Some Brunn-Minkowski type inequalities for $L_{p}$ radial Blaschke-Minkowski homomorphisms

Abstract

Schuster introduced radial Blaschke-Minkowski homomorphisms. Recently, they were generalized to $L_{p}$ radial Blaschke-Minkowski homomorphisms by Wang et al. In this paper, we first establish Brunn-Minkowski type inequalities for some $L_{q}$ radial sums of $L_{p}$ radial Blaschke-Minkowski homomorphisms. Further, we consider monotonic inequalities for $L_{p}$ radial Blaschke-Minkowski homomorphisms.

1 Introduction

The setting for this paper is the Euclidean n-space $\mathbb{R}^{n}$. Let $\mathcal{S}^{n}_{o}$ denote the set of star bodies in $\mathbb{R}^{n}$. Let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^{n}$. For the n-dimensional volume of body K, we write $V(K)$.

Intersection bodies first appeared in a paper by Busemann [1] and were explicitly defined and named by Lutwak in the important paper [2]. Intersection bodies have been becoming the central notion in the dual Brunn Minkowski theory (see, e.g., [2–16]). In 2006, Ludwig [14] characterized the intersection body operator, which is the only nontrivial $GL(n)$ contravariant radial valuation. Whereafter, Schuster [17] introduced radial Blaschke-Minkowski homomorphisms, which are more general intersection body operators:

Definition 1.1

A map $\Psi: \mathcal{S}^{n}_{o} \rightarrow \mathcal{S}^{n}_{o}$ is called a radial Blaschke-Minkowski homomorphism if it satisfies the following conditions:

• (1) Ψ is continuous;

• (2) for all $K, L\in\mathcal{S}^{n}_{o}$, $\Psi(K \widetilde{+}_{n-1} L)=\Psi K \widetilde{+}\Psi L$, that is, ΨK is a radial Blaschke-Minkowski sum, where $\widetilde{+}_{n-1}$ and $\widetilde{+}$ denote $L_{n-1}$ and $L_{1}$ radial Minkowski addition, respectively;

• (3) Ψ intertwines rotations, that is, $\Psi(\vartheta K)=\vartheta\Psi K$ for all $K\in\mathcal{S}^{n}_{o}$ and all $\vartheta\in SO(n)$.

Further, Schuster [17] showed that radial Blaschke-Minkowski homomorphisms satisfy the geometric inequalities of the Aleksandrov-Fenchel, Minkowski, and Brunn-Minkowski types and established the following Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms of star bodies.

Theorem 1.A

If $K,L\in\mathcal{S}^{n}_{o}$, then

$$V\bigl(\Psi(K \widetilde{+} L)\bigr)^{\frac{1}{n(n-1)}}\leq V(\Psi K )^{\frac {1}{n(n-1)}} +V(\Psi L)^{\frac{1}{n(n-1)}} $$

with equality if and only if K and L are dilates.

In recent years, many inequalities for the radial Blaschke-Minkowski homomorphisms were established (see, e.g., [18–27]). Later, by associating the $L_{q}$ harmonic radial sum with the $L_{q}$ radial Blaschke sum of star bodies Wei et al. [23] gave the following Brunn-Minkowski type inequalities for radial Blaschke-Minkowski homomorphisms.

Theorem 1.B

If $K, L\in\mathcal{S}^{n}_{o}$ and real $q\geq1$, then

$$V\bigl(\Psi(K \breve{+}_{q} L)\bigr)^{-\frac{q}{n(n-1)}}\geq V(\Psi K )^{-\frac{q}{n(n-1)}} +V(\Psi L)^{-\frac{q}{n(n-1)}} $$

with equality if and only if K and L are dilates, where $\breve {+}_{q}$ is the $L_{q}$ harmonic radial sum.

Theorem 1.C

If $K, L\in\mathcal{S}^{n}_{o}$ and real $n>q\geq1$, then

$$V\bigl(\Psi(K \hat{+}_{q} L)\bigr)^{\frac{n-q}{n(n-1)}}\leq V(\Psi K )^{\frac {n-q}{n(n-1)}} +V(\Psi L)^{\frac{n-q}{n(n-1)}} $$

with equality if and only if K and L are dilates, where $\hat{+}_{q}$ is the $L_{q}$ radial Blaschke sum.

In 2011, Wang et al. [28] introduced the concept of an $L_{p}$ radial Blaschke-Minkowski homomorphism.

Definition 1.2

Let $K, L$ be star bodies, $p\in \mathbb{R}$, $p\neq0$. A map $\Psi_{p} : \mathcal{S}^{n}_{o} \rightarrow \mathcal{S}^{n}_{o}$ is called an $L_{p}$ radial Blaschke-Minkowski homomorphism if it satisfies the following conditions:

• (1) $\Psi_{p}$ is continuous with respect to radial metric;

• (2) For all $K,L\in\mathcal{S}^{n}_{o}$, $\Psi_{p}(K \widetilde{+}_{n-p} L)=\Psi_{p} K \widetilde{+}_{p}\Psi_{p} L$, that is, $\Psi_{p}K$ is an $L_{p}$ radial Blaschke-Minkowski sum, where $\widetilde{+}_{q}$ denotes $L_{q}$ radial Minkowski addition;

• (3) $\Psi_{p}$ is $SO(n)$ equivariant, that is, $\Psi_{p}(\vartheta K)=\vartheta\Psi_{p}K$ for all $K\in\mathcal{S}^{n}_{o}$ and all $\vartheta\in SO(n)$.

Meanwhile, they [28] studied the Busemann-Petty type problem for $L_{p}$ radial Blaschke-Minkowski homomorphisms. These results are generalized to a large class of $L_{p}$ radial valuations.

The main goal of this paper is to establish Brunn-Minkowski type inequalities for the $L_{q}$ radial Minkowski sum, $L_{q}$ harmonic radial sum, $L_{q}$ radial Blaschke sum, and $L_{q}$ harmonic Blaschke sum of $L_{p}$ radial Blaschke-Minkowski homomorphisms. First, we obtain the following Brunn-Minkowski type inequality for an $L_{q}$ radial Minkowski sum.

Theorem 1.1

Let $K,L\in\mathcal{S}^{n}_{o}$ and $p, q\in\mathbb{R}$, $p, q\neq0$.

• (i) If $p>0$ and $0< q< n-p$, then

  $$ V\bigl(\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \leq V(\Psi_{p} K )^{\frac{pq}{n(n-p)}} +V(\Psi_{p} L)^{\frac{pq}{n(n-p)}}; $$

  (1.1)

• (ii) If $q>n-p>0>p$ or $q< n-p<0$ or $q<0<n-p$ and $p>0$, then

  $$ V\bigl(\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \geq V(\Psi_{p} K )^{\frac{pq}{n(n-p)}} +V(\Psi_{p} L)^{\frac{pq}{n(n-p)}}. $$

  (1.2)

  Equality holds in each inequality if and only if K and L are dilates.

Taking $p=q=1$ in Theorem 1.1, by (1.1) we obtain Theorem 1.A. As applications of Theorem 1.1, in Section 3, we give Brunn-Minkowski type inequalities for the $L_{q}$ harmonic radial sum and $L_{q}$ radial Blaschke sum of $L_{p}$ radial Blaschke-Minkowski homomorphisms, that is, Theorem 3.1 and Theorem 3.2. Taking $p=1$ in Theorems 3.1 and 3.2, we easily get Theorems 1.B and 1.C, respectively.

Further, a Brunn-Minkowski type inequality for the $L_{q}$ harmonic Blaschke sum of $L_{p}$ radial Blaschke-Minkowski homomorphisms can be given as follows.

Theorem 1.2

Let $K,L\in\mathcal{S}^{n}_{o}$, $p, q\in\mathbb{R}$, $p\neq0$, $q\neq -n$.

• (i) If $0< p<-q<n$, then

  $$ \frac{V(\Psi_{p} (K \mp_{q} L))^{\frac{p(n+q)}{n(n-p)}}}{V(K \mp_{q} L)}\leq \frac{V(\Psi_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(\Psi_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}; $$

  (1.3)

• (ii) If $-q< p<0$, then

  $$ \frac{V(\Psi_{p} (K \mp_{q} L))^{\frac{p(n+q)}{n(n-p)}}}{V(K \mp_{q} L)}\geq \frac{V(\Psi_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(\Psi_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}. $$

  (1.4)

  Equality holds in each inequality if and only if K and L are dilates.

Here $K\mp_{q}L$ denotes the $L_{q}$ harmonic Blaschke sum of K and L.

In 2006, Haberl and Ludwig [29] defined the $L_{p}$-intersection bodies as follows: For $K\in\mathcal{S}^{n}_{o}$, real $p<1$, $p\neq0$, the $L_{p}$-intersection body $I_{p}K$ of K is the origin-symmetric star body whose radial function is defined by

$$\begin{aligned} \rho^{p}_{I_{p}K}(u)&= \int_{K}|u\cdot x|^{-p}\,dx =\frac{1}{n-p} \int_{S^{n-1}}|u\cdot v|^{-p}\rho ^{n-p}_{K}(v)\,dS(v) \end{aligned}$$

(1.5)

for all $u\in\mathcal{S}^{n-1}$. For the studies of $L_{p}$-intersection bodies, also see [30–35].

According to Definition 1.2 and (1.5), we easily see that the $L_{p}$-intersection body operator $I_{p}$ is a particular $L_{p}$ radial Blaschke-Minkowski homomorphism. So from Theorems 1.1 and 1.2 we have the following results.

Corollary 1.1

For $K, L\in\mathcal{S}^{n}_{o}$, $p, q\in\mathbb{R}$, $q\neq0$, $p<1$, and $p\neq0$, we have:

• (i) If $p>0$ and $0< q< n-p$, then

  $$V\bigl(I_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \leq V(I_{p} K )^{\frac {pq}{n(n-p)}} +V(I_{p} L)^{\frac{pq}{n(n-p)}}; $$

• (ii) If $p<0$ and $q>n-p$, or $p>0$ and $q<0$, then

  $$V\bigl(I_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \geq V(I_{p} K )^{\frac {pq}{n(n-p)}} +V(I_{p} L)^{\frac{pq}{n(n-p)}}. $$

  Equality holds in each inequality if and only if K and L are dilates.

Corollary 1.2

For $K, L\in\mathcal{S}^{n}_{o}$, $p,q\in\mathbb{R}$, $q\neq -n$, $p<1$, $p\neq0$, we have:

• (i) If $0< p<-q$, then

  $$\frac{V(I_{p} (K \mp_{q} L))^{\frac{p(n+q)}{n(n-p)}}}{V(K \mp_{q} L)}\leq \frac{V(I_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(I_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}; $$

• (ii) If $0>p>-q$, then

  $$\frac{V(I_{p} (K \mp_{q} L))^{\frac{p(n+q)}{n(n-p)}}}{V(K \mp_{q} L)}\geq \frac{V(I_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(I_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}. $$

  Equality holds in each inequality if and only if K and L are dilates.

The proofs of Theorems 1.1 and 1.2 are completed in Section 3. Besides, in Section 4, we establish two monotonic inequalities for $L_{p}$ radial Blaschke-Minkowski homomorphisms.

2 Background materials

If K is a compact star-shaped (about the origin) set in $\mathbb{R}^{n}$, then its radial function $\rho_{K}=\rho(K, \cdot): R^{n} \backslash\{0\}\rightarrow[0,\infty)$ is defined as (see [4])

$$\rho(K,u)=\max\{\lambda\geq0: \lambda u\in K\} $$

for all $u\in S^{n-1}$. If $\rho(K,\cdot)$ is positive and continuous, K is called a star body.

2.1 $L_{p}$ radial Minkowski combination and $L_{p}$ dual mixed volume

For $K, L\in\mathcal{S}^{n}_{o}$, real $p\neq0$, and $\lambda,\mu \geq0$ (not both 0), the $L_{p}$ radial Minkowski combination $\lambda\cdot K\widetilde{+}_{p}\mu\cdot L\in\mathcal{S}^{n}_{o}$ of K and L is defined by (see [30])

$$ \rho(\lambda\cdot K\widetilde{+}_{p}\mu\cdot L,\cdot)^{p}= \lambda\rho (K,\cdot)^{p}+\mu\rho(L,\cdot)^{p}. $$

(2.1)

If $p=1$ in (2.1), then $\lambda\cdot K\widetilde{+}\mu\cdot L$ is called the radial Minkowski combination of K and L.

For $K, L\in\mathcal{S}^{n}_{o}$, real $p\neq0$, and $\varepsilon>0$, the $L_{p}$ dual mixed volume $\widetilde{V}_{p}(K,L)$ of K and L is defined by (see [30])

$$\frac{n}{p}\widetilde{V}_{p}(K, L) =\lim_{\varepsilon\longrightarrow 0^{+}} \frac{V(K\widetilde{+}_{p}\varepsilon\cdot L)-V(K)}{\varepsilon}. $$

This definition and the polar coordinate formula for volume give the following integral representation of the $L_{p}$ dual mixed volume (see [30]):

$$ \widetilde{V}_{p}(K,L)=\frac{1}{n} \int_{S^{n-1}}\rho^{n-p}_{K}(u)\rho ^{p}_{L}(u)\,du, $$

(2.2)

where the integration is with respect to the spherical Lebesgue measure on $S^{n-1}$.

From (2.2) it follows immediately that, for each $K\in\mathcal{S}^{n}_{o}$,

$$ \widetilde{V}_{p}(K, K)=\frac{1}{n} \int_{S^{n-1}}\rho^{n}_{K}(u)\,du=V(K). $$

(2.3)

As an application of the Hölder inequality, we get the $L_{p}$ dual Minkowski inequality for $L_{p}$ dual mixed volume (see [30]).

Lemma 2.1

For $K, L\in\mathcal{S}^{n}_{o}$, if $0< p<n$, then

$$ \widetilde{V}_{p}(K, L)\leq V(K)^{\frac{n-p}{n}}V(L)^{\frac{p}{n}}; $$

(2.4)

if $p<0$ or $p>n$, then

$$ \widetilde{V}_{p}(K, L)\geq V(K)^{\frac{n-p}{n}}V(L)^{\frac{p}{n}}. $$

(2.5)

Equality holds in each inequality if and only if K and L are dilates.

2.2 $L_{q}$ harmonic radial sum, $L_{q}$ radial Blaschke sum, and $L_{q}$ harmonic Blaschke sum

The notion of $L_{q}$ harmonic radial sum can be introduced as follows: For $K,L \in S_{o}^{n}$, real $q\geq1$, the $L_{q}$ harmonic radial sum $K \breve{+}_{q}L\in\mathcal{S}^{n}_{o}$ of K and L is defined by (see [36])

$$ \rho(K\breve{+}_{q}L,\cdot)^{-q}=\rho(K, \cdot)^{-q}+\rho(L,\cdot)^{-q}. $$

(2.6)

If $q=1$, then $K \breve{+}L$ is the harmonic radial sum of K and L (see [4]).

The notion of radial Blaschke sum was given by Lutwak [2]. For $K, L\in\mathcal{S}_{o}^{n}$, $n \geq2$, the radial Blaschke sum $K \hat{+}L\in\mathcal{S}_{o}^{n}$ of K and L is defined by

$$\rho(K \hat{+}L, \cdot)^{n-1}= \rho(K,\cdot)^{n-1}+\rho(L, \cdot)^{n-1}. $$

In 2015, Wang and Wang [37] introduced the notion of $L_{q}$ radial Blaschke sum as follows: For $K, L\in\mathcal{S}_{o}^{n}$, $q\in\mathbb{R}$, and $n > q>0$, the $L_{q}$ radial Blaschke sum $K \hat{+}_{q}L\in\mathcal{S}_{o}^{n}$ of K and L is defined by

$$ \rho(K\hat{+}_{q}L, \cdot)^{n-q}= \rho(K, \cdot)^{n-q}+\rho(L,\cdot)^{n-q}. $$

(2.7)

The harmonic Blaschke sum was introduced by Lutwak [38]. For $K, L\in\mathcal{S}_{o}^{n}$, the harmonic Blaschke sum $K \mp L\in\mathcal {S}^{n}_{o}$ of K and L is defined by

$$\frac{\rho(K\mp L,\cdot)^{n+1}}{V(K\mp L)}= \frac{\rho(K ,\cdot)^{n+1}}{V(K)}+\frac{\rho(L,\cdot)^{n+1}}{V(L)}. $$

Based on this definition, Feng and Wang [39] defined the $L_{q}$ harmonic Blaschke sum as follows: For $K, L\in\mathcal{S}_{o}^{n}$ and real $q\neq -n$, the $L_{q}$ harmonic Blaschke sum $K \mp_{q}L\in\mathcal{S}^{n}_{o}$ of K and L is given by

$$ \frac{\rho(K\mp_{q}L,\cdot)^{n+q}}{V(K\mp_{q}L)}= \frac{\rho(K ,\cdot)^{n+q}}{V(K)}+\frac{\rho(L,\cdot)^{n+q}}{V(L)}. $$

(2.8)

3 Brunn-Minkowski type inequalities for $L_{p}$ radial Blaschke-Minkowski homomorphisms

Theorems 1.1 and 1.2 show Brunn-Minkowski type inequalities for the $L_{q}$ radial Minkowski sum and $L_{q}$ harmonic Blaschke sum of $L_{p}$ radial Blaschke-Minkowski homomorphisms. In this section, we prove Theorems 1.1 and 1.2. As applications of Theorem 1.1, we yet give two Brunn-Minkowski type inequalities for both $L_{q}$ harmonic radial sum and $L_{q}$ radial Blaschke sum of $L_{p}$ radial Blaschke-Minkowski homomorphisms. In order to prove Theorem 1.1, the following lemmas shall be needed.

Lemma 3.1

([28])

Let $\Psi_{p}: \mathcal {S}^{n}_{o} \rightarrow \mathcal{S}^{n}_{o}$ be an $L_{p}$ radial Blaschke-Minkowski homomorphism with real $p\neq0$. Then, for $K,L\in\mathcal{S}^{n}_{o}$,

$$ \widetilde{V}_{p}(K,\Psi_{p}L)=\widetilde{V}_{p}(L, \Psi_{p}K). $$

(3.1)

Lemma 3.2

Let $K, L\in\mathcal{S}^{n}_{o}$, $p, q\in \mathbb{R}$, $p,q\neq0$.

• (i) If $\frac{n-p}{q}>1$, then, for any $Q\in\mathcal{S}^{n}_{o}$,

  $$ \widetilde{V}_{p}(K \widetilde{+}_{q} L,Q)^{\frac{q}{n-p}} \leq\widetilde {V}_{p}(K,Q)^{\frac{q}{n-p}}+\widetilde{V}_{p}(L,Q)^{\frac{q}{n-p}}. $$

  (3.2)

• (ii) If $\frac{n-p}{q}<1$, then, for any $Q\in\mathcal{S}^{n}_{o}$,

  $$ \widetilde{V}_{p}(K \widetilde{+}_{q} L,Q)^{\frac{q}{n-p}} \geq\widetilde {V}_{p}(K,Q)^{\frac{q}{n-p}}+\widetilde{V}_{p}(L,Q)^{\frac{q}{n-p}}. $$

  (3.3)

  Equality holds in each inequality if and only if K and L are dilates.

Proof

From (2.1) and the Minkowski integral inequality, which enforces the condition $\frac{n-p}{q}>1$, it follows that, for any $Q\in\mathcal{S}^{n}_{o}$,

$$\begin{aligned} \widetilde{V}_{p}(K \widetilde{+}_{q} L,Q)^{\frac{q}{n-p}}={}& \biggl[\frac {1}{n} \int_{S^{n-1}}\rho(K \widetilde{+}_{q} L,u)^{n-p} \rho (Q,u)^{p}\,du \biggr]^{\frac{q}{n-p}} \\ ={}& \biggl\{ \frac{1}{n} \int _{S^{n-1}}\bigl[\rho(K,u)^{q}+\rho(L,u)^{q} \bigr]^{\frac{n-p}{q}}\rho(Q,u)^{p}\,du \biggr\} ^{\frac{q}{n-p}} \\ \leq{}& \biggl[\frac{1}{n} \int_{S^{n-1}}\rho(K,u)^{n-p}\rho (Q,u)^{p}\,du \biggr]^{\frac{q}{n-p}} \\ &{} + \biggl[\frac{1}{n} \int_{S^{n-1}}\rho(L,u)^{n-p}\rho(Q,u)^{p}\,du \biggr]^{\frac{q}{n-p}} \\ ={}&\widetilde{V}_{p}(K,Q)^{\frac{q}{n-p}}+\widetilde{V}_{p}(L,Q)^{\frac {q}{n-p}}; \end{aligned}$$

this is just (3.2). According to the condition of equality in the Minkowski integral inequality, we see that equality holds in (3.2) if and only if $\rho(K,\cdot)$ and $\rho(L,\cdot)$ are positively proportional, that is, equality holds in (3.2) if and only if K and L are dilates.

Similarly, again using (2.1) and the Minkowski integral inequality, which now enforces the condition $\frac{n-p}{q}<1$, we obtain inequality (3.3) with the equality condition. □

Proof of Theorem 1.1

(i) If $p>0$ and $0< q< n-p$, then $\frac{n-p}{q}>1$ and $0< p< n$. Thus, by (3.1), (3.2), and (2.4) we have, for any $N\in\mathcal{S}^{n}_{o}$,

$$\begin{aligned} \widetilde{V}_{p}\bigl(N,\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac {q}{n-p}}&=\widetilde{V}_{p}(K \widetilde{+}_{q} L ,\Psi_{p} N)^{\frac {q}{n-p}} \\ &\leq\widetilde{V}_{p}(K,\Psi_{p} N)^{\frac {q}{n-p}}+ \widetilde{V}_{p}(L,\Psi_{p} N)^{\frac{q}{n-p}} \end{aligned}$$

(3.4)

$$\begin{aligned} &=\widetilde{V}_{p}(N,\Psi_{p} K)^{\frac {q}{n-p}}+\widetilde{V}_{p}(N,\Psi_{p} L)^{\frac{q}{n-p}} \\ &\leq V(N)^{\frac {q}{n}}\bigl[V(\Psi_{p} K)^{\frac{pq}{n(n-p)}}+V( \Psi_{p} L)^{\frac {pq}{n(n-p)}}\bigr]. \end{aligned}$$

(3.5)

Setting $N=\Psi_{p} (K \widetilde{+}_{q} L)$, by (2.3) we obtain

$$V\bigl(\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \leq V(\Psi_{p} K )^{\frac{pq}{n(n-p)}} +V(\Psi_{p} L)^{\frac{pq}{n(n-p)}}. $$

This gives inequality (1.1).

By the equality conditions of (3.4) and (3.5) we know that equality in (1.1) holds if and only if K, L, $\Psi_{p} K$, $\Psi_{p} L$, and $\Psi _{p} (K \widetilde{+}_{q} L)$ all are dilates. But if K and L are dilates, then $\Psi_{p} (K \widetilde{+}_{q} L)$, $\Psi_{p} K$, and $\Psi_{p} L$ all are dilates. Thus, equality in (1.1) holds if and only if K and L are dilates.

(ii) For $q>n-p>0>p$ or $q< n-p<0$, we know that $0<\frac {n-p}{q}<1$, $p<0$ or $p>n$ (for $q<0<n-p$ and $p>0$, we get $\frac{n-p}{q}<0$ and $0< p< n$). From this, using (3.1), (3.3), and (2.5) (or (2.4)), we have, for any $N\in\mathcal{S}^{n}_{o}$,

$$\begin{aligned} \widetilde{V}_{p}\bigl(N,\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac {q}{n-p}}&=\widetilde{V}_{p}(K \widetilde{+}_{q} L ,\Psi_{p} N)^{\frac {q}{n-p}} \\ &\geq\widetilde{V}_{p}(K,\Psi_{p} N)^{\frac {q}{n-p}}+ \widetilde{V}_{p}(L,\Psi_{p} N)^{\frac{q}{n-p}} \\ &=\widetilde{V}_{p}(N,\Psi_{p} K)^{\frac {q}{n-p}}+ \widetilde{V}_{p}(N,\Psi_{p} L)^{\frac{q}{n-p}} \\ &\geq V(N)^{\frac {q}{n}}\bigl[V(\Psi_{p} K)^{\frac{pq}{n(n-p)}}+V( \Psi_{p} L)^{\frac{pq}{n(n-p)}}\bigr]. \end{aligned}$$

Setting $N=\Psi_{p} (K \widetilde{+}_{q} L)$ and using (2.3), we have

$$V\bigl(\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \geq V(\Psi_{p} K )^{\frac{pq}{n(n-p)}} +V(\Psi_{p} L)^{\frac{pq}{n(n-p)}}. $$

This yields (1.2), and equality holds in (1.2) if and only if K and L are dilates. □

As an application of Theorem 1.1, from the $L_{q}$ harmonic radial sum (2.6) we obtain the following:

Theorem 3.1

Let $K,L\in\mathcal{S}^{n}_{o}$, $p, q\in \mathbb{R}$, $p\neq0$, $q\geq1$. If $-q< n-p<0$ or $0< p< n$, then

$$ V\bigl(\Psi_{p} (K \breve{+}_{q} L)\bigr)^{-\frac{pq}{n(n-p)}}\geq V(\Psi_{p} K )^{-\frac{pq}{n(n-p)}} +V(\Psi_{p} L)^{-\frac{pq}{n(n-p)}}, $$

(3.6)

where equality holds if and only if K and L are dilates.

Proof

By (2.1) and (2.6) we see that, for $q\geq1$, $K \breve {+}_{q} L=K\widetilde{+}_{-q}L$. Hence, if $-q< n-p<0$, then (3.6) is true by (1.2); if $0< p< n$, then since $q\geq1$, we have $-q<0<n-p$, which, together with (1.2), shows that (3.6) also holds. □

Similarly, as another application of Theorem 1.1, by the $L_{q}$ radial Blaschke sum (2.7) we have the following:

Theorem 3.2

Let $K,L\in\mathcal{S}^{n}_{o}$, $p, q\in \mathbb{R}$, $p\neq0$, $0< q< n$.

• (i) If $n>q>p$>0, then

  $$ V\bigl(\Psi_{p} (K \hat{+}_{q} L)\bigr)^{\frac{p(n-q)}{n(n-p)}}\leq V(\Psi_{p} K )^{\frac{p(n-q)}{n(n-p)}} +V(\Psi_{p} L)^{\frac{p(n-q)}{n(n-p)}} $$

  (3.7)

  with equality if and only if K and L are dilates.

Proof

From (2.1) and (2.7) we know that, for $0< q< n$, $K\hat {+}_{q} L=K\widetilde{+}_{n-q}L$. Thus, if $n>q>p>0$, then $0< n-q< n-p$ and $p>0$. This, together with (1.1), yields (3.7). □

The proof of Theorem 1.2 requires the following lemma.

Lemma 3.3

Let $K,L\in\mathcal{S}^{n}_{o}$, $p,q\in \mathbb{R}$, $p\neq0$, $q\neq-n$.

• (i) If $\frac{n-p}{n+q}>1$, then, for any $Q\in\mathcal{S}^{n}_{o}$,

  $$ \frac{\widetilde{V}_{p}(K\mp_{q}L,Q)^{\frac{{n+q}}{n-p}}}{V(K\mp_{q}L)}\leq \frac{\widetilde{V}_{p}(K,Q)^{\frac{{n+q}}{n-p}}}{V(K)}+\frac{ \widetilde {V}_{p}(L,Q)^{\frac{{n+q}}{n-p}}}{V(L)}. $$

  (3.8)

• (ii) If $\frac{n-p}{n+q}<1$, then, for any $Q\in\mathcal{S}^{n}_{o}$,

  $$ \frac{\widetilde{V}_{p}(K\mp_{q}L,Q)^{\frac{{n+q}}{n-p}}}{V(K\mp_{q}L)}\geq \frac{\widetilde{V}_{p}(K,Q)^{\frac{{n+q}}{n-p}}}{V(K)}+\frac{ \widetilde {V}_{p}(L,Q)^{\frac{{n+q}}{n-p}}}{V(L)}. $$

  (3.9)

  Equality holds in each inequality if and only if K and L are dilates.

Proof

By (2.2), (2.8), and the Minkowski integral inequality, which enforces the condition $\frac{n-p}{n+q}>1$, we have, for any $Q\in \mathcal{S}^{n}_{o}$,

$$\begin{aligned} \frac{\widetilde{V}_{p}(K\mp_{q}L,Q)^{\frac{{n+q}}{n-p}}}{V(K\mp_{q}L)}={}&\frac { [\frac{1}{n}\int_{S^{n-1}}\rho(K\mp_{q}L,u)^{n-p}\rho (Q,u)^{p}\,du ]^{\frac{{n+q}}{n-p}}}{V(K\mp_{q}L)} \\ ={}& \biggl[\frac {1}{n} \int_{S^{n-1}} \biggl(\frac{\rho(K,u)^{n+q}}{V(K)}+ \frac{\rho(L,u)^{n+q}}{V(L)} \biggr)^{\frac{n-p}{{n+q}}}\rho (Q,u)^{p}\,du \biggr]^{\frac{{n+q}}{n-p}} \\ \leq{}& \frac{1}{V(K)}\biggl[\frac{1}{n} \int_{S^{n-1}}\rho(K,u)^{n-p}\rho (Q,u)^{p}\,du \biggr]^{\frac{{n+q}}{n-p}} \\ &{}+ \frac{1}{V(L)}\biggl[\frac{1}{n} \int_{S^{n-1}}\rho(L,u)^{n-p}\rho (Q,u)^{p}\,du \biggr]^{\frac{n+q}{n-p}} \\ ={}&\frac{\widetilde{V}_{p}(K,Q)^{\frac{{n+q}}{n-p}}}{V(K)}+\frac{ \widetilde {V}_{p}(L,Q)^{\frac{{n+q}}{n-p}}}{V(L)} \end{aligned}$$

with equality if and only if K and L are dilates. This inequality gives (3.8).

Similarly, again using (2.2) and the Minkowski integral inequality, which now enforces the condition $\frac{n-p}{n+q}<1$, we obtain inequality (3.9) with the equality condition. □

Proof of Theorem 1.2

(i) For $K,L\in\mathcal {S}^{n}_{o}$, since $q\neq -n$, if $0< p<-q<n$, then $\frac{n-p}{n+q}>1$ and $0< p< n$. So by (3.1), (3.8), and (2.4) we have, for any $N\in\mathcal{S}^{n}_{o}$,

$$\begin{aligned} \frac{\widetilde{V}_{p}(N,\Psi_{p} (K\mp_{q}L))^{\frac{n+q}{n-p}}}{V(K\mp _{q}L)}&=\frac{\widetilde{V}_{p}(K\mp_{q}L ,\Psi_{p} N)^{\frac{n+q}{n-p}}}{V(K\mp _{q}L)} \\ &\leq\frac{\widetilde{V}_{p}(K,\Psi_{p} N)^{\frac {n+q}{n-p}}}{V(K)}+\frac{\widetilde{V}_{p}(L, \Psi_{p} N)^{\frac{n+q}{n-p}}}{V(L)} \\ &=\frac{\widetilde{V}_{p}(N,\Psi_{p} K)^{\frac{n+q}{n-p}}}{V(K)}+\frac{\widetilde{V}_{p}(N,\Psi_{p} L)^{\frac {n+q}{n-p}}}{V(L)} \\ &\leq V(N)^{\frac{n+q}{n}} \biggl[\frac{V(\Psi_{p} K)^{\frac {p(n+q)}{n(n-p)}}}{V(K)}+\frac{V(\Psi_{p} L)^{\frac {p(n+q)}{n(n-p)}}}{V(L)} \biggr]. \end{aligned}$$

(3.10)

Setting $N=\Psi_{p} (K \mp_{q} L)$ in (3.10), by (2.3) we get

$$\frac{V(\Psi_{p} (K\mp_{q}L))^{\frac{p(n+q)}{n(n-p)}}}{V(K\mp_{q}L)}\leq \frac{V(\Psi_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(\Psi_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}, $$

and equality holds if and only if K and L are dilates. Therefore, inequality (1.3) is obtained.

(ii) If $-q< p<0$, then $0<\frac{n-p}{n+q}<1$. This, together with (3.1), (3.9), and (2.5), yields

$$\frac{V(\Psi_{p} (K\mp_{q}L))^{\frac{p(n+q)}{n(n-p)}}}{V(K\mp_{q}L)}\geq \frac{V(\Psi_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(\Psi_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}, $$

and equality holds if and only if K and L are dilates. This is just inequality (1.4). □

4 Monotonic inequalities for the $L_{p}$ radial Blaschke-Minkowski homomorphisms

In this section, we establish monotonic inequalities for the $L_{p}$ radial Blaschke-Minkowski homomorphisms.

Theorem 4.1

Let $\Phi_{p}: \mathcal{S}^{n}_{o} \rightarrow \mathcal{S}^{n}_{o}$ be an $L_{p}$ radial Blaschke-Minkowski homomorphism, $p\neq0$, $K, L\in\mathcal{S}^{n}_{o}$, and $\Phi_{p}K\subseteq\Phi_{p}L$. If $p>0$, then, for any $Q\in\Phi_{p}\mathcal{S}^{n}_{o}$,

$$ \widetilde{V}_{p}(K, Q)\leq\widetilde{V}_{p}(L, Q); $$

(4.1)

if $p<0$, then, for any $Q\in\Phi_{p}\mathcal{S}^{n}_{o}$,

$$ \widetilde{V}_{p}(K, Q)\geq\widetilde{V}_{p}(L, Q). $$

(4.2)

Equality holds in (4.1) or (4.2) if and only if $\Phi_{p}K=\Phi_{p}L$.

Proof

Since $\Phi_{p}K\subseteq\Phi_{p}L$, by (2.2) we know that, for $p>0$ and any $N\in\mathcal{S}^{n}_{o}$,

$$ \widetilde{V}_{p}(N, \Phi_{p}K)\leq\widetilde{V}_{p}(N, \Phi_{p}L). $$

(4.3)

This, together with (3.1), gives

$$\widetilde{V}_{p}(K, \Phi_{p}N)\leq\widetilde{V}_{p}(L, \Phi_{p}N). $$

Let $Q=\Phi_{p}N$. Then $Q\in\Phi_{p}\mathcal{S}^{n}_{o}$ and $\widetilde {V}_{p}(K, Q)\leq\widetilde{V}_{p}(L, Q)$. From the equality condition for (4.3), we see that equality holds in (4.1) if and only if $\Phi_{p}K=\Phi_{p}L$.

Similarly, if $p<0$ and $\Phi_{p}K\subseteq\Phi_{p}L$, by (2.2) we easily obtain that, for any $Q\in\Phi_{p}\mathcal{S}^{n}_{o}$, $\widetilde{V}_{p}(K, Q)\geq\widetilde{V}_{p}(L, Q)$, and equality holds if and only if $\Phi_{p}K=\Phi_{p}L$. □

For $0< p< n$, let $K\in\Phi_{p}\mathcal{S}^{n}_{o}$ in (4.1) (for $p>n$, let $L\in\Phi_{p}\mathcal{S}^{n}_{o}$ in (4.1); for $p<0$, let $K\in\Phi_{p}\mathcal {S}^{n}_{o}$ in (4.2)). Using inequality (2.4) (or inequality (2.5)), we may get a positive form of Busemann-Petty type problem for the $L_{p}$ radial Blaschke-Minkowski homomorphisms given by Wang et al. [28].

Corollary 4.1

Let $\Phi_{p}: \mathcal{S}^{n}_{o} \rightarrow \mathcal{S}^{n}_{o}$ be an $L_{p}$ radial Blaschke-Minkowski homomorphism, $p\neq0$, $K, L\in\mathcal{S}^{n}_{o}$, and $\Phi_{p}K\subseteq\Phi_{p}L$. If $n>p>0$ and $K\in\Phi_{p}\mathcal{S}^{n}_{o}$, then

$$\Phi_{p}K\subseteq\Phi_{p}L\quad\Longrightarrow\quad V(K)\leq V(L), $$

and $V(K)=V(L)$ if and only if $K=L$.

If $p>n$ and $L\in\Phi_{p}\mathcal{S}^{n}_{o}$ or $p<0$ and $K\in\Phi _{p}\mathcal{S}^{n}_{o}$, then

$$\Phi_{p}K\subseteq\Phi_{p}L\quad\Longrightarrow\quad V(K)\geq V(L), $$

and $V(K)=V(L)$ if and only if $K=L$.

Theorem 4.2

Let $\Phi_{p}: \mathcal{S}^{n}_{o} \rightarrow \mathcal{S}^{n}_{o}$ be an $L_{p}$ radial Blaschke-Minkowski homomorphism, $p\neq0$. If $K, L\in\mathcal{S}^{n}_{o}$, and for any $Q\in\mathcal{S}^{n}_{o}$,

$$ \widetilde{V}_{p}(K, Q)\leq\widetilde{V}_{p}(L, Q), $$

(4.4)

then, for $p>0$,

$$ V(\Phi_{p}K)\leq V(\Phi_{p}L), $$

(4.5)

and, for $p<0$,

$$ V(\Phi_{p}K)\geq V(\Phi_{p}L). $$

(4.6)

Equality holds in (4.5) or (4.6) only if $K=L$.

Proof

For $0< p< n$, let $Q=\Phi_{p}\Phi_{p}K$ in (4.4). Then

$$\widetilde{V}_{p}(K, \Phi_{p}\Phi_{p}K)\leq \widetilde{V}_{p}(L, \Phi_{p}\Phi_{p}K). $$

Using inequality (2.4) and equalities (3.1) and (2.3), we have

$$V(\Phi_{p}K)=\widetilde{V}_{p}(\Phi_{p}K, \Phi_{p}K)\leq\widetilde{V}_{p}(\Phi _{p}K, \Phi_{p}L)\leq V(\Phi_{p}K)^{\frac{n-p}{n}}V( \Phi_{p}L)^{\frac{p}{n}}, $$

which yields (4.5), and equality holds only if $K=L$.

Similarly, for $p>n$ (or $p<0$), let $Q=\Phi_{p}\Phi_{p}L$ in (4.4). By inequality (2.5) and equalities (3.1) and (2.3), we can obtain (4.5) (or (4.6)). □