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:
Anzeige
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
with equality if and only if
K
and
L
are dilates.
$$V\bigl(\Psi(K \widetilde{+} L)\bigr)^{\frac{1}{n(n-1)}}\leq V(\Psi K )^{\frac {1}{n(n-1)}} +V(\Psi L)^{\frac{1}{n(n-1)}} $$
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.
Anzeige
Theorem 1.B
If
\(K, L\in\mathcal{S}^{n}_{o}\)
and real
\(q\geq1\), then
with equality if and only if
K
and
L
are dilates, where
\(\breve {+}_{q}\)
is the
\(L_{q}\)
harmonic radial sum.
$$V\bigl(\Psi(K \breve{+}_{q} L)\bigr)^{-\frac{q}{n(n-1)}}\geq V(\Psi K )^{-\frac{q}{n(n-1)}} +V(\Psi L)^{-\frac{q}{n(n-1)}} $$
Theorem 1.C
If
\(K, L\in\mathcal{S}^{n}_{o}\)
and real
\(n>q\geq1\), then
with equality if and only if
K
and
L
are dilates, where
\(\hat{+}_{q}\)
is the
\(L_{q}\)
radial Blaschke sum.
$$V\bigl(\Psi(K \hat{+}_{q} L)\bigr)^{\frac{n-q}{n(n-1)}}\leq V(\Psi K )^{\frac {n-q}{n(n-1)}} +V(\Psi L)^{\frac{n-q}{n(n-1)}} $$
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
Equality holds in each inequality if and only if
K
and
L
are dilates.
$$ V\bigl(\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \geq V(\Psi_{p} K )^{\frac{pq}{n(n-p)}} +V(\Psi_{p} L)^{\frac{pq}{n(n-p)}}. $$
(1.2)
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
\(K,L\in\mathcal{S}^{n}_{o}\), \(p, q\in\mathbb{R}\), \(p\neq0\), \(q\neq -n\).
Here
\(K\mp_{q}L\)
denotes the
\(L_{q}\)
harmonic Blaschke sum of
K
and
L.
(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
Equality holds in each inequality if and only if
K
and
L
are dilates.
$$ \frac{V(\Psi_{p} (K \mp_{q} L))^{\frac{p(n+q)}{n(n-p)}}}{V(K \mp_{q} L)}\geq \frac{V(\Psi_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(\Psi_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}. $$
(1.4)
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 for all \(u\in\mathcal{S}^{n-1}\). For the studies of \(L_{p}\)-intersection bodies, also see [30‐35].
$$\begin{aligned} \rho^{p}_{I_{p}K}(u)&= \int_{K}|u\cdot x|^{-p}\,dx =\frac{1}{n-p} \int_{S^{n-1}}|u\cdot v|^{-p}\rho ^{n-p}_{K}(v)\,dS(v) \end{aligned}$$
(1.5)
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
Equality holds in each inequality if and only if
K
and
L
are dilates.
$$V\bigl(I_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \geq V(I_{p} K )^{\frac {pq}{n(n-p)}} +V(I_{p} L)^{\frac{pq}{n(n-p)}}. $$
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
Equality holds in each inequality if and only if
K
and
L
are dilates.
$$\frac{V(I_{p} (K \mp_{q} L))^{\frac{p(n+q)}{n(n-p)}}}{V(K \mp_{q} L)}\geq \frac{V(I_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(I_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}. $$
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]) for all \(u\in S^{n-1}\). If \(\rho(K,\cdot)\) is positive and continuous, K is called a star body.
$$\rho(K,u)=\max\{\lambda\geq0: \lambda u\in K\} $$
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]) If \(p=1\) in (2.1), then \(\lambda\cdot K\widetilde{+}\mu\cdot L\) is called the radial Minkowski combination of K and L.
$$ \rho(\lambda\cdot K\widetilde{+}_{p}\mu\cdot L,\cdot)^{p}= \lambda\rho (K,\cdot)^{p}+\mu\rho(L,\cdot)^{p}. $$
(2.1)
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]) This definition and the polar coordinate formula for volume give the following integral representation of the \(L_{p}\) dual mixed volume (see [30]): where the integration is with respect to the spherical Lebesgue measure on \(S^{n-1}\).
$$\frac{n}{p}\widetilde{V}_{p}(K, L) =\lim_{\varepsilon\longrightarrow 0^{+}} \frac{V(K\widetilde{+}_{p}\varepsilon\cdot L)-V(K)}{\varepsilon}. $$
$$ \widetilde{V}_{p}(K,L)=\frac{1}{n} \int_{S^{n-1}}\rho^{n-p}_{K}(u)\rho ^{p}_{L}(u)\,du, $$
(2.2)
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
if
\(p<0\)
or
\(p>n\), then
Equality holds in each inequality if and only if
K
and
L
are dilates.
$$ \widetilde{V}_{p}(K, L)\leq V(K)^{\frac{n-p}{n}}V(L)^{\frac{p}{n}}; $$
(2.4)
$$ \widetilde{V}_{p}(K, L)\geq V(K)^{\frac{n-p}{n}}V(L)^{\frac{p}{n}}. $$
(2.5)
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]) If \(q=1\), then \(K \breve{+}L\) is the harmonic radial sum of K and L (see [4]).
$$ \rho(K\breve{+}_{q}L,\cdot)^{-q}=\rho(K, \cdot)^{-q}+\rho(L,\cdot)^{-q}. $$
(2.6)
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 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
$$\frac{\rho(K\mp L,\cdot)^{n+1}}{V(K\mp L)}= \frac{\rho(K ,\cdot)^{n+1}}{V(K)}+\frac{\rho(L,\cdot)^{n+1}}{V(L)}. $$
$$ \frac{\rho(K\mp_{q}L,\cdot)^{n+q}}{V(K\mp_{q}L)}= \frac{\rho(K ,\cdot)^{n+q}}{V(K)}+\frac{\rho(L,\cdot)^{n+q}}{V(L)}. $$
(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
\(Q\in\mathcal{S}^{n}_{o}\),
Equality holds in each inequality if and only if
K
and
L
are dilates.
$$ \widetilde{V}_{p}(K \widetilde{+}_{q} L,Q)^{\frac{q}{n-p}} \geq\widetilde {V}_{p}(K,Q)^{\frac{q}{n-p}}+\widetilde{V}_{p}(L,Q)^{\frac{q}{n-p}}. $$
(3.3)
Proof
From (2.1) and the Minkowski integral inequality, which enforces the condition \(\frac{n-p}{q}>1\), it follows that, for any \(Q\in\mathcal{S}^{n}_{o}\), 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.
$$\begin{aligned} \widetilde{V}_{p}(K \widetilde{+}_{q} L,Q)^{\frac{q}{n-p}}={}& \biggl[\frac {1}{n} \int_{S^{n-1}}\rho(K \widetilde{+}_{q} L,u)^{n-p} \rho (Q,u)^{p}\,du \biggr]^{\frac{q}{n-p}} \\ ={}& \biggl\{ \frac{1}{n} \int _{S^{n-1}}\bigl[\rho(K,u)^{q}+\rho(L,u)^{q} \bigr]^{\frac{n-p}{q}}\rho(Q,u)^{p}\,du \biggr\} ^{\frac{q}{n-p}} \\ \leq{}& \biggl[\frac{1}{n} \int_{S^{n-1}}\rho(K,u)^{n-p}\rho (Q,u)^{p}\,du \biggr]^{\frac{q}{n-p}} \\ &{} + \biggl[\frac{1}{n} \int_{S^{n-1}}\rho(L,u)^{n-p}\rho(Q,u)^{p}\,du \biggr]^{\frac{q}{n-p}} \\ ={}&\widetilde{V}_{p}(K,Q)^{\frac{q}{n-p}}+\widetilde{V}_{p}(L,Q)^{\frac {q}{n-p}}; \end{aligned}$$
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 \(N\in\mathcal{S}^{n}_{o}\),
Setting \(N=\Psi_{p} (K \widetilde{+}_{q} L)\), by (2.3) we obtain This gives inequality (1.1).
$$\begin{aligned} \widetilde{V}_{p}\bigl(N,\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac {q}{n-p}}&=\widetilde{V}_{p}(K \widetilde{+}_{q} L ,\Psi_{p} N)^{\frac {q}{n-p}} \\ &\leq\widetilde{V}_{p}(K,\Psi_{p} N)^{\frac {q}{n-p}}+ \widetilde{V}_{p}(L,\Psi_{p} N)^{\frac{q}{n-p}} \end{aligned}$$
(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)
$$V\bigl(\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \leq V(\Psi_{p} K )^{\frac{pq}{n(n-p)}} +V(\Psi_{p} L)^{\frac{pq}{n(n-p)}}. $$
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 \(N\in\mathcal{S}^{n}_{o}\), Setting \(N=\Psi_{p} (K \widetilde{+}_{q} L)\) and using (2.3), we have This yields (1.2), and equality holds in (1.2) if and only if K and L are dilates. □
$$\begin{aligned} \widetilde{V}_{p}\bigl(N,\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac {q}{n-p}}&=\widetilde{V}_{p}(K \widetilde{+}_{q} L ,\Psi_{p} N)^{\frac {q}{n-p}} \\ &\geq\widetilde{V}_{p}(K,\Psi_{p} N)^{\frac {q}{n-p}}+ \widetilde{V}_{p}(L,\Psi_{p} N)^{\frac{q}{n-p}} \\ &=\widetilde{V}_{p}(N,\Psi_{p} K)^{\frac {q}{n-p}}+ \widetilde{V}_{p}(N,\Psi_{p} L)^{\frac{q}{n-p}} \\ &\geq V(N)^{\frac {q}{n}}\bigl[V(\Psi_{p} K)^{\frac{pq}{n(n-p)}}+V( \Psi_{p} L)^{\frac{pq}{n(n-p)}}\bigr]. \end{aligned}$$
$$V\bigl(\Psi_{p} (K \widetilde{+}_{q} L)\bigr)^{\frac{pq}{n(n-p)}} \geq V(\Psi_{p} K )^{\frac{pq}{n(n-p)}} +V(\Psi_{p} L)^{\frac{pq}{n(n-p)}}. $$
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
where equality holds if and only if
K
and
L
are dilates.
$$ V\bigl(\Psi_{p} (K \breve{+}_{q} L)\bigr)^{-\frac{pq}{n(n-p)}}\geq V(\Psi_{p} K )^{-\frac{pq}{n(n-p)}} +V(\Psi_{p} L)^{-\frac{pq}{n(n-p)}}, $$
(3.6)
Proof
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
with equality if and only if
K
and
L
are dilates.
$$ V\bigl(\Psi_{p} (K \hat{+}_{q} L)\bigr)^{\frac{p(n-q)}{n(n-p)}}\leq V(\Psi_{p} K )^{\frac{p(n-q)}{n(n-p)}} +V(\Psi_{p} L)^{\frac{p(n-q)}{n(n-p)}} $$
(3.7)
Proof
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
\(Q\in\mathcal{S}^{n}_{o}\),
Equality holds in each inequality if and only if
K
and
L
are dilates.
$$ \frac{\widetilde{V}_{p}(K\mp_{q}L,Q)^{\frac{{n+q}}{n-p}}}{V(K\mp_{q}L)}\geq \frac{\widetilde{V}_{p}(K,Q)^{\frac{{n+q}}{n-p}}}{V(K)}+\frac{ \widetilde {V}_{p}(L,Q)^{\frac{{n+q}}{n-p}}}{V(L)}. $$
(3.9)
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 \(Q\in \mathcal{S}^{n}_{o}\), with equality if and only if K and L are dilates. This inequality gives (3.8).
$$\begin{aligned} \frac{\widetilde{V}_{p}(K\mp_{q}L,Q)^{\frac{{n+q}}{n-p}}}{V(K\mp_{q}L)}={}&\frac { [\frac{1}{n}\int_{S^{n-1}}\rho(K\mp_{q}L,u)^{n-p}\rho (Q,u)^{p}\,du ]^{\frac{{n+q}}{n-p}}}{V(K\mp_{q}L)} \\ ={}& \biggl[\frac {1}{n} \int_{S^{n-1}} \biggl(\frac{\rho(K,u)^{n+q}}{V(K)}+ \frac{\rho(L,u)^{n+q}}{V(L)} \biggr)^{\frac{n-p}{{n+q}}}\rho (Q,u)^{p}\,du \biggr]^{\frac{{n+q}}{n-p}} \\ \leq{}& \frac{1}{V(K)}\biggl[\frac{1}{n} \int_{S^{n-1}}\rho(K,u)^{n-p}\rho (Q,u)^{p}\,du \biggr]^{\frac{{n+q}}{n-p}} \\ &{}+ \frac{1}{V(L)}\biggl[\frac{1}{n} \int_{S^{n-1}}\rho(L,u)^{n-p}\rho (Q,u)^{p}\,du \biggr]^{\frac{n+q}{n-p}} \\ ={}&\frac{\widetilde{V}_{p}(K,Q)^{\frac{{n+q}}{n-p}}}{V(K)}+\frac{ \widetilde {V}_{p}(L,Q)^{\frac{{n+q}}{n-p}}}{V(L)} \end{aligned}$$
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 \(N\in\mathcal{S}^{n}_{o}\), Setting \(N=\Psi_{p} (K \mp_{q} L)\) in (3.10), by (2.3) we get and equality holds if and only if K and L are dilates. Therefore, inequality (1.3) is obtained.
$$\begin{aligned} \frac{\widetilde{V}_{p}(N,\Psi_{p} (K\mp_{q}L))^{\frac{n+q}{n-p}}}{V(K\mp _{q}L)}&=\frac{\widetilde{V}_{p}(K\mp_{q}L ,\Psi_{p} N)^{\frac{n+q}{n-p}}}{V(K\mp _{q}L)} \\ &\leq\frac{\widetilde{V}_{p}(K,\Psi_{p} N)^{\frac {n+q}{n-p}}}{V(K)}+\frac{\widetilde{V}_{p}(L, \Psi_{p} N)^{\frac{n+q}{n-p}}}{V(L)} \\ &=\frac{\widetilde{V}_{p}(N,\Psi_{p} K)^{\frac{n+q}{n-p}}}{V(K)}+\frac{\widetilde{V}_{p}(N,\Psi_{p} L)^{\frac {n+q}{n-p}}}{V(L)} \\ &\leq V(N)^{\frac{n+q}{n}} \biggl[\frac{V(\Psi_{p} K)^{\frac {p(n+q)}{n(n-p)}}}{V(K)}+\frac{V(\Psi_{p} L)^{\frac {p(n+q)}{n(n-p)}}}{V(L)} \biggr]. \end{aligned}$$
(3.10)
$$\frac{V(\Psi_{p} (K\mp_{q}L))^{\frac{p(n+q)}{n(n-p)}}}{V(K\mp_{q}L)}\leq \frac{V(\Psi_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(\Psi_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}, $$
(ii) If \(-q< p<0\), then \(0<\frac{n-p}{n+q}<1\). This, together with (3.1), (3.9), and (2.5), yields and equality holds if and only if K and L are dilates. This is just inequality (1.4). □
$$\frac{V(\Psi_{p} (K\mp_{q}L))^{\frac{p(n+q)}{n(n-p)}}}{V(K\mp_{q}L)}\geq \frac{V(\Psi_{p} K )^{\frac{p(n+q)}{n(n-p)}}}{V(K)} +\frac{V(\Psi_{p} L)^{\frac{p(n+q)}{n(n-p)}}}{V(L)}, $$
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
\(Q\in\Phi_{p}\mathcal{S}^{n}_{o}\),
if
\(p<0\), then, for any
\(Q\in\Phi_{p}\mathcal{S}^{n}_{o}\),
Equality holds in (4.1) or (4.2) if and only if
\(\Phi_{p}K=\Phi_{p}L\).
$$ \widetilde{V}_{p}(K, Q)\leq\widetilde{V}_{p}(L, Q); $$
(4.1)
$$ \widetilde{V}_{p}(K, Q)\geq\widetilde{V}_{p}(L, Q). $$
(4.2)
Proof
Since \(\Phi_{p}K\subseteq\Phi_{p}L\), by (2.2) we know that, for \(p>0\) and any \(N\in\mathcal{S}^{n}_{o}\), This, together with (3.1), gives 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 \(\Phi_{p}K=\Phi_{p}L\).
$$ \widetilde{V}_{p}(N, \Phi_{p}K)\leq\widetilde{V}_{p}(N, \Phi_{p}L). $$
(4.3)
$$\widetilde{V}_{p}(K, \Phi_{p}N)\leq\widetilde{V}_{p}(L, \Phi_{p}N). $$
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
and
\(V(K)=V(L)\)
if and only if
\(K=L\).
$$\Phi_{p}K\subseteq\Phi_{p}L\quad\Longrightarrow\quad V(K)\leq V(L), $$
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
and
\(V(K)=V(L)\)
if and only if
\(K=L\).
$$\Phi_{p}K\subseteq\Phi_{p}L\quad\Longrightarrow\quad V(K)\geq V(L), $$
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
\(Q\in\mathcal{S}^{n}_{o}\),
then, for
\(p>0\),
and, for
\(p<0\),
Equality holds in (4.5) or (4.6) only if
\(K=L\).
$$ \widetilde{V}_{p}(K, Q)\leq\widetilde{V}_{p}(L, Q), $$
(4.4)
$$ V(\Phi_{p}K)\leq V(\Phi_{p}L), $$
(4.5)
$$ V(\Phi_{p}K)\geq V(\Phi_{p}L). $$
(4.6)
Proof
For \(0< p< n\), let \(Q=\Phi_{p}\Phi_{p}K\) in (4.4). Then Using inequality (2.4) and equalities (3.1) and (2.3), we have which yields (4.5), and equality holds only if \(K=L\).
$$\widetilde{V}_{p}(K, \Phi_{p}\Phi_{p}K)\leq \widetilde{V}_{p}(L, \Phi_{p}\Phi_{p}K). $$
$$V(\Phi_{p}K)=\widetilde{V}_{p}(\Phi_{p}K, \Phi_{p}K)\leq\widetilde{V}_{p}(\Phi _{p}K, \Phi_{p}L)\leq V(\Phi_{p}K)^{\frac{n-p}{n}}V( \Phi_{p}L)^{\frac{p}{n}}, $$
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.