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On p-radial Blaschke and harmonic Blaschke additions

Journal of Inequalities and Applications20172017:308

https://doi.org/10.1186/s13660-017-1581-y

  • Received: 22 September 2017
  • Accepted: 4 December 2017
  • Published:

Abstract

In the paper, we first improve the radial Blaschke and harmonic Blaschke additions and introduce the p-radial Blaschke and p-harmonic Blaschke additions. Following this, Dresher type inequalities for the radial Blaschke-Minkowski homomorphisms with respect to p-radial Blaschke and p-harmonic Blaschke additions are established.

Keywords

  • radial Blaschke addition
  • harmonic Blaschke addition
  • p-radial Blaschke addition
  • p-harmonic Blaschke addition
  • radial Blaschke-Minkowski homomorphisms
  • Brunn-Minkowski inequality

MSC

  • 46E27
  • 52A20

1 Notation and preliminaries

The setting for this paper is an n-dimensional Euclidean space \({\Bbb {R}}^{n}\). We reserve the letter u for unit vectors, and the letter B is reserved for the unit ball centered at the origin. The surface of B is \(S^{n-1}\). The volume of the unit n-ball is denoted by \(\omega_{n}\). We use \(V(K)\) for the n-dimensional volume of a body K. Associated with a compact subset K of \({\Bbb {R}}^{n}\), which is star-shaped with respect to the origin, is its radial function \(\rho(K,\cdot): S^{n-1}\rightarrow{\Bbb {R}}\) defined for \(u\in S^{n-1}\) by
$$\rho(K,u)=\max\{\lambda\geq0: \lambda u\in K\}. $$
If \(\rho(K,\cdot)\) is positive and continuous, K will be called a star body. Let \({\mathcal {S}}^{n}\) denote the set of star bodies in \({\Bbb {R}}^{n}\). Let δ̃ denote the radial Hausdorff metric, i.e., if \(K, L\in{\mathcal {S}}^{n}\), then \(\tilde{\delta}(K,L)=|\rho(K,u)-\rho(L,u)|_{\infty}\), where \(|\cdot|_{\infty}\) denotes the sup-norm on the space of continuous functions \(C(S^{n-1})\).

1.1 Dual mixed volumes

The radial Minkowski linear combination, \(\lambda_{1}K_{1}\widetilde {+}\cdots\widetilde{+}\lambda_{r} K_{r}\) is defined by
$$\begin{aligned} \lambda_{1}K_{1}\widetilde{+}\cdots\widetilde{+} \lambda_{r} K_{r}=\{\lambda_{1}x_{1} \widetilde{+}\cdots\widetilde{+}\lambda_{r} x_{r}: x_{i}\in K_{i}, i=1,\ldots,r\} \end{aligned}$$
(1.1)
for \(K_{1},\ldots,K_{r}\in{\mathcal {S}}^{n}\) and \(\lambda_{1},\ldots,\lambda_{r}\in{\Bbb {R}}\). It has the following important property (see [1]):
$$\begin{aligned} \rho(\lambda K\widetilde{+}\mu L,\cdot)=\lambda\rho(K,\cdot)+\mu\rho(L, \cdot) \end{aligned}$$
(1.2)
for \(K, L\in{\mathcal {S}}^{n}\) and \(\lambda, \mu\geq0\). For \(K_{1},\ldots,K_{r}\in{\mathcal {S}}^{n}\) and \(\lambda_{1},\ldots,\lambda_{r}\geq0\), the volume of the radial Minkowski linear combination \(\lambda_{1}K_{1}\widetilde{+}\cdots\widetilde{+}\lambda_{r}K_{r}\) is a homogeneous polynomial of degree n in the \(\lambda_{i}\),
$$\begin{aligned} V(\lambda_{1}K_{1}\widetilde{+}\cdots\widetilde{+}\lambda _{r}K_{r})=\sum_{i_{1},\ldots,i_{n}=1}^{r} \widetilde{V}(K_{i_{1}},\ldots,K_{i_{n}})\lambda_{i_{1}} \cdots \lambda_{i_{n}}. \end{aligned}$$
(1.3)
If we require the coefficients of the polynomial in (1.3) to be symmetric in their arguments, then they are uniquely determined. The coefficient \(\widetilde{V}(K_{i_{1}},\ldots,K_{i_{n}})\) is nonnegative and depends only on the bodies \(K_{i_{1}},\ldots,K_{i_{n}}\). It is called the dual mixed volume of \(K_{i_{1}},\ldots,K_{i_{n}}\).
If \(K_{1},\ldots,K_{n}\in{\mathcal {S}}^{n}\), then the dual mixed volume \(\widetilde{V}(K_{1},\ldots,K_{n})\) can be represented in the form (see [2])
$$\begin{aligned} \widetilde{V}(K_{1},\ldots,K_{n})=\frac{1}{n} \int_{S^{n-1}}\rho (K_{1},u)\cdots\rho(K_{n},u)\,dS(u). \end{aligned}$$
(1.4)
If \(K_{1}=\cdots=K_{n-i}=K\), \(K_{n-i+1}=\cdots=K_{n}=L\), then the dual mixed volume is written as \(\widetilde{V}_{i}(K,L)\). If \(L=B\), then the dual mixed volume \(\widetilde{V}_{i}(K,L)=\widetilde{V}_{i}(K,B)\) is written as \(\widetilde{W}_{i}(K)\). For \(K,L\in{\mathcal {S}}^{n}\), the ith dual mixed volume of K and L, \(\widetilde{V}_{i}(K,L)\) can be extended to all \(i\in{\Bbb {R}}\) by
$$\begin{aligned} \widetilde{V}_{i}(K,L)=\frac{1}{n} \int_{S^{n-1}}\rho(K,u)^{n-i}\rho (L,u)^{i}\,dS(u), \end{aligned}$$
(1.5)
where \(i\in{\Bbb {R}}\). Thus, if \(K\in{\mathcal {S}}^{n}\), then
$$\begin{aligned} \widetilde{W}_{i}(K)=\frac{1}{n} \int_{S^{n-1}}\rho (K,u)^{n-i}\,dS(u). \end{aligned}$$
(1.6)

1.2 Mixed intersection bodies

For \(K\in{\mathcal {S}}^{n}\), there is a unique star body I K whose radial function satisfies, for \(u\in S^{n-1}\),
$$\rho(\mathbf{I}K,u)=v(K\cap E_{u}), $$
where v is \((n-1)\)-dimensional dual volume. It is called the intersection body of K. The volume of the intersection body of K is given by (see [1])
$$V(\mathbf{I}K)=\frac{1}{n} \int_{S^{n-1}}v(K\cap E_{u})^{n}\,dS(u). $$
The mixed intersection body of \(K_{1},\ldots,K_{n-1}\in{\mathcal {S} }^{n}\), denoted by \(\mathbf{I}(K_{1},\ldots,K_{n-1})\), is defined by
$$\rho\bigl(\mathbf{I}(K_{1},\ldots,K_{n-1}),u\bigr)= \tilde{v}(K_{1}\cap E_{u},\ldots,K_{n-1}\cap E_{u}), $$
where is the \((n-1)\)-dimensional dual mixed volume. If \(K_{1}=\cdots=K_{n-i-1}=K, K_{n-i}=\cdots=K_{n-1}=L\), then \({\bf I}(K_{1},\ldots,K_{n-1})\) is written as \(\mathbf{I}_{i}(K,L)\). If \(L=B\), then \(\mathbf{I}_{i}(K,L)\) is written as \(\mathbf{I}_{i}K\) and called the ith intersection body of K. For \(\mathbf{I}_{0}K\), we simply write I K.

2 Improvement of the radial Blaschke addition

Let us recall the concept of radial Blaschke addition defined by Lutwak [1]. Suppose that K and L are star bodies in \({\Bbb {R}}^{n}\), the radial Blaschke addition denoted by \(K\widehat{+}L\) is a star body whose radial function is
$$\begin{aligned} \rho(K\widehat{+}L, \cdot)^{n-1}=\rho(K,\cdot)^{n-1}+\rho(L, \cdot)^{n-1}. \end{aligned}$$
(2.1)
The dual Knesser-Süss inequality for the radial Blaschke addition was established by Lutwak [1]. If \(K,L\in{\mathcal {S}}^{n}\), then
$$\begin{aligned} V(K\widehat{+}L)^{(n-1)/n}\leq V(K)^{(n-1)/n}+V(L)^{(n-1)/n}, \end{aligned}$$
(2.2)
with equality if and only if K and L are dilates.

In the section, we give a generalized concept of the radial Blaschke addition.

Definition 2.1

If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq p< n-1\) and \(\lambda, \mu>0\) (not both zero), the p-radial Blaschke linear combination of K and L denoted by \(\lambda\diamond K\widehat{+}_{p}\mu\diamond L\) is a star body whose radial function is defined by
$$\begin{aligned} \rho(\lambda\diamond K\widehat{+}_{p}\mu\diamond L,\cdot )^{n-p-1}=\lambda\rho(K,\cdot)^{n-p-1}+\mu\rho(L,\cdot )^{n-p-1}. \end{aligned}$$
(2.3)
From (2.3), it is easy to see that
$$\lambda\diamond K=\lambda^{1/(n-p-1)} K. $$
When \(\lambda=\mu=1\), the p-radial Blaschke combination becomes the p-radial Blaschke addition \(K\widehat{+}_{p}L\) and
$$\begin{aligned} \rho(K\widehat{+}_{p}L,\cdot)^{n-p-1}=\rho(K, \cdot)^{n-p-1}+\rho (L,\cdot)^{n-p-1}. \end{aligned}$$
(2.4)
Obviously, when \(p=0\), (2.4) becomes (2.1).

In the following, we define the dual mixed quermassintegral with respect to the p-radial Blaschke addition. First, we show two propositions. The following proposition follows immediately from (2.3) with L’Hôpital’s rule.

Proposition 2.2

Let \(0\leq p< n-1\), \(0\leq i< n\) and \(\varepsilon>0\). If \(K,L\in{\mathcal {S}}^{n}\), then
$$\begin{aligned} \lim_{\varepsilon\rightarrow0^{+}}\frac{\rho(K\widehat {+}_{p}\varepsilon\diamond L,u)^{n-i}-\rho(K,u)^{n-i}}{\varepsilon}=\frac{n-i}{n-p-1}\rho (K,u)^{p-i+1}\rho(L,u)^{n-p-1}. \end{aligned}$$
(2.5)

The following proposition follows immediately from Proposition 2.2 and (1.6).

Proposition 2.3

Let \(0\leq p< n-1\), \(0\leq i< n\) and \(\varepsilon>0\). If \(K,L\in{\mathcal {S}}^{n}\), then
$$\begin{aligned} &\frac{n-p-1}{n-i}\lim_{\varepsilon\rightarrow0^{+}}\frac {\widetilde{W}_{i}(K\widehat{+}_{p}\varepsilon\diamond L,u)-\widetilde{W}_{i}(K)}{\varepsilon} \\ &\quad=\frac{1}{n} \int _{S^{n-1}}\rho(K,u)^{p-i+1}\rho(L,u)^{n-p-1}\,dS(u). \end{aligned}$$
(2.6)

Definition 2.4

For \(0\leq p< n-1\), \(0\leq i< n\) and \(K,L\in {\mathcal {S}}^{n}\), the p-dual mixed quermassintegral of star bodies K and L, denoted by \(\widetilde{W}_{p,i}(K,L)\), is defined by
$$\begin{aligned} \widetilde{W}_{p,i}(K,L)=\frac{1}{n} \int_{S^{n-1}}\rho (K,u)^{p-i+1}\rho(L,u)^{n-p-1}\,dS(u). \end{aligned}$$
(2.7)
Obviously, when \(K=L\), \(\widetilde{W}_{p,i}(K,L)\) becomes the dual quermassintegral of star body K, i.e., \(\widetilde{W}_{p,i}(K,K)=\widetilde{W}_{i}(K)\). Taking \(i=0\) in (2.7), \(\widetilde{W}_{p,i}(K,L)\) becomes the p-dual mixed volume \(\widetilde{V}_{p}(K,L)\) and
$$\begin{aligned} \widetilde{V}_{p}(K,L)=\frac{1}{n} \int_{S^{n-1}}\rho(K,u)^{p+1}\rho (L,u)^{n-p-1}\,dS(u). \end{aligned}$$
(2.8)

From (2.7), combining Hölder’s integral inequality (see [3]) gives the following.

Proposition 2.5

(Minkowski type inequality)

If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\) and \(0\leq p< n-1\), then
$$\begin{aligned} \widetilde{W}_{p,i}(K,L)^{n-i}\leq\widetilde {W}_{i}(K)^{p-i+1}\widetilde{W}_{i}(L)^{n-p-1}, \end{aligned}$$
(2.9)
with equality if and only if K and L are dilates.
Taking \(i=0\) in (2.9), we have: If \(K,L\in{\mathcal {S}}^{n}\) and \(0\leq p< n-1\), then
$$\begin{aligned} \widetilde{V}_{p}(K,L)^{n}\leq V(K)^{p+1}V(L)^{n-p-1}, \end{aligned}$$
(2.10)
with equality if and only if K and L are dilates. In the following, we establish the Brunn-Minkowski inequality for the p-radial Blaschke addition.

Proposition 2.6

If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\) and \(0\leq p< n-1\), then
$$\begin{aligned} \widetilde{W}_{i}(K\widehat{+}_{p}L)^{(n-p-1)/(n-i)}\leq \widetilde {W}_{i}(K)^{(n-p-1)/(n-i)}+\widetilde{W}_{i}(L)^{(n-p-1)/(n-i)}, \end{aligned}$$
(2.11)
with equality if and only if K and L are dilates.

Proof

From (2.3) and (2.7), it is easily seen that the p-dual mixed quermassintegral \(\widehat{W}_{p,i}(K,L)\) is linear with respect to the p-radial Blaschke addition and together with inequality (2.9) shows that
$$\begin{aligned} \widetilde{W}_{p,i}(Q, K\widehat{+}_{p}L)&=\widetilde {W}_{p,i}(Q,K)+\widetilde {W}_{p,i}(Q,L) \\ &\leq\widetilde{W}_{i}(Q)^{(p-i+1)/(n-i)}\bigl(\widetilde {W}_{i}(K)^{(n-p-1)/(n-i)}+\widetilde {W}_{i}(L)^{(n-p-1)/(n-i)} \bigr), \end{aligned}$$
(2.12)
with equality if and only if K and L are dilates of Q. Take \(K\widehat{+}_{p}L\) for Q in (2.12), recall that \(\widetilde{W}_{p,i}(Q,Q)=\widetilde{W}_{i}(Q)\), inequality (2.11) follows easy.
Taking \(i=0\) in (2.11), we obtain that if \(K,L\in{\mathcal {S}}^{n}\) and \(0\leq p< n-1\), then
$$V(K\widehat{+}_{p} L)^{(n-p-1)/n}\leq V(K)^{(n-p-1)/n}+V(L)^{(n-p-1)/n}, $$
with equality if and only if K and L are dilates. Taking \(p=0\) and \(i=0\) in (2.11), (2.11) becomes the well-known dual Knesser-Süss inequality (2.2). □

3 Improvement of the harmonic Blaschke addition

Let us recall the concept of harmonic Blaschke addition defined by Lutwak [4]. Suppose that K and L are star bodies in \({\Bbb {R}}^{n}\), the harmonic Blaschke addition denoted by \(K\breve{+}L\) is defined by
$$\begin{aligned} \frac{\rho(K\breve{+}L, \cdot)^{n+1}}{V(K\breve{+}L)}=\frac{\rho(K,\cdot )^{n+1}}{V(K)}+\frac{\rho(L,\cdot)^{n+1}}{V(L)}. \end{aligned}$$
(3.1)
Lutwak’s Brunn-Minkowski inequality for the harmonic Blaschke addition was established (see [4]). If \(K,L\in{\mathcal {S}}^{n}\), then
$$\begin{aligned} V(K\breve{+}L)^{1/n}\geq V(K)^{1/n}+V(L)^{1/n}, \end{aligned}$$
(3.2)
with equality if and only if K and L are dilates.

In the section, we give an improved concept of the harmonic Blaschke addition.

Definition 3.1

For \(0\leq i< n\), \(p< i-1\) and \(K,L\in{\mathcal {S}}^{n}\), we define the p-harmonic Blaschke addition of star bodies K and L denoted by \(K\breve{+}_{p}L\) and defined by
$$\begin{aligned} \frac{\rho(K\breve{+}_{p}L,\cdot)^{n-p-1}}{\tilde{W}_{i}(K\breve {+}_{p}L)}=\frac{\rho(K,\cdot)^{n-p-1}}{ \widetilde{W}_{i}(K)}+\frac{\rho(L,\cdot)^{n-p-1}}{\tilde {W}_{i}(L)}. \end{aligned}$$
(3.3)

The Brunn-Minkowski inequality for the p-harmonic Blaschke addition follows immediately from (1.6), (3.3) and Minkowski’s integral inequality (see [3]).

Proposition 3.2

If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\) and \(p< i-1\), then
$$\begin{aligned} \widetilde{W}_{i}(K\breve{+}_{p}L)^{-(p+1-i)/(n-i)}\leq \widetilde {W}_{i}(K)^{-(p+1-i)/(n-i)} +\widetilde{W}_{i}(L)^{-(p+1-i)/(n-i)}, \end{aligned}$$
(3.4)
with equality if and only if K and L are dilates.

4 Radial Blaschke-Minkowski homomorphisms

Definition 4.1

([5])

A map \(\Psi: {\mathcal {S}}^{n}\rightarrow {\mathcal {S}}^{n}\) is called a radial Blaschke-Minkowski homomorphism if it satisfies the following conditions:
  1. (a)

    Ψ is continuous.

     
  2. (b)
    For all \(K,L\in{\mathcal {S}}^{n}\),
    $$\Psi(K\ddot{+}L)=\Psi(K)\widetilde{+}\Psi(L). $$
     
  3. (c)
    For all \(K,L\in{\mathcal {S}}^{n}\) and every \(\vartheta\in SO(n)\),
    $$\Psi(\vartheta K)=\vartheta\Psi(K), $$
    where \(SO(n)\) is the group of rotations in n dimensions.
     
Radial Blaschke-Minkowski homomorphisms are important examples of star body valued valuations. Their natural duals, Blaschke-Minkowski homomorphisms, are an important notion in the theory of convex body valued valuations (see, e.g., [612] and [1320]). In 2006, Schuster [5] established the following Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms of star bodies. If K and L are star bodies in \({\Bbb {R}}^{n}\), then
$$\begin{aligned} V\bigl(\Psi(K\widetilde{+}L)\bigr)^{1/n(n-1)}\leq V(\Psi K)^{1/n(n-1)}+V(\Psi L)^{1/n(n-1)}, \end{aligned}$$
(4.1)
with equality if and only if K and L are dilates.
If K and L are star bodies in \({\Bbb {R}}^{n}\), \(p\neq0\) and \(\lambda, \mu\geq0\), then \(\lambda\cdot K\widetilde{+}_{p}\mu\cdot L\) is the star body whose radial function is given by (see, e.g., [21])
$$\begin{aligned} \rho(\lambda\cdot K\widetilde{+}_{p}\mu\cdot L,\cdot)^{p}= \lambda\rho(K,\cdot)^{p}+\mu\rho(L,\cdot )^{p}. \end{aligned}$$
(4.2)
The addition \(\widetilde{+}_{p}\) is called \(L_{p}\)-radial addition. The \(L_{p}\) dual Brunn-Minkowski inequality states: If \(K,L\in{\mathcal {S}}^{n}\) and \(0< p\leq n\), then
$$V(K\widetilde{+}_{p}L)^{p/n}\leq V(K)^{p/n}+V(L)^{p/n}, $$
with equality when \(p\neq n\) if and only if K and L are dilates. The inequality is reversed when \(p>n\) or \(p<0\) (see [21]).
In 2013, an \(L_{p}\) Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms was established in [22]: If K and L are star bodies in \({\Bbb {R}}^{n}\) and \(0< p< n-1\), then
$$\begin{aligned} V\bigl(\Psi(K\widetilde{+}_{p}L)\bigr)^{p/n(n-1)}\leq V(\Psi K)^{p/n(n-1)}+V(\Psi L)^{p/n(n-1)}, \end{aligned}$$
(4.3)
with equality if and only if K and L are dilates. Taking \(p=1\), (4.3) reduces to (4.1).

Theorem 4.2

(see [5])

Let \(\Psi: {\mathcal {S}}^{n}\rightarrow{\mathcal {S}}^{n}\) be a radial Blaschke-Minkowski homomorphism. There is a continuous operator \(\Psi: \underbrace{{{\mathcal {S}}^{n}\times\cdots\times\mathcal {S}}^{n}}_{n-1}\rightarrow{\mathcal {S}}^{n}\) symmetric in its arguments such that, for \(K_{1},\ldots,K_{m}\in{\mathcal {S}}^{n}\) and \(\lambda_{1},\ldots,\lambda_{m}\geq0\),
$$\begin{aligned} \Psi(\lambda_{1}K_{1}\widetilde{+}\cdots\widetilde{+} \lambda _{m}K_{m})=\sum_{i_{1},\ldots,i_{n-1}} \lambda_{i_{1}}\cdots\lambda _{i_{n-1}} \Psi(K_{i_{1}}, \ldots,K_{i_{n-1}}). \end{aligned}$$
(4.4)

Clearly, Theorem 4.2 generalizes the notion of radial Blaschke-Minkowski homomorphisms. We call \(\Psi: {\mathcal {S}}^{n}\times\cdots\times{\mathcal {S}}^{n}\rightarrow{\mathcal {S}}^{n}\) a mixed radial Blaschke-Minkowski homomorphism induced by Ψ. Mixed radial Blaschke-Minkowski homomorphisms were first studied in more detail in [23, 24]. If \(K_{1}=\cdots=K_{n-i-1}=K, K_{n-i}=\cdots=K_{n-1}=L\), we write \(\Psi_{i}(K,L)\) for \(\Psi(\underbrace{K,\ldots,K}_{n-i-1},\underbrace{L,\ldots,L}_{i})\). If \(K_{1}=\cdots=K_{n-i-1}=K, K_{n-i}=\cdots=K_{n-1}=B\), we write \(\Psi_{i} K\) for \(\Psi(\underbrace{K,\ldots,K}_{n-i-1},\underbrace{B,\ldots,B}_{i})\) and call \(\Psi_{i} K\) the mixed Blaschke-Minkowski homomorphism of order i of K. \(\Psi_{0} K\) is written simply as ΨK.

Lemma 4.3

(see [5])

A map \(\Psi: {\mathcal {S}}^{n}\rightarrow{\mathcal {S}}^{n}\) is a radial Blaschke-Minkowski homomorphism if and only if there is a measure \(\mu\in{\mathcal {M}}_{+}(S^{n-1},\hat{e})\) such that
$$\begin{aligned} \rho(\Psi K,\cdot)=\rho(K,\cdot)^{n-1}\ast\mu, \end{aligned}$$
(4.5)
where \({\mathcal {M}}_{+}(S^{n-1},\hat{e})\) denotes the set of nonnegative zonal measures on \(S^{n-1}\).
For the mixed radial Blaschke-Minkowski homomorphism induced by Ψ, Schuster [5] proved that
$$\rho\bigl(\Psi(K_{1},\ldots,K_{n-1}),\cdot\bigr)= \rho(K_{1},\cdot)\cdots \rho(K_{n-1},\cdot)\ast\mu. $$
Obviously, a special case is the following:
$$\rho(\Psi_{i}K,\cdot)=\rho(K,\cdot)^{n-1-i}\ast\mu, $$
where i are integers. We now extend the integers i to real numbers, define the Blaschke-Minkowski homomorphism of order p of K.

Definition 4.4

Let \(K\in{\mathcal {S}}^{n}\), the Blaschke-Minkowski homomorphism of order p of K, denoted by \(\Psi_{p}K\), is defined for all \(p\in{\Bbb {R}}\) by
$$\begin{aligned} \rho(\Psi_{p}K,\cdot)=\rho(K,\cdot)^{n-1-p}\ast \mu. \end{aligned}$$
(4.6)
This extended definition will be required to prove our main results.

5 Inequalities for the radial Blaschke-Minkowski homomorphism

Theorem 5.1

Let \(K,L\in{\mathcal {S}}^{n}\). If \(0\leq p< n-1\) and \(i\leq n-1\leq j\leq n\), then
$$\begin{aligned} \biggl(\frac{\widetilde{W}_{i}(\Psi_{p}(K\widehat {+}_{p}L))}{\widetilde{W}_{j}(\Psi_{p}(K\widehat{+}_{p}L))} \biggr)^{1/(j-i)} \leq \biggl(\frac{\widetilde{W}_{i}(\Psi_{p}K)}{\widetilde {W}_{j}(\Psi_{p}K)} \biggr) ^{1/(j-i)}+ \biggl(\frac{\widetilde{W}_{i}(\Psi_{p} L)}{\widetilde{W}_{j}(\Psi_{p}L)} \biggr)^{1/(j-i)}, \end{aligned}$$
(5.1)
with equality if and only if \(\Psi_{p} K\) and \(\Psi_{p} L\) are dilates.

Remark 5.2

Taking \(j=n\) in (5.1) and noting that \(\widetilde{W}_{n}(K)=\int_{S^{n-1}}\,dS(u)=n\omega_{n}\), (5.1) becomes the following inequality: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq p< n-1\) and \(i\leq n-1\), then
$$\begin{aligned} \widetilde{W}_{i}\bigl(\Psi_{p}(K\widehat{+}_{p}L) \bigr)^{1/(n-i)} \leq\widetilde{W}_{i}(\Psi_{p} K)^{1/(n-i)}+\widetilde{W}_{i}(\Psi_{p}L)^{1/(n-i)}, \end{aligned}$$
(5.2)
with equality if and only if \(\Psi_{p} K\) and \(\Psi_{p} L\) are dilates. Taking \(p=0\) in (5.1), (5.1) becomes the following inequality: If \(K,L\in{\mathcal {S}}^{n}\) and \(i\leq n-1\leq j\leq n\), then
$$\begin{aligned} \biggl(\frac{\widetilde{W}_{i}(\Psi(K\widehat{+}L))}{\widetilde {W}_{j}(\Psi(K\widehat{+}L))} \biggr)^{1/(j-i)} \leq \biggl(\frac{\widetilde{W}_{i}(\Psi K)}{\widetilde{W}_{j}(\Psi K)} \biggr)^{1/(j-i)}+ \biggl(\frac{\widetilde{W}_{i}(\Psi L)}{\widetilde{W}_{j}(\Psi L)} \biggr)^{1/(j-i)}, \end{aligned}$$
(5.3)
with equality if and only if ΨK and ΨL are dilates.

Theorem 5.3

Let \(K,L\in{\mathcal {S}}^{n}\). If \(0\leq i< n \), \(p< i-1\) and \(k,j\in{\Bbb {R}}\) satisfy \(j\leq n-1\leq k\leq n\), then
$$\begin{aligned} &\frac{1}{\widetilde{W}_{i}(K\breve{+}_{p} L)} \biggl(\frac{\widetilde{W}_{j}(\Psi_{p}(K\breve {+}_{p}L))}{\widetilde{W}_{k}(\Psi_{p}(K\breve{+}_{p}L))} \biggr)^{1/(k-j)} \\ &\quad\leq\frac{1}{\widetilde{W}_{i}(K)} \biggl(\frac {\widetilde{W}_{j}(\Psi_{p}K)}{\widetilde{W}_{k}(\Psi_{p}K)} \biggr) ^{1/(k-j)}+ \frac{1}{\widetilde{W}_{i}(L)} \biggl(\frac{\widetilde {W}_{j}(\Psi_{p}L)}{\widetilde{W}_{k}(\Psi_{p}L)} \biggr)^{1/(k-j)}, \end{aligned}$$
(5.4)
with equality if and only if \(\Psi_{p}K\) and \(\Psi_{p}L\) are dilates.

Remark 5.4

Taking \(k=n\) in (5.4) and noting that \(\widetilde{W}_{n}(K)=\int_{S^{n-1}}\,dS(u)=n\omega_{n}\), (5.4) becomes the following inequality: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\), \(p< i-1\) and \(j\leq n-1\), then
$$\begin{aligned} \frac{\widetilde{W}_{j}(\Psi_{p}(K\breve {+}_{p}L))^{1/(n-j)}}{\widetilde{W}_{i}(K\breve{+}_{p} L)}\leq\frac{\widetilde{W}_{j}(\Psi_{p}K)^{1/(n-j)}}{\widetilde {W}_{i}(K)}+\frac{\widetilde{W}_{j}(\Psi_{p}L)^{1/(n-j)}}{\widetilde {W}_{i}(L)}, \end{aligned}$$
(5.5)
with equality if and only if \(\Psi_{p}K\) and \(\Psi_{p}L\) are dilates. Taking \(i=0\), \(j=0\) and \(k=n\) in (5.4), we have: If \(K,L\in{\mathcal {S}}^{n}\) and \(p<-1\), then
$$\begin{aligned} \frac{V(\Psi_{p}(K\breve{+}_{p}L))^{1/n}}{V(K\breve{+}_{p} L)}\leq\frac{V(\Psi_{p}K)^{1/n}}{V(K)}+\frac{V(\Psi _{p}L)^{1/n}}{V(L)}, \end{aligned}$$
(5.6)
with equality if and only if \(\Psi_{p}K\) and \(\Psi_{p}L\) are dilates.

6 Dresher’s inequalities for p-radial Blaschke and harmonic Blaschke additions

An extension of Beckenbach’s inequality (see [3], p. 27) was obtained by Dresher [25] by means of moment-space techniques.

Lemma 6.1

(Dresher’s inequality)

If \(p\geq1\geq r\geq0\), \(f,g\geq0\) and ϕ is a distribution function, then
$$\begin{aligned} \biggl(\frac{\int(f+g)^{p}\,d\phi}{\int(f+g)^{r}\,d\phi} \biggr)^{1/(p-r)}\leq \biggl(\frac{\int f^{p}\,d\phi}{\int f^{r}\,d\phi} \biggr) ^{1/(p-r)}+ \biggl(\frac{\int g^{p}\,d\phi}{\int g^{r}\,d\phi} \biggr)^{1/(p-r)}, \end{aligned}$$
(6.1)
with equality if and only if the functions f and g are proportional.

We are now in a position to prove Theorem 5.1. The following statement is just a slight reformulation of it.

Theorem 6.2

Let \(K,L\in{\mathcal {S}}^{n}\). If \(0\leq p< n-1\) and \(s,t\in{\Bbb {R}}\) satisfy \(s\geq1\geq t\geq0\), then
$$\begin{aligned} \biggl(\frac{\widetilde{W}_{n-s}(\Psi_{p}(K\widehat {+}_{p}L))}{\widetilde{W}_{n-t}(\Psi_{p}(K\widehat{+}_{p}L))} \biggr)^{1/(s-t)} \leq \biggl(\frac{\widetilde{W}_{n-s}(\Psi_{p}K)}{\widetilde {W}_{n-t}(\Psi_{p}K)} \biggr) ^{1/(s-t)}+ \biggl(\frac{\widetilde{W}_{n-s}(\Psi_{p}L)}{\widetilde {W}_{n-t}(\Psi_{p}L)} \biggr)^{1/(s-t)}, \end{aligned}$$
(6.2)
with equality if and only if \(\Psi_{p} K\) and \(\Psi_{p} L\) are dilates.

Proof

From (2.4), we obtain
$$\rho(K\widehat{+}_{p} L,\cdot)^{n-p-1}\ast\mu=\rho(K, \cdot)^{n-p-1}\ast\mu+\rho (L,\cdot)^{n-p-1}\ast\mu, $$
where μ is the generating measure of Ψ from Lemma 4.3. Hence, from (4.6), we obtain
$$\rho\bigl(\Psi_{p}(K\widehat{+}_{p}L),\cdot\bigr)=\rho( \Psi_{p}K,\cdot )+\rho(\Psi_{p}L,\cdot). $$
Therefore, by (1.6), we have
$$\begin{aligned} \widetilde{W}_{n-s}\bigl(\Psi_{p}(K\widehat{+}_{p}L) \bigr)=\frac{1}{n} \int _{S^{n-1}} \bigl(\rho(\Psi_{p}K,u)+\rho( \Psi_{p}L,u) \bigr)^{s}\,dS(u) \end{aligned}$$
(6.3)
and
$$\begin{aligned} \widetilde{W}_{n-t}\bigl(\Psi_{p}(K\widehat{+}_{p}L) \bigr)=\frac{1}{n} \int _{S^{n-1}} \bigl(\rho(\Psi_{p}K,u)+\rho( \Psi_{p}L,u) \bigr)^{t}\,dS(u). \end{aligned}$$
(6.4)
From (6.3), (6.4) and Lemma 6.1, we obtain
$$\begin{aligned} &\biggl(\frac{\widetilde{W}_{n-s}(\Psi_{p}(K\widehat {+}_{p}L))}{\widetilde{W}_{n-t}(\Psi_{p}(K\widehat{+}_{p}L))} \biggr)^{1/(s-t)} \\ &\quad= \biggl(\frac{\int_{S^{n-1}} (\rho(\Psi_{p}K,u)+\rho(\Psi _{p}L,u) )^{s}\,dS(u)}{ \int_{S^{n-1}} (\rho(\Psi_{p}K,u)+\rho(\Psi_{p}L,u) )^{t}\,dS(u)} \biggr)^{1/(s-t)} \\ &\quad\leq \biggl(\frac{\int_{S^{n-1}}\rho (\Psi_{p}K,u)^{s}\,dS(u)}{ \int_{S^{n-1}}\rho(\Psi_{p}K,u)^{t}\,dS(u)} \biggr)^{1/(s-t)}+ \biggl(\frac {\int_{S^{n-1}}\rho(\Psi_{p}L,u)^{s}\,dS(u)}{\int_{S^{n-1}}\rho(\Psi _{p}L,u)^{t}\,dS(u)} \biggr)^{1/(s-t)} \\ &\quad= \biggl(\frac{\widetilde{W}_{n-s}(\Psi_{p}K)}{\widetilde {W}_{n-t}(\Psi_{p}K)} \biggr) ^{1/(s-t)}+ \biggl(\frac{\widetilde{W}_{n-s}(\Psi_{p}L)}{\widetilde {W}_{n-t}(\Psi_{p}L)} \biggr)^{1/(s-t)}. \end{aligned}$$
Equality holds if and only if the functions \(\rho(\Psi_{p}K,u)\) and \(\rho(\Psi_{p}L,u)\) are proportional.
Taking \(s=n-i\) and \(t=n-j\) in Theorem 6.2, Theorem 6.2 becomes Theorem 5.1 stated in Section 5. If \(\Psi:\underbrace{{\mathcal {S}}^{n}\times\cdots\times{\mathcal {S}}^{n}}_{n-1}\rightarrow {\mathcal {S}}^{n}\) is the mixed intersection operator \({\bf I}:\underbrace{{\mathcal {S}}^{n}\times\cdots\times{\mathcal {S}}^{n}}_{n-1}\rightarrow{\mathcal {S}}^{n}\) in (6.2) and \(n-s=i\) and \(n-t=j\), we obtain the following result: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq p< n-1\) and \(i\leq n-1\leq j\leq n\), then
$$\begin{aligned} \biggl(\frac{\widetilde{W}_{i}(\mathbf{I}_{p}(K\widehat {+}_{p}L))}{\widetilde{W}_{j}(\mathbf{I}_{p}(K\widehat{+}_{p}L))} \biggr)^{1/(j-i)} \leq \biggl(\frac{\widetilde{W}_{i}({\bf I}_{p}K)}{\widetilde{W}_{j}(\mathbf{I}_{p}K)} \biggr) ^{1/(j-i)}+ \biggl(\frac{\widetilde{W}_{i}({\bf I}_{p}L)}{\widetilde{W}_{j}({\bf I}_{p}L)} \biggr)^{1/(j-i)}, \end{aligned}$$
(6.5)
with equality if and only if \(\mathbf{I}_{p}K\) and \(\mathbf{I}_{p}L\) are dilates. Taking \(j=n\) in (6.5) and noting that \(\widetilde{W}_{n}(K)=\int_{S^{n-1}}\,dS(u)=n\omega_{n}\), (6.5) becomes the following inequality: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq p< n-1\) and \(i\leq n-1\), then
$$\widetilde{W}_{i}\bigl(\mathbf{I}_{p}(K \widehat{+}_{p}L)\bigr)^{1/(n-i)} \leq\widetilde{W}_{i}({ \bf I}_{p}K)^{1/(n-i)}+\widetilde{W}_{i}( \mathbf{I}_{p}L)^{1/(n-i)}, $$
with equality if and only if \(\mathbf{I}_{p}K\) and \(\mathbf{I}_{p}L\) are dilates. □

We are now in a position to prove Theorem 5.3. The following statement is just a slight reformulation of it.

Theorem 6.3

Let \(K,L\in{\mathcal {S}}^{n}\). If \(0\leq i< n \), \(p< i-1\) and \(s,t\in{\Bbb {R}}\) satisfy \(s\geq1\geq t\geq0\), then
$$\begin{aligned} &\frac{1}{\widetilde{W}_{i}(K\breve{+}_{p} L)} \biggl(\frac{\widetilde{W}_{n-s}(\Psi_{p}(K\breve {+}_{p}L))}{\widetilde{W}_{n-t}(\Psi_{p}(K\breve{+}_{p}L))} \biggr)^{1/(s-t)} \\ &\quad\leq\frac{1}{\widetilde{W}_{i}(K)} \biggl(\frac {\widetilde{W}_{n-s}(\Psi_{p}K)}{\widetilde{W}_{n-t}(\Psi _{p}K)} \biggr) ^{1/(s-t)}+ \frac{1}{\widetilde{W}_{i}(L)} \biggl(\frac{\widetilde {W}_{n-s}(\Psi_{p}L)}{\widetilde{W}_{n-t}(\Psi_{p}L)} \biggr)^{1/(s-t)}, \end{aligned}$$
(6.6)
with equality if and only if \(\Psi_{p}K\) and \(\Psi_{p}L\) are dilates.

Proof

From (3.3), we obtain
$$\frac{\rho(K\breve{+}_{p} L,\cdot)^{n-p-1}\ast\mu}{\widetilde{W}_{i}(K\breve{+}_{p} L)}=\frac{\rho(K,\cdot)^{n-p-1}\ast\mu}{\widetilde {W}_{i}(K)}+\frac{\rho(L,\cdot)^{n-p-1}\ast\mu}{\widetilde{W}_{i}(L)}. $$
Hence, from (4.6), we obtain
$$\frac{\rho(\Psi_{p}(K\breve{+}_{p}L),\cdot)}{\widetilde {W}_{i}(K\breve{+}_{p} L)}=\frac{\rho(\Psi_{p}K,\cdot)}{\widetilde{W}_{i}(K)}+\frac{\rho (\Psi_{p}L,\cdot)}{\widetilde{W}_{i}(L)}. $$
By (1.6), we have
$$\begin{aligned} \frac{\widetilde{W}_{n-s}(\Psi_{p}(K\breve{+}_{p}L))}{\widetilde {W}_{i}(K\breve{+}_{p} L)^{s}}=\frac{1}{n} \int_{S^{n-1}} \biggl(\frac{\rho(\Psi _{p}K,u)}{\widetilde{W}_{i}(K)}+\frac{\rho(\Psi_{p}L,u)}{\widetilde {W}_{i}(L)} \biggr)^{s}\,dS(u) \end{aligned}$$
(6.7)
and
$$\begin{aligned} \frac{\widetilde{W}_{n-t}(\Psi_{p}(K\breve{+}_{p}L))}{\widetilde {W}_{i}(K\breve{+}_{p} L)^{t}}=\frac{1}{n} \int_{S^{n-1}} \biggl(\frac{\rho(\Psi _{p}K,u)}{\widetilde{W}_{i}(K)}+\frac{\rho(\Psi_{p}L,u)}{\widetilde {W}_{i}(L)} \biggr)^{t}\,dS(u). \end{aligned}$$
(6.8)
From (6.7), (6.8) and Lemma 6.1, we obtain
$$\begin{aligned} &\frac{1}{\widetilde{W}_{i}(K\breve{+}_{p} L)} \biggl(\frac{\widetilde{W}_{n-s}(\Psi_{p}(K\breve {+}_{p}L))}{\widetilde{W}_{n-t}(\Psi_{p}(K\breve{+}_{p}L))} \biggr)^{1/(s-t)} \\ &\quad= \biggl( \frac{\int_{S^{n-1}} (\frac{\rho(\Psi _{p}K,u)}{\widetilde{W}_{i}(K)}+\frac{\rho(\Psi_{p}L,u)}{\widetilde {W}_{i}(L)} )^{s}\,dS(u)}{ \int_{S^{n-1}} (\frac{\rho(\Psi_{p}K,u)}{\widetilde {W}_{i}(K)}+\frac{\rho(\Psi_{p}L,u)}{\widetilde{W}_{i}(L)} )^{t}\,dS(u)} \biggr)^{1/(s-t)} \\ &\quad\leq \biggl(\frac{\int_{S^{n-1}} (\frac{\rho(\Psi_{p}K,u)}{\widetilde {W}_{i}(K)} )^{s}\,dS(u)}{ \int_{S^{n-1}} (\frac{\rho(\Psi_{p}K,u)}{\widetilde {W}_{i}(K)} )^{t}\,dS(u)} \biggr)^{1/(s-t)}+ \biggl(\frac{\int _{S^{n-1}} (\frac{\rho(\Psi_{p}L,u)}{\widetilde {W}_{i}(L)} )^{s}\,dS(u)}{ \int_{S^{n-1}} (\frac{\rho(\Psi_{p}L,u)}{\widetilde {W}_{i}(L)} )^{t}\,dS(u)} \biggr)^{1/(s-t)} \\ &\quad=\frac{1}{\widetilde{W}_{i}(K)} \biggl(\frac {\widetilde{W}_{n-s}(\Psi_{p}K)}{\widetilde{W}_{n-t}(\Psi _{p}K)} \biggr) ^{1/(s-t)}+ \frac{1}{\widetilde{W}_{i}(L)} \biggl(\frac{\widetilde {W}_{n-s}(\Psi_{p}L)}{\widetilde{W}_{n-t}(\Psi_{p}L)} \biggr)^{1/(s-t)}, \end{aligned}$$
with equality if and only if \(\Psi_{p}K\) and \(\Psi_{p}L\) are dilates.
Taking \(s=n-j\) and \(t=n-k\) in Theorem 6.3, Theorem 6.3 becomes Theorem 5.3 stated in Section 5. If \(\Psi:\underbrace{{\mathcal {S}}^{n}\times\cdots\times{\mathcal {S}}^{n}}_{n-1}\rightarrow {\mathcal {S}}^{n}\) is the mixed intersection operator \({\bf I}:\underbrace{{\mathcal {S}}^{n}\times\cdots\times{\mathcal {S}}^{n}}_{n-1}\rightarrow{\mathcal {S}}^{n}\) in (6.6) and \(j=n-s\) and \(k=n-t\), we obtain the following result: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\), \(p< i-1\) and \(j\leq n-1\leq k\leq n\), then
$$\begin{aligned} &\frac{1}{\widetilde{W}_{i}(K\breve{+}_{p} L)} \biggl(\frac{\widetilde{W}_{j}({\bf I}_{p}(K\breve{+}_{p}L))}{\widetilde{W}_{k}({\bf I}_{p}(K\breve{+}_{p}L))} \biggr)^{1/(k-j)} \\ &\quad \leq \frac{1}{\widetilde{W}_{i}(K)} \biggl(\frac{\widetilde {W}_{j}({\bf I}_{p}K)}{\widetilde{W}_{k}(\mathbf{I}_{p}K)} \biggr) ^{1/(k-j)}+ \frac{1}{\widetilde{W}_{i}(L)} \biggl(\frac{\widetilde {W}_{j}({\bf I}_{p}L)}{\widetilde{W}_{k}({\bf I}_{p}L)} \biggr)^{1/(k-j)}, \end{aligned}$$
(6.9)
with equality if and only if \(\mathbf{I}_{p}K\) and \(\mathbf{I}_{p}L\) are dilates. Taking \(k=n\) in (6.9) and noting that \(\widetilde{W}_{n}(K)=\int_{S^{n-1}}\,dS(u)=n\omega_{n}\), (6.9) becomes the following inequality: If \(K,L\in{\mathcal {S}}^{n}\), \(0\leq i< n\), \(p< i-1\) and \(j\leq n-1\), then
$$\begin{aligned} \frac{\widetilde{W}_{j}({\bf I}_{p}(K\breve{+}_{p}L))^{1/(n-j)}}{\widetilde{W}_{i}(K\breve{+}_{p} L)}\leq\frac{\widetilde{W}_{j}({\bf I}_{p}K)^{1/(n-j)}}{\widetilde{W}_{i}(K)}+\frac{\widetilde {W}_{j}({\bf I}_{p}L)^{1/(n-j)}}{\widetilde{W}_{i}(L)}, \end{aligned}$$
(6.10)
with equality if and only if \(\mathbf{I}_{p}K\) and \(\mathbf{I}_{p}L\) are dilates. □

7 Conclusions

In the present study, we first revised and improved the concepts of radial Blaschke addition and harmonic Blaschke addition in an \(L_{p}\) space. Following this, we established Dresher’s inequalities (Brunn-Minkowski type) for the radial Blaschke-Minkowski homomorphisms with respect to the p-radial addition and the p-harmonic Blaschke addition.

Declarations

Funding

The author’s research is supported by the Natural Science Foundation of China (11371334).

Authors’ contributions

C-JZ provided the questions and gave the proof for the main results. He read and approved the manuscript.

Competing interests

The author declares that he has no competing interests.

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.

Authors’ Affiliations

(1)
Department of Mathematics, China Jiliang University, Hangzhou, 310018, P.R. China

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