(u)}{\int_{S^{n-1}}(\rho_{K}(u)\rho^{\frac{1}{r}} _{L}(u)+\rho_{K'}(u)\rho^{\frac{1}{r}} _{L}(u))^{r}\,dS(u)} \biggr)^{\frac{1}{p-r}} \\ \leq& \biggl(\frac{\int_{S^{n-1}}(\rho_{K}(u)\rho^{\frac{1}{p}} _{L}(u))^{p}\,dS(u)}{\int_{S^{n-1}}(\rho_{K}(u)\rho^{\frac{1}{r}} _{L}(u))^{r}\,dS(u)} \biggr)^{\frac{1}{p-r}} \\ &{}+ \biggl( \frac{\int_{S^{n-1}}(\rho_{K'}(u)\rho^{\frac{1}{p}} _{L}(u))^{p}\,dS(u)}{\int_{S^{n-1}}(\rho_{K'}(u)\rho^{\frac{1}{r}} _{L}(u))^{r}\,dS(u)} \biggr)^{\frac{1}{p-r}} \\ =& \biggl(\frac{\int_{S^{n-1}}\rho_{K}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K}^{r}(u)\rho_{L}(u)\,dS(u)} \biggr)^{\frac{1}{p-r}} \\ &{}+ \biggl(\frac{\int_{S^{n-1}}\rho_{K'}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K'}^{r}(u)\rho_{L}(u)\,dS(u)} \biggr)^{\frac{1}{p-r}} \\ =& \biggl(\frac{\widetilde{W}_{n-p-1}(K,L)}{\widetilde {W}_{n-r-1}(K,L)} \biggr)^{\frac{1}{p-r}}+ \biggl(\frac{\widetilde{W}_{n-p-1}(K',L)}{\widetilde {W}_{n-r-1}(K',L)} \biggr)^{\frac{1}{p-r}}. \end{aligned}
(3.7)

According to the equality condition of inequality (3.1), we see that equality holds in (3.7) if and only if K and L, $$K'$$, and L are dilates, respectively. So K and $$K'$$ are dilates.

Let $$i=n-p-1$$, $$j=n-r-1$$, then $$p\geq1\geq r>0$$ and $$i\leq n-2\leq j< n-1$$ are equivalent. This and (3.7) yield inequality (1.7) and its equality condition.

Similarly, if $$1\geq p>0>r$$, according to (3.2), (3.5), and (3.6), we have

$$\biggl(\frac{\widetilde{W}_{n-p-1}(K\,\tilde{+}\,K',L)}{\widetilde {W}_{n-r-1}(K\,\tilde{+}\,K',L)} \biggr) ^{\frac{1}{p-r}}\geq \biggl(\frac{\widetilde{W}_{n-p-1}(K,L)}{\widetilde{W}_{n-r-1}(K,L)} \biggr)^{\frac{1}{p-r}}+ \biggl(\frac{\widetilde{W}_{n-p-1}(K',L)}{\widetilde {W}_{n-r-1}(K',L)} \biggr)^{\frac{1}{p-r}},$$
(3.8)

and equality holds if and only if K and $$K'$$ are dilates.

Let $$i=n-p-1$$, $$j=n-r-1$$, then (3.8) gives inequality (1.8) and its equality condition. □

### Proof of Theorem 1.2

From (2.2), for $$K, K', L \in\mathcal {S}^{n}_{o}$$, we have

\begin{aligned} \widetilde{W}_{n-p-1}\bigl(K\,\breve{+}\,K',L\bigr) =& \frac{1}{n}\int_{S^{n-1}}\rho _{K\,\breve{+}\,K'}^{p}(u) \rho_{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}}\bigl(\rho_{K}^{n-1}(u)+ \rho _{K'}^{n-1}(u)\bigr)^{\frac{p}{n-1}}\rho_{L}(u) \,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}} \bigl(\rho_{K}^{n-1}(u) \rho_{L}^{\frac {n-1}{p}}(u)+\rho_{K'}^{n-1}(u) \rho_{L}^{\frac{n-1}{p}}(u) \bigr)^{\frac{p}{n-1}}\,dS(u) \end{aligned}
(3.9)

and

\begin{aligned} \widetilde{W}_{n-r-1}\bigl(K\,\breve{+}\,K',L\bigr) =& \frac{1}{n}\int_{S^{n-1}}\rho _{K\,\breve{+}\,K'}^{r}(u) \rho_{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}} \bigl(\rho_{K}^{n-1}(u) \rho_{L}^{\frac {n-1}{r}}(u)+\rho_{K'}^{n-1}(u) \rho_{L}^{\frac{n-1}{r}}(u) \bigr)^{\frac{r}{n-1}}\,dS(u). \end{aligned}
(3.10)

According to (3.1), (3.9), and (3.10), for $$p\geq n-1\geq r>0$$,

\begin{aligned} Q^{\frac{n-1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K\,\breve{+}\,K',L)} =& \biggl[\frac{\widetilde{W}_{n-p-1}(K\,\breve{+}\,K',L)}{\widetilde {W}_{n-r-1}(K\,\breve{+}\,K',L)} \biggr]^{\frac{n-1}{p-r}} \\ =& \biggl[\frac{\int_{S^{n-1}} (\rho_{K}^{n-1}(u)\rho_{L}^{\frac {n-1}{p}}(u)+\rho_{K'}^{n-1}(u)\rho_{L}^{\frac{n-1}{p}}(u) )^{\frac{p}{n-1}}\,dS(u)}{\int_{S^{n-1}} (\rho_{K}^{n-1}(u)\rho _{L}^{\frac{n-1}{r}}(u)+\rho_{K'}^{n-1}(u)\rho_{L}^{\frac {n-1}{r}}(u) )^{\frac{r}{n-1}}\,dS(u)} \biggr]^{\frac{n-1}{p-r}} \\ \leq& \biggl[\frac{\int_{S^{n-1}} (\rho_{K}^{n-1}(u)\rho_{L}^{\frac {n-1}{p}}(u) )^{\frac{p}{n-1}}\,dS(u)}{\int_{S^{n-1}} (\rho _{K}^{n-1}(u)\rho_{L}^{\frac{n-1}{r}}(u) )^{\frac {r}{n-1}}\,dS(u)} \biggr]^{\frac{n-1}{p-r}} \\ &{}+ \biggl[\frac{\int_{S^{n-1}} (\rho_{K'}^{n-1}(u)\rho_{L}^{\frac {n-1}{p}}(u) )^{\frac{p}{n-1}}\,dS(u)}{\int_{S^{n-1}} (\rho _{K'}^{n-1}(u)\rho_{L}^{\frac{n-1}{r}}(u) )^{\frac {r}{n-1}}\,dS(u)} \biggr]^{\frac{n-1}{p-r}} \\ =& \biggl[\frac{\int_{S^{n-1}}\rho_{K}^{p}(u)\rho _{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K}^{r}(u)\rho_{L}(u)\,dS(u)} \biggr]^{\frac{n-1}{p-r}}+ \biggl[\frac{\int_{S^{n-1}}\rho_{K'}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K'}^{r}(u)\rho_{L}(u)\,dS(u)} \biggr]^{\frac{n-1}{p-r}} \\ =& \biggl[\frac{\widetilde{W}_{n-p-1}(K,L)}{\widetilde {W}_{n-r-1}(K,L)} \biggr]^{\frac{n-1}{p-r}} + \biggl[\frac{\widetilde{W}_{n-p-1}(K',L)}{ \widetilde{W}_{n-r-1}(K',L)} \biggr]^{\frac{n-1}{p-r}} \\ =&Q^{\frac{n-1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K,L)}+Q^{\frac {n-1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K',L)}. \end{aligned}

Then

$$Q^{\frac{n-1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K\breve {+}K',L)}\leq Q^{\frac{n-1}{p-r}}_{\widetilde {W}_{n-p-1,n-r-1}(K,L)}+Q^{\frac{n-1}{p-r}}_{\widetilde {W}_{n-p-1,n-r-1}(K',L)}.$$
(3.11)

According to the equality condition of inequality (3.1), we see that equality holds in (3.11) if and only if K and $$K'$$ are dilates.

Let $$i=n-p-1$$ and $$j=n-r-1$$, then $$p\geq n-1\geq r>0$$ and $$i\leq0\leq j< n-1$$ are equivalent. This and (3.11) yield inequality (1.9) and its equality condition.

Similarly, if $$n-1\geq p>0>r$$, according to (3.2), (3.9), and (3.10), we have

$$\biggl(\frac{\widetilde{W}_{n-p-1}(K\,\tilde{+}\,K',L)}{\widetilde {W}_{n-r-1}(K\,\tilde{+}\,K',L)} \biggr) ^{\frac{1}{p-r}}\geq \biggl(\frac{\widetilde{W}_{n-p-1}(K,L)}{\widetilde{W}_{n-r-1}(K,L)} \biggr)^{\frac{1}{p-r}}+ \biggl(\frac{\widetilde{W}_{n-p-1}(K',L)}{\widetilde {W}_{n-r-1}(K',L)} \biggr)^{\frac{1}{p-r}},$$
(3.12)

and equality holds if and only if K and $$K'$$ are dilates.

Let $$i=n-p-1$$ and $$j=n-r-1$$, then (3.12) gives inequality (1.10) and its equality condition. □

### Proof of Theorem 1.3

From (2.3), for $$K, K', L \in\mathcal {S}^{n}_{o}$$,

\begin{aligned} \frac{\widetilde{W}_{n-p-1}(K\,\hat{+}\,K', L)}{V(K\hat {+}K')^{p/(n+1)}} =&\frac{1}{n}\int_{S^{n-1}} \frac{\rho_{K\,\hat{+}\,K'}^{p}(u)\rho _{L}(u)}{V(K\,\hat{+}\,K')^{p/(n+1)}}\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}} \biggl(\frac{\rho_{K\hat {+}K'}^{n+1}(u)}{V(K\,\hat{+}\,K')} \biggr)^{\frac{p}{n+1}}\rho _{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}} \biggl(\frac{\rho_{K}^{n+1}(u)}{V(K)}+ \frac {\rho_{K'}^{n+1}(u)}{V(K')} \biggr)^{\frac{p}{n+1}}\rho_{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}} \biggl(\frac{\rho_{K}^{n+1}(u)\rho _{L}^{\frac{n+1}{p}}(u)}{V(K)}+ \frac{\rho_{K'}^{n+1}(u)\rho_{L}^{\frac{n+1}{p}}(u)}{V(K')} \biggr)^{\frac{p}{n+1}}\,dS(u) \end{aligned}
(3.13)

and

$$\frac{\widetilde{W}_{n-r-1}(K\,\hat{+}\,K', L)}{V(K\hat {+}K')^{r/(n+1)}} =\frac{1}{n}\int_{S^{n-1}} \biggl(\frac{\rho_{K}^{n+1}(u)\rho _{L}^{\frac{n+1}{r}}(u)}{V(K)}+ \frac{\rho_{K'}^{n+1}(u)\rho_{L}^{\frac{n+1}{r}}(u)}{V(K')} \biggr)^{\frac{r}{n+1}}\,dS(u).$$
(3.14)

According to (3.1), (3.13), and (3.14), for $$p\geq n+1>r>0$$,

\begin{aligned} Q^{\frac{n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K\,\hat{+}\,K',L)} =& \biggl(\frac{\widetilde{W}_{n-p-1}(K\,\hat{+}\,K',L)}{ \widetilde{W}_{n-r-1}(K\,\hat{+}\,K',L)} \biggr)^{\frac{n+1}{p-r}} \\ =&V\bigl(K\,\hat{+}\,K'\bigr) \biggl[\frac{\int_{S^{n-1}} (\frac{\rho_{K}^{n+1}(u)}{V(K)}\rho_{L}^{\frac {n+1}{p}}(u)+\frac{\rho_{K'}^{n+1}(u)}{V(K')}\rho_{L}^{\frac {n+1}{p}}(u) )^{\frac{p}{n+1}}\,dS(u)}{\int_{S^{n-1}} (\frac{\rho _{K}^{n+1}(u)}{V(K)}\rho_{L}^{\frac{n+1}{r}}(u)+\frac {\rho_{K'}^{n+1}(u)}{V(K')}\rho_{L}^{\frac{n+1}{r}}(u) )^{\frac{r}{n+1}}\,dS(u)} \biggr]^{\frac{n+1}{p-r}} \\ \leq& V\bigl(K\,\hat{+}\,K'\bigr) \biggl[\frac{\int_{S^{n-1}} (V(K)^{-1}\rho_{K}^{n+1}(u)\rho_{L}^{\frac{n+1}{p}}(u) )^{\frac {p}{n+1}}\,dS(u)}{\int_{S^{n-1}} (V(K)^{-1}\rho_{K}^{n+1}(u)\rho _{L}^{\frac{n+1}{r}}(u) )^{\frac{r}{n+1}}\,dS(u)} \biggr]^{\frac{n+1}{p-r}} \\ &{}+V\bigl(K\,\hat{+}\,K'\bigr) \biggl[\frac{\int_{S^{n-1}} (V(K')^{-1}\rho_{K'}^{n+1}(u)\rho_{L}^{\frac{n+1}{p}}(u) )^{\frac{p}{n+1}}\,dS(u)}{\int_{S^{n-1}} (V(K')^{-1}\rho _{K'}^{n+1}(u)\rho_{L}^{\frac{n+1}{r}}(u) )^{\frac {r}{n+1}}\,dS(u)} \biggr]^{\frac{n+1}{p-r}} \\ =&V\bigl(K\,\hat{+}\,K'\bigr) \biggl[\frac{\int_{S^{n-1}}(V(K)^{-1})^{\frac {p}{n+1}}\rho_{K}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}(V(K)^{-1})^{\frac{r}{n+1}}\rho_{K}^{r}(u)\rho _{L}(u)\,dS(u)} \biggr]^{\frac{n+1}{p-r}} \\ &{}+V\bigl(K\,\hat{+}\,K'\bigr) \biggl[\frac{\int_{S^{n-1}}(V(K')^{-1})^{\frac{p}{n+1}}\rho_{K'}^{p}(u)\rho _{L}(u)\,dS(u)}{\int_{S^{n-1}}(V(K')^{-1})^{\frac{r}{n+1}}\rho _{K'}^{r}(u)\rho_{L}(u)\,dS(u)} \biggr]^{\frac{n+1}{p-r}} \\ =&\frac{V(K\,\hat{+}\,K')}{V(K)} \biggl(\frac{\int_{S^{n-1}}\rho _{K}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K}^{r}(u)\rho _{L}(u)\,dS(u)} \biggr)^{\frac{n+1}{p-r}} \\ &{}+\frac{V(K\,\hat{+}\,K')}{V(K')} \biggl(\frac{\int_{S^{n-1}}\rho _{K'}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K'}^{r}(u)\rho _{L}(u)\,dS(u)} \biggr)^{\frac{n+1}{p-r}} \\ =&\frac{V(K\,\hat{+}\,K')}{V(K)} \biggl(\frac{\widetilde {W}_{n-p-1}(K,L)}{\widetilde{W}_{n-r-1}(K,L)} \biggr) ^{\frac{n+1}{p-r}} \\ &{}+\frac{V(K\,\hat{+}\,K')}{V(K')} \biggl(\frac{\widetilde {W}_{n-p-1}(K',L)}{\widetilde{W}_{n-r-1}(K',L)} \biggr) ^{\frac{n+1}{p-r}} \\ =&V\bigl(K\,\hat{+}\,K'\bigr) \biggl(\frac{Q^{\frac {n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K,L)}}{ V(K)}+ \frac{Q^{\frac{n+1}{p-r}}_{\widetilde {W}_{n-p-1,n-r-1}(K',L)}}{V(K')} \biggr), \end{aligned}

i.e.,

$$\frac{Q^{\frac{n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K\hat {+}K',L)} }{V(K\,\hat{+}\,K')}\leq\frac{Q^{\frac {n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K,L)}}{V(K)} +\frac{Q^{\frac{n+1}{p-r}}_{\widetilde {W}_{n-p-1,n-r-1}(K',L)}}{V(K')}.$$
(3.15)

According to the equality condition of inequality (3.1), we see that equality holds in (3.15) if and only if K and $$K'$$ are dilates.

Let $$i=n-p-1$$ and $$j=n-r-1$$, then $$p\geq n+1\geq r>0$$ and $$i\leq-2\leq j< n-1$$ are equivalent. This and (3.15) yield inequality (1.11) and its equality condition.

If $$n+1\geq p>0>r$$, according to (3.2), (3.13), and (3.14), we have

$$\frac{Q^{\frac{n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K\hat {+}K',L)} }{V(K\,\hat{+}\,K')}\geq\frac{Q^{\frac {n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K,L)}}{V(K)} +\frac{Q^{\frac{n+1}{p-r}}_{\widetilde {W}_{n-p-1,n-r-1}(K',L)}}{V(K')},$$
(3.16)

with equality if and only if K and $$K'$$ are dilates.

Let $$i=n-p-1$$ and $$j=n-r-1$$, then (3.16) gives inequality (1.12) and its equality condition. □

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## Acknowledgements

This work was supported by the Natural Science Foundation of China (Grant Nos. 11371224 and 11102101).

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Correspondence to Xiaohua Zhang.

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The main idea of this paper was proposed by the second author. All authors contributed equally to the writing of the paper. All authors read and approved the final manuscript.

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