Open Access

General \(L_{p}\)-mixed-brightness integrals

Journal of Inequalities and Applications20152015:190

https://doi.org/10.1186/s13660-015-0708-2

Received: 31 December 2014

Accepted: 21 May 2015

Published: 12 June 2015

Abstract

The notion of mixed-brightness integrals was introduced by Li and Zhu. In this paper, motivated by the notion of general \(L_{p}\)-projection bodies, introduced by Haberl and Schuster, we define general \(L_{p}\)-mixed-brightness integrals and determine their extremal values, as well as several other inequalities for them.

Keywords

mixed-brightness integrals general \(L_{p}\)-mixed-brightness integrals general \(L_{p}\)-projection body

MSC

52A20 52A40

1 Introduction

Let \(\mathcal{ K}^{n}\) denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space \(\mathbb {R}^{n}\). For the set of convex bodies containing the origin in their interiors and the set of origin-symmetric convex bodies in \(\mathbb {R}^{n}\), we write \(\mathcal{ K}^{n}_{o}\) and \(\mathcal{ K}^{n}_{os}\), respectively. Let \(\mathcal{ S}^{n}_{o}\) denote the set of star bodies (about the origin) in \(\mathbb{R}^{n}\) and let \(S^{n-1}\) denote the unit sphere in \(\mathbb {R}^{n}\). By \(V(K)\) we denote the n-dimensional volume of a body K and for the standard unit ball B in \(\mathbb{R}^{n}\), we write \(\omega _{n}\) for its volume.

If \(K\in\mathcal{ K}^{n}\), then its support function, \(h_{K}=h(K, \cdot ):\mathbb{R}^{n}\rightarrow(-\infty, \infty)\), is defined by [1, 2]
$$h(K,x)=\max\{x \cdot y: y\in K\}, \quad x\in\mathbb{R}^{n}, $$
where \(x\cdot y\) denotes the standard inner product of x and y.
Projection bodies of convex bodies were introduced at the turn of the previous century by Minkowski [1]. For \(K\in\mathcal{ K}^{n}\), the projection body, ΠK, of K is the origin-symmetric convex body, defined by
$$h(\Pi K, u)=\frac{1}{2}\int_{S^{n-1}}|u\cdot v|\,dS(K, v) $$
for all \(u\in S^{n-1}\). Here \(S(K, \cdot)\) denotes the surface area measure of K.
Using the classical notion of projection bodies, Li and Zhu [3] recently introduced the mixed-brightness integral: For \(K_{1}, \ldots, K_{n}\in\mathcal{ K}^{n}\), the mixed-brightness integral, \(D(K_{1}, \ldots, K_{n})\), is defined by
$$ D(K_{1}, \ldots, K_{n})=\frac{1}{n}\int _{S^{n-1}}\delta(K_{1}, u)\cdots \delta(K_{n}, u)\,dS(u), $$
(1.1)
where \(\delta(K, u)=\frac{1}{2}h(\Pi K, u)\) is the half brightness of \(K\in\mathcal{ K}^{n}\) in the direction u. Convex bodies \(K_{1},\ldots,K_{n}\) are said to have similar brightness if there exist constants \(\lambda _{1},\ldots,\lambda_{n}>0\) such that \(\lambda_{1}\delta(K_{1},u)=\lambda _{2}\delta(K_{2},u)=\cdots=\lambda_{n}\delta(K_{n},u)\) for all \(u\in S^{n-1}\).

Further, Li and Zhu [3] established the following Fenchel-Aleksandrov type inequality for mixed-brightness integrals.

Theorem 1.A

If \(K_{1},\ldots,K_{n} \in\mathcal{ K}^{n}\) and \(1< m\leq n\), then
$$ D(K_{1}, \ldots, K_{n})^{m}\leq\prod ^{m-1}_{i=0}D(K_{1}, \ldots, K_{n-m},K_{n-i}, \ldots, K_{n-i}), $$
(1.2)
with equality if and only if \(K_{n-m+1},K_{n-m+2},\ldots,K_{n}\) are all of similar brightness.

More recently, Zhou et al. [4] obtained Brunn-Minkowski type inequalities for mixed-brightness integrals.

The notion of \(L_{p}\)-projection bodies was introduced by Lutwak et al. [5]. For each \(K \in\mathcal{ K}^{n}_{o}\) and \(p\geq1\), the \(L_{p}\)-projection body, \(\Pi_{p} K\), is the origin-symmetric convex body whose support function is defined by
$$ h_{\Pi_{p} K}^{p}(u)=\alpha_{n,p}\int _{S^{n-1}}|u,v|^{p}\,dS_{p}(K,\upsilon ), $$
(1.3)
for all \(u\in S^{n-1}\), where \(\alpha_{n,p}=1/n\omega_{n}c_{n-2,p}\) with \(c_{n,p}=\omega_{n+p}/\omega_{2}\omega_{n}\omega_{p-1}\), and \(S_{p}(K, \cdot)\) is the \(L_{p}\)-surface measure of K. The normalization in definition (1.3) is chosen such that \(\Pi_{p}B = B\).
As part of the tremendous progress in the theory of Minkowski valuations (see [614]), Ludwig [15] discovered more general \(L_{p}\)-projection bodies \(\Pi_{p}^{\tau}K\in\mathcal{ K}^{n}_{o}\), which can be defined using the function \(\varphi _{\tau}:\mathbb{R}\rightarrow[0,\infty)\) given by
$$\varphi_{\tau}(t)=|t|+\tau t, $$
where \(\tau\in[-1,1]\). Now for \(K\in\mathcal{ K}^{n}_{o}\) and \(p\geq1\), let \(\Pi_{p}^{\tau}K\in\mathcal{ K}^{n}_{o}\) with support function
$$ h_{\Pi_{p}^{\tau}K}^{p}(u)=\alpha_{n,p}(\tau)\int _{S^{n-1}}\varphi_{\tau}(u\cdot v)^{p}\,dS_{p}(K, \upsilon), $$
(1.4)
where
$$\alpha_{n,p}(\tau)=\frac{\alpha_{n,p}}{(1+\tau)^{p}+(1-\tau)^{p}}. $$
The normalization is again chosen such that \(\Pi^{\tau}_{p}B=B\) for every \(\tau\in[-1,1]\). Obviously, if \(\tau=0\), then \(\Pi^{\tau}_{p}K=\Pi_{p}K\).

For general \(L_{p}\)-projection bodies, Haberl and Schuster [16] proved the general \(L_{p}\)-Petty projection inequality and determined the extremal values of volume for polars of general \(L_{p}\)-projection bodies. Wang and Wan [17] investigated Shephard type problems for general \(L_{p}\)-projection bodies. Wang and Feng [18] established general \(L_{p}\)-Petty affine projection inequality. These investigations were the starting point of a new and rapidly evolving asymmetric \(L_{p}\)-Brunn-Minkowski theory (see [1332]).

In this article, using the notion of general \(L_{p}\)-projection bodies, we define general \(L_{p}\)-mixed-brightness integrals as follows: For \(K_{1}, \ldots, K_{n}\in\mathcal{ K}_{o}^{n}\), \(p\geq1\) and \(\tau\in[-1,1]\), the general \(L_{p}\)-mixed-brightness integral, \(D^{(\tau)}_{p}(K_{1}, \ldots , K_{n})\), of \(K_{1}, \ldots, K_{n}\) is defined by
$$ D^{(\tau)}_{p}(K_{1}, \ldots, K_{n})=\frac{1}{n}\int_{S^{n-1}}\delta ^{(\tau)}_{p} (K_{1}, u)\cdots\delta^{(\tau)}_{p}(K_{n}, u)\,dS(u), $$
(1.5)
where \(\delta^{(\tau)}_{p}(K, u)=\frac{1}{2}h(\Pi^{\tau}_{p} K, u)\) denotes the half general \(L_{p}\)-brightness of \(K\in\mathcal{ K}_{o}^{n}\) in the direction u. Convex bodies \(K_{1},\ldots,K_{n}\) are said to have similar general \(L_{p}\)-brightness if there exist constants \(\lambda_{1},\ldots ,\lambda_{n}>0\) such that, for all \(u\in S^{n-1}\),
$$\lambda_{1}\delta^{(\tau)}_{p}(K_{1},u)= \lambda_{2}\delta^{(\tau )}_{p}(K_{2},u)= \cdots=\lambda_{n}\delta^{(\tau)}_{p}(K_{n},u). $$

Remark 1.1

For \(\tau=0\) in (1.5), we write \(D^{(\tau)}_{p}(K_{1}, \ldots, K_{n})=D_{p}(K_{1}, \ldots, K_{n})\) and \(\delta^{(\tau)}_{p}(K,u)=\delta_{p}(K,u)\) for all \(u\in S^{n-1}\). Then
$$ D_{p}(K_{1}, \ldots, K_{n})= \frac{1}{n}\int_{S^{n-1}}\delta_{p} (K_{1}, u)\cdots\delta_{p}(K_{n}, u)\,dS(u), $$
(1.6)
where \(\delta_{p}(K, u)=\frac{1}{2}h(\Pi_{p} K, u)\). Here \(D_{p}(K_{1}, \ldots, K_{n})\) is called the \(L_{p}\)-mixed-brightness integral of \(K_{1}, \ldots, K_{n}\in\mathcal{ K}_{o}^{n}\). Obviously, for \(p=1\), (1.6) is just the mixed-brightness integral from (1.1).
Let \(\underbrace{K_{1}=\cdots=K_{n-i}}_{n-i}=K\) and \(\underbrace {K_{n-i+1}=\cdots=K_{n}}_{i}=L\) (\(i=0, 1, \ldots, n\)) in (1.5), we denote \(D^{\tau}_{p,i}(K,L)=D^{(\tau)}_{p}(\underbrace{K, \ldots, K}_{n-i}, \underbrace{L, \ldots, L}_{i})\). More general, if i is any real, we define for \(K, L\in\mathcal{ K}_{o}^{n}\), \(p\geq1\), and \(\tau\in [-1,1]\), the general \(L_{p}\)-mixed-brightness integral, \(D^{\tau}_{p,i}(K, L)\), of K and L by
$$ D^{(\tau)}_{p,i}(K,L)=\frac{1}{ n}\int _{S^{n-1}}\delta^{(\tau)}_{p} (K, u)^{n-i} \delta^{(\tau)}_{p} (L, u)^{i}\,dS(u). $$
(1.7)
For \(L=B\) in (1.7), we write \(D^{(\tau)}_{p,i}(K, B)=\frac {1}{2^{i}}D^{(\tau)}_{p,i}(K)\) and notice that \(\delta^{(\tau)}_{p} (B, u)=\frac{1}{2}h(\Pi^{\tau}_{p} B, u)=\frac{1}{2}\) for all \(u\in S^{n-1}\), which together with (1.7) yields
$$ D^{(\tau)}_{p,i}(K)=\frac{1}{2^{i}\cdot n}\int _{S^{n-1}}\delta^{(\tau )}_{p} (K, u)^{n-i}\,dS(u), $$
(1.8)
where \(D^{(\tau)}_{p,i}(K)\) is called the ith general \(L_{p}\)-mixed-brightness integral of K. If \(\tau=0\), then \(D^{(\tau )}_{p,i}(K)=D_{p,i}(K)\). For \(\tau= \pm1\), we write \(D^{(\tau)}_{p,i}(K)=D^{\pm}_{p,i}(K)\).
For \(L=K\) in (1.7), write \(D^{(\tau)}_{p,i}(K, K)=D^{(\tau)}_{p}(K)\), which is called the general \(L_{p}\)-brightness integral of K. Clearly,
$$ D^{(\tau)}_{p}(K)=\frac{1}{ n}\int _{S^{n-1}}\delta^{(\tau)}_{p} (K, u)^{n}\,dS(u). $$
(1.9)
Obviously, by (1.5), (1.7), (1.8), and (1.9), we have
$$\begin{aligned}& D^{(\tau)}_{p}(K, \ldots, K)=D^{(\tau)}_{p}(K); \end{aligned}$$
(1.10)
$$\begin{aligned}& D^{(\tau)}_{p,0}(K)=D^{(\tau)}_{p}(K); \\& D^{(\tau)}_{p,0}(K, L)=D^{(\tau)}_{p}(K),\quad\quad D^{(\tau)}_{p,n}(K, L)=D^{(\tau)}_{p}(L). \end{aligned}$$
(1.11)

In this paper, we establish several inequalities for general \(L_{p}\)-mixed-brightness integrals. First, we determine the extremal values of general \(L_{p}\)-mixed-brightness integrals.

Theorem 1.1

If \(K\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1,1]\), then
$$ D_{p,2n}(K)\leq D^{(\tau)}_{p,2n}(K)\leq D^{\pm}_{p,2n}(K). $$
(1.12)
If K is not origin-symmetric and p is not an odd integer, there is equality in the left inequality if and only if \(\tau= 0\) and equality in the right inequality if and only if \(\tau= \pm1\).

Next, we obtain a Brunn-Minkowski type inequality for general \(L_{p}\)-mixed-brightness integrals.

Theorem 1.2

If \(K, L\in\mathcal{ K}^{n}_{os}\), \(p\geq1\), \(\tau\in[-1,1]\), and \(i\in \mathbb{R}\), and such that \(i\neq n\), then for \(i< n-p\),
$$ D^{(\tau)}_{p,i}(\lambda\circ K\oplus_{p} \mu\circ L)^{\frac {p}{n-i}}\leq\lambda D^{(\tau)}_{p,i}(K)^{\frac{p}{n-i}}+ \mu D^{(\tau)}_{p,i}(L)^{\frac{p}{n-i}}. $$
(1.13)
For \(n-p< i< n\) or \(i>n\), we have
$$ D^{(\tau)}_{p,i}(\lambda\circ K\oplus_{p} \mu\circ L)^{\frac {p}{n-i}}\geq\lambda D^{(\tau)}_{p,i}(K)^{\frac{p}{n-i}}+ \mu D^{(\tau)}_{p,i}(L)^{\frac{p}{n-i}}. $$
(1.14)
In each case, equality holds if and only if K and L have similar general \(L_{p}\)-brightness. For \(i=n-p\), equality always holds in (1.13) or (1.14).

Here, \(\lambda\circ K\oplus_{p} \mu\circ L\) denotes the \(L_{p}\)-Blaschke combination of K and L.

Next, we extend inequality (1.2) to general \(L_{p}\)-mixed-brightness integrals.

Theorem 1.3

If \(K_{1}, \ldots, K_{n}\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), \(\tau\in[-1,1]\), and \(1< m\leq n\), then
$$ D^{(\tau)}_{p}(K_{1}, \ldots, K_{n})^{m}\leq\prod^{m}_{i=1}D^{(\tau )}_{p}(K_{1}, \ldots, K_{n-m},\underbrace{K_{n-i+1}, \ldots, K_{n-i+1}}_{m}), $$
(1.15)
with equality if and only if \(K_{n-m+1},K_{n-m+2},\ldots,K_{n}\) are all of similar general \(L_{p}\)-brightness.

Taking \(m=n\) in Theorem 1.3 and using (1.10), we obtain the following corollary.

Corollary 1.1

If \(K_{1}, \ldots, K_{n}\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\tau\in [-1,1]\), then
$$D^{(\tau)}_{p}(K_{1}, \ldots, K_{n})^{n} \leq D^{(\tau)}_{p}(K_{1})\cdots D^{(\tau)}_{p}(K_{n}), $$
with equality if and only if \(K_{1},K_{2},\ldots,K_{n}\) are all of similar general \(L_{p}\)-brightness.

Moreover, we also establish the following cyclic inequality for general \(L_{p}\)-mixed-brightness integrals.

Theorem 1.4

If \(K,L\in\mathcal {K}^{n}_{o}\), \(p\geq1\), \(\tau\in[-1,1]\), and \(i,j,k\in\mathbb{R}\) such that \(i< j< k\), then
$$ D^{(\tau)}_{p,j}(K,L)^{k-i}\leq D^{(\tau)}_{p,i}(K,L)^{k-j}D^{(\tau )}_{p,k}(K,L)^{j-i}, $$
(1.16)
with equality if and only if K and L have similar general \(L_{p}\)-brightness.

Taking \(i=0\), \(k=n\) in Theorem 1.4 and using (1.11), we obtain the following result.

Corollary 1.2

If \(K,L\in\mathcal{K}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1,1]\), then for \(0< j< n\),
$$ D^{(\tau)}_{p,j}(K,L)^{n}\leq D^{(\tau)}_{p}(K)^{n-j}D^{(\tau )}_{p}(L)^{j}, $$
(1.17)
with equality if and only if K and L have similar general \(L_{p}\)-brightness. For \(j=0\) or \(j=n\), equality always holds in (1.17).

Let \(L=B\) in Theorem 1.4, we also have the following result.

Corollary 1.3

If \(K\in\mathcal{K}^{n}_{o}\), \(p\geq 1\), \(\tau\in[-1,1]\), and \(i,j,k\in\mathbb{R}\) such that \(i< j< k\), then
$$D^{(\tau)}_{p,j}(K)^{k-i}\leq D^{(\tau)}_{p,i}(K)^{k-j}D^{(\tau )}_{p,k}(K)^{j-i}, $$
with equality if and only if K and L have similar general \(L_{p}\)-brightness, i.e., K has constant general \(L_{p}\)-brightness.

2 Notation and background material

2.1 Radial function and polars of convex bodies

If K is a compact star-shaped set (about the origin) in \(\mathbb{R}^{n}\), then its radial function, \(\rho_{K}=\rho(K,\cdot):\mathbb{R}^{n}\rightarrow[0,\infty)\), is defined by (see [1])
$$ \rho(K,x)=\max\{\lambda\geq0: \lambda x\in K\},\quad x\in\mathbb {R}^{n}. $$
(2.1)
If \(\rho_{K}\) is positive and continuous, K will be called a star body (with respect to the origin). Two star bodies K and L are said to be dilates (of one another) if \(\rho_{K}(u)/\rho_{L}(u)\) is independent of \(u\in S^{n-1}\).
If E is a nonempty set in \(\mathbb{R}^{n}\), then the polar set of E, \(E^{\ast}\), is defined by (see [1])
$$E^{\ast}=\{x: x\cdot y\leq1, y\in E \},\quad x\in\mathbb{R}^{n}. $$
From this, we see that (see [1]) if \(K\in \mathcal{ K}^{n}_{o}\), then \((K^{\ast})^{\ast}=K\) and
$$ h_{K^{\ast}}=\frac{1}{\rho_{K}},\quad\quad \rho_{K^{\ast}}= \frac{1}{h_{K}}. $$
(2.2)
Lutwak in [33] defined dual quermassintegrals as follows. For \(K\in S^{n}_{o}\) and any real i, the dual quermassintegral, \(\widetilde{ W}_{i}(K)\), of K is defined by
$$ \widetilde{W}_{i}(K)=\frac{1}{n}\int _{S^{n-1}}\rho(K,u)^{n-i}\,du. $$
(2.3)
Obviously, (2.3) implies that
$$ V(K)=\widetilde{W}_{0}(K)=\frac{1}{n}\int _{S^{n-1}}\rho (K,u)^{n}\,du. $$
(2.4)

2.2 \(L_{p}\)-combinations of convex and star bodies

For \(K, L\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), the Firey \(L_{p}\)-combination, \(\lambda\cdot K+_{p}\mu\cdot L\in\mathcal{ K}^{n}_{o}\), of K and L is defined by (see [34, 35])
$$ h(\lambda\cdot K+_{p}\mu\cdot L, \cdotp)^{p} = \lambda h(K, \cdot)^{p} + \mu h(L, \cdot)^{p}, $$
(2.5)
where the symbol in \(\lambda\cdot K\) denotes the Firey scalar multiplication. Note that \(\lambda\cdot K=\lambda^{1/p}K\).
For \(K, L\in\mathcal{S}^{n}_{o}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), the \(L_{p}\)-harmonic radial combination, \(\lambda\star K+_{-p}\mu\star L\in\mathcal{S}^{n}_{o}\), of K and L is defined by (see [36])
$$ \rho(\lambda\star K+_{-p}\mu\star L, \cdot)^{-p} = \lambda\rho(K, \cdot)^{-p} + \mu\rho(L, \cdot)^{-p}, $$
(2.6)
where \(\lambda\star K=\lambda^{-1/p}K\).
From (2.2), (2.5), and (2.6), we easily find that if \(K, L\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), then
$$ (\lambda\cdot K+_{p}\mu\cdot L)^{\ast}=\lambda \star K^{\ast}+_{-p}\mu \star L^{\ast}. $$
(2.7)

In [37] Wang and Leng established the following Brunn-Minkowski type inequality for dual quermassintegrals with respect to an \(L_{p}\)-harmonic radial combination of star bodies.

Theorem 2.A

If \(K,L\in\mathcal{S}^{n}_{o}\), \(p\geq1\), \(i\in\mathbb{R}\) and such that \(i\neq n\), and \(\lambda, \mu\geq0\) (not both zero), then for \(i< n\) or \(n< i< n+p\),
$$ \widetilde{W}_{i}(\lambda\star K+_{-p}\mu\star L)^{-\frac {p}{n-i}}\geq\lambda\widetilde{W}_{i}(K)^{-\frac{p}{n-i}}+ \mu \widetilde{W}_{i}(L)^{-\frac{p}{n-i}}; $$
(2.8)
for \(i>n+p\),
$$ \widetilde{W}_{i}(\lambda\star K+_{-p}\mu\star L)^{-\frac {p}{n-i}}\leq\lambda\widetilde{W}_{i}(K)^{-\frac{p}{n-i}}+ \mu \widetilde{W}_{i}(L)^{-\frac{p}{n-i}}. $$
(2.9)
In each inequality, equality holds if and only if K and L are dilates. For \(i=n+p\), equality always holds in (2.8) and (2.9).
The \(L_{p}\)-Blaschke combination of origin-symmetric convex bodies was introduced by Lutwak [35]. For \(K, L\in\mathcal{ K}^{n}_{os}\), \(p\geq 1\), and \(\lambda, \mu\geq0\) (not both zero), the \(L_{p}\)-Blaschke combination, \(\lambda\circ K\oplus_{p}\mu\circ L\in \mathcal{ K}^{n}_{os}\), of K and L is defined by
$$ dS_{p}(\lambda\circ K\oplus_{p}\mu\circ L, \cdot)=\lambda \,dS_{p}(K,\cdot )+\mu \,dS_{p}(L,\cdot), $$
(2.10)
where \(\lambda\circ K=\lambda^{1/(n-p)}K\). For more information on these and other binary operations between convex and star bodies, see [3842].

2.3 General \(L_{p}\)-projection bodies

For \(p\geq1\), Ludwig [15] discovered the asymmetric \(L_{p}\)-projection body, \(\Pi^{+}_{p}K\), of \(K\in\mathcal{ K}^{n}_{o}\), whose support function is defined by
$$h^{p}_{\Pi^{+}_{p}K}(u)= \alpha_{n,p}\int _{S^{n-1}} (u\cdot v)_{+}^{p}\,dS_{p}(K,v), $$
where \((u\cdot v)_{+}=\max\{u\cdot v, 0\}\). In [16], Haberl and Schuster also defined
$$\Pi^{-}_{p}K=\Pi^{+}_{p}(-K). $$
Using definition (1.4) of general \(L_{p}\)-projection bodies, Haberl and Schuster [16] showed that, for \(K\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1,1]\),
$$\Pi^{\tau}_{p}K=f_{1}(\tau)\cdot \Pi^{+}_{p}K+_{p}f_{2}(\tau)\cdot \Pi^{-}_{p}K, $$
where
$$f_{1}(\tau)=\frac{(1+\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}},\quad\quad f_{2}(\tau)= \frac{(1-\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}}. $$

Moreover, they [16] determined the following extremal values of the volume for polars of general \(L_{p}\)-projection bodies.

Theorem 2.B

If \(K\in\mathcal{ K}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then
$$ V\bigl(\Pi^{\ast}_{p}K\bigr)\leq V\bigl( \Pi^{\tau,\ast}_{p}K\bigr)\leq V\bigl(\Pi^{\pm,\ast }_{p}K \bigr). $$
(2.11)
If K is not origin-symmetric and p is not an odd integer, there is equality in the left inequality if and only if \(\tau=0\) and equality in the right inequality if and only if \(\tau=\pm1\).

Here, \(\Pi^{\tau,\ast}_{p}K\) denotes the polar of the general \(L_{p}\)-projection body \(\Pi^{\tau}_{p}K\).

3 Proofs of the main theorems

In this section, we will prove Theorems 1.1-1.3.

To complete the proofs of Theorems 1.1-1.2, we require the following a lemma.

Lemma 3.1

If \(K\in\mathcal{ K}_{o}^{n}\), \(p\geq1\), \(\tau\in[-1,1]\), and i is any real, then
$$ D^{(\tau)}_{p,i}(K)=\frac{1}{2^{n}} \widetilde{W}_{2n-i}\bigl(\Pi_{p}^{\tau ,\ast} K\bigr). $$
(3.1)

Proof

By (1.8), (2.2), and (2.3), we have
$$\begin{aligned} D^{(\tau)}_{p,i}(K) =&\frac{1}{2^{i}\cdot n}\int_{S^{n-1}} \delta^{(\tau )}_{p}(K,u)^{n-i}\,dS(u) \\ =&\frac{1}{2^{n}\cdot n}\int_{S^{n-1}}h\bigl(\Pi ^{\tau}_{p} K, u\bigr)^{n-i}\,dS(u) \\ =&\frac{1}{2^{n}\cdot n}\int_{S^{n-1}}\rho \bigl(\Pi^{\tau,*}_{p} K, u\bigr)^{i-n}\,dS(u) \\ =&\frac{1}{2^{n}}\widetilde{W}_{2n-i}\bigl(\Pi_{p}^{\tau,\ast} K\bigr). \end{aligned}$$
 □

Proof of Theorem 1.1

Taking \(i=2n\) in (3.1) and using (2.4), we obtain
$$ D^{(\tau)}_{p,2n}(K)=\frac{1}{2^{n}}V\bigl( \Pi_{p}^{\tau,\ast} K\bigr). $$
(3.2)
Therefore, by inequality (2.11) together with (3.2), we immediately obtain
$$D_{p,2n}(K)\leq D^{(\tau)}_{p,2n}(K)\leq D^{\pm}_{p,2n}(K). $$
This is inequality (1.12).

According to the equality conditions of inequality (2.11), we know that if K is not origin-symmetric and p is not an odd integer, there is equality in the left inequality of (1.12) if and only if \(\tau= 0\) and equality in the right inequality of (1.12) if and only if \(\tau= \pm 1\). □

Proof of Theorem 1.2

By (1.4) and (2.10), we have, for all \(u\in S^{n-1}\),
$$h\bigl(\Pi_{p}^{\tau}(\lambda\circ K\oplus_{p}\mu \circ L), u\bigr)^{p}=\lambda h\bigl(\Pi_{p}^{\tau}K, u \bigr)^{p}+\mu h\bigl(\Pi_{p}^{\tau}L, u \bigr)^{p}, $$
i.e.,
$$\Pi_{p}^{\tau}(\lambda\circ K\oplus_{p}\mu\circ L)= \lambda\cdot\Pi _{p}^{\tau}K+_{p}\mu\cdot \Pi_{p}^{\tau}L. $$
This together with (2.7), yields
$$ \Pi_{p}^{\tau,\ast}(\lambda\circ K \oplus_{p}\mu\circ L)=\bigl(\lambda \cdot\Pi_{p}^{\tau}K+_{p} \mu\cdot\Pi_{p}^{\tau}L\bigr)^{\ast}=\lambda\star \Pi_{p}^{\tau,\ast}K+_{-p}\mu\star\Pi_{p}^{\tau,\ast }L. $$
(3.3)
Hence, if \(i< n-p\), then \(2n-i>n+p\). From this, (3.1), (3.3), and inequality (2.9), we obtain
$$\begin{aligned}& \bigl(2^{n}D^{(\tau)}_{p,i}(\lambda\circ K \oplus_{p}\mu\circ L) \bigr)^{\frac{p}{n-i}} \\& \quad=\widetilde{W}_{2n-i} \bigl(\Pi_{p}^{\tau,\ast}(\lambda \circ K\oplus _{p}\mu\circ L) \bigr)^{-\frac{p}{n-(2n-i)}} \\& \quad=\widetilde{W}_{2n-i}\bigl(\lambda\star\Pi_{p}^{\tau,\ast}K+_{-p} \mu \star\Pi_{p}^{\tau,\ast}L\bigr) ^{-\frac{p}{n-(2n-i)}} \\& \quad\leq\lambda\widetilde{W}_{2n-i}\bigl(\Pi _{p}^{\tau,\ast}K \bigr)^{-\frac{p}{n-(2n-i)}} +\mu\widetilde{W}_{2n-i}\bigl(\Pi_{p}^{\tau,\ast}L \bigr)^{-\frac{p}{n-(2n-i)}} \\& \quad=\lambda \bigl(2^{n}D^{(\tau)}_{p,i}(K) \bigr)^{\frac{p}{n-i}} +\mu \bigl(2^{n}D^{(\tau)}_{p,i}(L) \bigr)^{\frac{p}{n-i}}. \end{aligned}$$
This yields inequality (1.13).

From the equality conditions of inequality (2.9), we see that equality holds in (1.13) if and only if \(\Pi_{p}^{\tau,\ast}K\) and \(\Pi _{p}^{\tau,\ast}L\) are dilates, i.e., \(\Pi_{p}^{\tau}K\) and \(\Pi_{p}^{\tau}L\) are dilates. This means equality holds in (1.13) if and only if K and L have similar general \(L_{p}\)-brightness.

Similarly, if \(n-p< i< n\) or \(i>n\), then \(2n-i< n\) or \(n<2n-i<n+p\). Thus, using (3.1), (3.3), and inequality (2.8), we obtain inequality (1.14).

If \(i=n-p\), then \(2n-i=n+p\). This combined with Theorem 2.A, shows that equality always holds in (1.13) or (1.14). □

The proof of Theorem 1.3 requires the following inequality [3].

Lemma 3.2

If \(f_{0},f_{1},\ldots, f_{m}\) are (strictly) positive continuous functions defined on \(S^{n-1}\) and \(\lambda_{1},\ldots,\lambda_{m}\) are positive constants the sum of whose reciprocals is unity, then
$$ \int_{S^{n-1}}f_{0}(u)\cdots f_{m}(u)\,dS(u)\leq\prod^{m}_{i=1} \biggl(\int_{S^{n-1}}f_{0}(u) f_{i}^{\lambda_{i}}(u)\,dS(u) \biggr)^{\frac{1}{\lambda _{i}}}, $$
(3.4)
with equality if and only if there exist positive constants \(\alpha _{1},\alpha_{2},\ldots,\alpha_{m}\) such that \(\alpha_{1}f_{1}^{\lambda _{1}}(u)=\cdots=\alpha_{m}f_{m}^{\lambda_{m}}(u)\) for all \(u\in S^{n-1}\).

Proof of Theorem 1.3

For \(K_{1},\ldots,K_{n}\in\mathcal{ K}^{n}_{o}\), take \(\lambda_{i}=m\) in (3.4) (\(1\leq i\leq n\)), and
$$\begin{aligned}& f_{0}=\delta^{(\tau)}_{p}(K_{1},u)\cdots \delta^{(\tau)}_{p}(K_{n-m},u) \quad(f_{0}=1 \mbox{ if } m=n), \\& f_{i}=\delta^{(\tau)}_{p}(K_{n-i+1},u) \quad(1\leq i \leq m). \end{aligned}$$
Then we have
$$\begin{aligned}& \int_{S^{n-1}}\delta^{(\tau)}_{p}(K_{1},u) \cdots\delta^{(\tau)}_{p}(K_{n},u)\,dS(u) \\& \quad\leq\prod^{m}_{i=1} \biggl( \int_{S^{n-1}}\delta^{(\tau )}_{p}(K_{1},u) \cdots\delta^{(\tau)}_{p}(K_{n-m},u)\delta^{(\tau )}_{p}(K_{n-i+1},u)^{m} \,dS(u) \biggr)^{\frac{1}{m}}, \end{aligned}$$
(3.5)
i.e.
$$D^{(\tau)}_{p}(K_{1}, \ldots, K_{n})^{m} \leq\prod^{m}_{i=1}D^{(\tau )}_{p}(K_{1}, \ldots, K_{n-m},\underbrace{K_{n-i+1}, \ldots, K_{n-i+1}}). $$
According to the equality conditions of Lemma 3.2, we see that equality holds in (3.5) if and only if there exist positive constants \(\lambda _{1},\lambda_{2},\ldots,\lambda_{m}\) such that
$$\lambda_{1}\delta^{(\tau)}_{p}(K_{n-m+1},u)^{m}= \lambda_{2}\delta^{(\tau )}_{p}(K_{n-m+2},u)^{m}= \cdots =\lambda_{m}\delta^{(\tau)}_{p}(K_{n},u)^{m} $$
for all \(u\in S^{n-1}\). Thus equality holds in (1.15) if and only if \(K_{n-m+1},K_{n-m+2},\ldots,K_{n}\) are all of similar general \(L_{p}\)-brightness. □

Proof of Theorem 1.4

From (1.7) and the Hölder inequality, we obtain
$$\begin{aligned}& D^{(\tau)}_{p,i}(K,L)^{\frac{k-j}{k-i}}D^{(\tau)}_{p,k}(K,L)^{\frac {j-i}{k-i}} \\& \quad= \biggl[\frac{1}{n}\int_{S^{n-1}}\delta^{(\tau)}_{p}(K,u)^{n-i} \delta ^{(\tau)}_{p}(L,u)^{i}\,dS(u) \biggr]^{\frac{k-j}{k-i}} \\& \quad\quad{}\times \biggl[\frac{1}{n}\int_{S^{n-1}} \delta^{(\tau )}_{p}(K,u)^{n-k}\delta^{(\tau)}_{p}(L,u)^{k}\,dS(u) \biggr]^{\frac{j-i}{k-i}} \\& \quad= \biggl[\frac{1}{n}\int_{S^{n-1}}\bigl[ \delta^{(\tau)}_{p}(K,u)^{\frac{(n-i) (k-j)}{(k-i)}}\delta ^{(\tau)}_{p}(L,u)^{\frac{i(k-j)}{k-i}} \bigr]^{\frac{k-i}{k-j}}\,dS(u) \biggr]^{\frac{k-j}{k-i}} \\& \quad\quad{}\times \biggl[\frac{1}{n}\int_{S^{n-1}}\bigl[ \delta^{(\tau)}_{p}(K,u)^{\frac {(n-k) (j-i)}{k-i}}\delta^{(\tau)}_{p}(L,u)^{\frac{k (j-i)}{k-i}} \bigr]^{\frac{k-i}{j-i}}\,dS(u) \biggr]^{\frac{j-i}{k-i}} \\& \quad\geq\frac{1}{n}\int_{S^{n-1}}\delta^{(\tau)}_{p}(K,u)^{n-j} \delta ^{(\tau)}_{p}(L,u)^{j}\,dS(u) \\& \quad=D^{(\tau)}_{p,j}(K,L). \end{aligned}$$
This gives the desired inequality (1.16). According to the equality conditions of the Hölder inequality, we know that equality holds in (1.16) if and only if there exists a constant \(\lambda>0\) such that
$$\bigl[\delta^{(\tau)}_{p}(K,u)^{\frac{(n-i) (k-j)}{(k-i)}}\delta ^{(\tau)}_{p}(L,u)^{\frac{i(k-j)}{k-i}} \bigr]^{\frac{k-i}{k-j}} =\lambda \bigl[\delta^{(\tau)}_{p}(K,u)^{\frac{(n-k) (j-i)}{k-i}}\delta^{(\tau)}_{p}(L,u)^{\frac{k (j-i)}{k-i}} \bigr]^{\frac{k-i}{j-i}}, $$
i.e. \(\delta^{(\tau)}_{p}(K,u)=\lambda\delta^{(\tau )}_{p}(L,u)\) for all \(u\in S^{n-1}\). Thus equality holds in (1.16) if and only if K and L have similar general \(L_{p}\)-brightness. □

Declarations

Acknowledgements

The authors would like to sincerely thank the referees for all valuable and helpful comments and suggestions which made the paper more accurate and readable. Research is supported in part by the Natural Science Foundation of China (Grant No. 11371224) and Foundation of Degree Dissertation of Master of China Three Gorges University (Grant No. 2015PV070).

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 Three Gorges University

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