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 [6–14]), 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 [13–32]).

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 [38–42].

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. □