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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.

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., [216]). 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. (1)

    Ψ is continuous;

  2. (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. (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., [1827]). 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. (1)

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

  2. (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. (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 \(p, q\in\mathbb{R}\), \(p, q\neq0\).

  1. (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)
  2. (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 \(K,L\in\mathcal{S}^{n}_{o}\), \(p, q\in\mathbb{R}\), \(p\neq0\), \(q\neq -n\).

  1. (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)
  2. (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 [3035].

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:

  1. (i)

    If \(p>0\) and \(0< q< n-p\), then

    $$V\bigl(I_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \leq V(I_{p} K )^{\frac {pq}{n(n-p)}} +V(I_{p} L)^{\frac{pq}{n(n-p)}}; $$
  2. (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:

  1. (i)

    If \(0< p<-q\), then

    $$\frac{V(I_{p} (K \mp_{q} L))^{\frac{p(n+q)}{n(n-p)}}}{V(K \mp_{q} L)}\leq \frac{V(I_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(I_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}; $$
  2. (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.

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.

\(L_{p}\) 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.

\(L_{q}\) 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)

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

  1. (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)
  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 \(K,L\in\mathcal{S}^{n}_{o}\), \(p, q\in \mathbb{R}\), \(p\neq0\), \(0< q< n\).

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

  1. (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)
  2. (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). □

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)). □

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Acknowledgements

The authors would like to sincerely thank the referees for very valuable and helpful comments and suggestions, which made the paper more accurate and readable. Research is supported by the Natural Science Foundation of China (Grant No. 11371224).

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Zhou, Y., Wang, W. Some Brunn-Minkowski type inequalities for \(L_{p}\) radial Blaschke-Minkowski homomorphisms. J Inequal Appl 2016, 183 (2016). https://doi.org/10.1186/s13660-016-1122-0

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MSC

  • 52A20
  • 52A40
  • 52A39

Keywords

  • \(L_{p}\) radial Blaschke-Minkowski homomorphism
  • Brunn-Minkowski inequality
  • \(L_{q}\) radial sum
  • monotonic inequality