1 Introduction

Let \(\mathcal{K}^{n}\) denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the n-dimensional 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 \({\mathcal {K}}^{n}_{\mathrm {o}}\) and \({\mathcal {K}}_{\mathrm{os}}^{n}\), respectively. Let \(S^{n-1}\) denote the unit sphere in \(\mathbb{R}^{n}\) and denote by \(V(K)\) the n-dimensional volume of the body K. For the standard unit ball B in \(\mathbb{R}^{n}\), write \(V(B)=\omega_{n}\).

For \(K\in{\mathcal {K}}^{n}\), its support function \(h_{K}=h(K,\cdot): \mathbb{R}^{n} \longrightarrow (-\infty,+\infty)\) is defined by (see [1])

$$h(K, x)=\max\{x\cdot y: y\in K\},\quad x\in \mathbb{R}^{n}, $$

where \(x\cdot y\) denotes the standard inner product of x and y.

The projection body of a convex body was introduced by Minkowski at the turn of the previous century. For \(K\in{\mathcal {K}}^{n}\), the projection body ΠK of K is the origin-symmetric convex body whose support function is defined by (see [1])

$$h_{\Pi K}(u)=\frac{1}{2} \int_{S^{n-1}}| u\cdot v|\, dS(K, v) $$

for all \(u\in{S}^{n-1}\). Here, \(S(K, \cdot)\) denotes the surface area measure of the convex body K. Classical projection bodies are a very important notion in the Brunn-Minkowski theory. During the past four decades, a number of important results regarding classical projection bodies were obtained (see [1–12]).

The notion of an \(L_{p}\)-projection body was introduced by Lutwak, Yang, and Zhang [13]. For \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\) and \(p\geq1\), the \(L_{p}\)-projection body \(\Pi_{p}K\) of K is the origin-symmetric convex body whose support function is given by

$$ h^{p}_{\Pi_{p}K}(u)= \alpha_{n,p} \int_{S^{n-1}} | u\cdot v|^{p}\,dS_{p}(K,v) $$
(1.1)

for all \(u\in{S}^{n-1}\). Here

$$ \alpha_{n,p}=\frac{1}{n\omega_{n}c_{n-2,p}} $$
(1.2)

with \(c_{n,p}=\omega_{n+p}/\omega_{2}\omega_{n}\omega_{p-1}\), and \(S_{p}(K,\cdot)\) is the \(L_{p}\)-surface area measure of K that has the Radon-Nikodym derivative

$$ \frac{dS_{p}(K,\cdot)}{dS(K,\cdot)}= h(K,\cdot)^{1-p}. $$
(1.3)

The unusual normalization of definition (1.1) is chosen so that for the unit ball B, we have \(\Pi_{p}B = B\). In particular, for \(p=1\), \(\Pi_{1}K\) is just the classical projection body ΠK of K under the different normalization of definition (1.1).

\(L_{p}\)-projection bodies belong to the \(L_{p}\)-Brunn-Minkowski theory, which is an extension of the classical Brunn-Minkowski theory. Apart from [13], \(L_{p}\)-projection bodies have been investigated intensively in recent years (see [6, 14–21]).

Through the characterization of so-called \(L_{p}\)-Minkowski valuations, Ludwig [15] discovered (see also [22–29] for related results) an asymmetric \(L_{p}\)-projection body \(\Pi^{+}_{p}K\) of \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), whose support function is defined by

$$ h^{p}_{\Pi^{+}_{p}K}(u)= 2\alpha_{n,p} \int_{S^{n-1}} (u\cdot v)_{+}^{p}\,dS_{p}(K,v), $$
(1.4)

where \((u\cdot v)_{+}=\max\{u\cdot v, 0\}\). From (1.2) and (1.4) we see \(\Pi^{+}_{p}B=B\).

Moreover, Ludwig [15] introduced the function \(\varphi_{\tau}: \mathbb{R}\longrightarrow[0, +\infty)\) given by

$$\varphi_{\tau}(t)=\vert t\vert +\tau t $$

for \(\tau\in[-1,1]\). For \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), let \(\Pi^{\tau}_{p}K\in{\mathcal {K}}^{n}_{\mathrm {o}}\) with support function

$$ h^{p}_{\Pi^{\tau}_{p}K}(u)= \alpha_{n,p}(\tau) \int_{S^{n-1}}\varphi _{\tau}(u\cdot v)^{p}\,dS_{p}(K,v), $$
(1.5)

where

$$\alpha_{n,p}(\tau)=\frac{2\alpha_{n,p}}{(1+\tau)^{p}+(1-\tau)^{p}}. $$

The normalization is chosen such that \(\Pi^{\tau}_{p}B=B\) for every \(\tau\in[-1,1]\). Here \(\Pi^{\tau}_{p}K\) is called the general \(L_{p}\)-projection body of K. Obviously, if \(\tau=0\), then \(\Pi^{\tau}_{p}K=\Pi^{0}_{p}K=\Pi_{p}K\).

Following the discovery of Ludiwg, Haberl and Schuster [30] defined

$$ \Pi^{-}_{p}K=\Pi^{+}_{p}(-K). $$
(1.6)

From (1.4), (1.5), and (1.6) they (see [30]) deduced that for \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), \(\tau\in[-1,1]\), and all \(u\in S^{n-1}\),

$$ h^{p}_{\Pi^{\tau}_{p}K}(u)=f_{1}(\tau)h^{p}_{\Pi^{+}_{p}K}(u)+f_{2}( \tau)h^{p}_{\Pi ^{-}_{p}K}(u), $$
(1.7)

that is,

$$ \Pi^{\tau}_{p}K=f_{1}(\tau)\cdot \Pi^{+}_{p}K+_{p}f_{2}(\tau)\cdot \Pi^{-}_{p}K, $$
(1.8)

where ‘\(+_{p}\)’ denotes the \(L_{p}\)-Minkowski addition of convex bodies, and

$$ f_{1}(\tau)=\frac{(1+\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}},\qquad f_{2}(\tau)= \frac{(1-\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}}. $$
(1.9)

If \(\tau=\pm1\), then \(\Pi^{\tau}_{p}K=\Pi^{\pm}_{p}K\).

For general \(L_{p}\)-projection bodies, Haberl and Schuster [30] not only established a general version of the \(L_{p}\)-Petty projection inequality but also determined the following extremum of volume for their polars.

Theorem 1.A

If \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq 1\), and \(\tau\in[-1, 1]\), then

$$ V\bigl(\Pi^{\ast}_{p}K\bigr)\leq V\bigl(\Pi^{\tau,\ast}_{p}K \bigr)\leq V\bigl(\Pi^{\pm,\ast }_{p}K\bigr). $$
(1.10)

If K is not origin-symmetric and p is not an odd integer, then 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 \(\Pi^{\tau}_{p}K\) of \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\).

Apart from [30], general \(L_{p}\)-projection bodies were studied by various authors; for example, Wang and Wan [31] investigated related Shephard-type problems, Wang and Feng [32] established Petty’s affine projection inequality for them. General \(L_{p}\)-projection bodies are a central notion in a new and rapidly evolving asymmetric \(L_{p}\)-Brunn-Minkowski theory (see [14, 15, 30–47]).

In this paper, we first extend inequality (1.10) to dual quermassintegrals forms, that is, the extremums of dual quermassintegrals for the polars of general \(L_{p}\)-projection bodies are obtained.

Theorem 1.1

If \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), \(\tau\in[-1, 1]\), and real \(i\neq n\), then, for \(i< n\) or \(i>n+p\),

$$ \widetilde{W}_{i}\bigl(\Pi^{\ast}_{p}K\bigr)\leq \widetilde{W}_{i}\bigl(\Pi^{\tau,\ast }_{p}K\bigr)\leq \widetilde{W}_{i}\bigl(\Pi^{\pm,\ast}_{p}K\bigr), $$
(1.11)

and, for \(n< i< n+p\),

$$ \widetilde{W}_{i}\bigl(\Pi^{\ast}_{p}K\bigr)\geq \widetilde{W}_{i}\bigl(\Pi^{\tau,\ast }_{p}K\bigr)\geq \widetilde{W}_{i}\bigl(\Pi^{\pm,\ast}_{p}K\bigr). $$
(1.12)

In each case, if K is not origin-symmetric and p is not an odd integer, then there is equality in the left inequality if and only if \(\tau=0\) and equality in the right inequality if and only if \(\tau=\pm1\). For \(i=n+p\), (1.11) and (1.12) become equalities. Here \(\widetilde{W}_{i}(Q)\) (i is any real) denote the dual quermassintegrals of the star body Q.

If \(i=0\), then since \(\widetilde{W}_{0}(Q)=V(Q)\), Theorem 1.1 reduces to Theorem 1.A.

Next, we obtain the extremums of quermassintegrals of general \(L_{p}\)-projection bodies.

Theorem 1.2

If \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), \(\tau\in[-1, 1]\), and \(i=0, 1, \ldots, n-1\), then

$$ W_{i}(\Pi_{p}K)\geq W_{i}\bigl( \Pi^{\tau}_{p}K\bigr)\geq W_{i}\bigl( \Pi^{\pm}_{p}K\bigr). $$
(1.13)

If K is not origin-symmetric and p is not an odd integer, then 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 \(W_{i}(Q)\) (\(i=0, 1, \ldots, n-1\)) denote the quermassintegrals of \(Q\in{\mathcal {K}}^{n}_{\mathrm {o}}\).

Taking \(i=0\) in Theorem 1.2, we obtain the following:

Corollary 1.1

If \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq 1\), and \(\tau\in[-1, 1]\), then

$$ V(\Pi_{p}K)\geq V\bigl(\Pi^{\tau}_{p}K\bigr)\geq V \bigl(\Pi^{\pm}_{p}K\bigr). $$
(1.14)

If K is not origin-symmetric and p is not an odd integer, then there is equality in the left inequality if and only if \(\tau=0\) and equality in the right inequality if and only if \(\tau=\pm1\).

Inequality (1.14) can be viewed as a dual version of inequality (1.10).

Finally, we determine the extremal values of the \(L_{q}\)-dual affine surface area (see Section 2) of the polars of general \(L_{p}\)-projection bodies.

Theorem 1.3

If \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq 1\), \(0< q< n\), and \(\tau\in[-1, 1]\), then

$$ \widetilde{\Omega}_{q}\bigl(\Pi^{\ast}_{p}K\bigr) \leq\widetilde{\Omega}_{q}\bigl(\Pi ^{\tau,\ast}_{p}K\bigr) \leq\widetilde{\Omega}_{q}\bigl(\Pi^{\pm,\ast}_{p}K \bigr). $$
(1.15)

If K is not origin-symmetric and p is not an odd integer, then 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 \(\widetilde{\Omega}_{q}(Q)\) denotes the \(L_{q}\)-dual affine surface area of the star body Q.

This paper is organized as follows. In Section 2, we provide some preliminary results. Then, in Section 3, we recall some basic properties of general \(L_{p}\)-projection bodies. Section 4 contains the proofs of Theorems 1.1-1.3.

2 Basic notions

2.1 Radial functions and polar bodies

If K is a compact star-shaped (about the origin) set in \(\mathbb{R}^{n} \), then its radial function \(\rho_{K}=\rho(K,\cdot): \mathbb{R}^{n}\backslash\{0\}\longrightarrow[0,+\infty)\) is defined by (see [1])

$$\rho(K, x)=\max\{\lambda\geq0:\lambda x\in K \},\quad x\in\mathbb{R}^{n} \backslash\{0\}. $$

If \(\rho_{K}\) is positive and continuous, then K is called a star body (about the origin). For the set of star bodies containing the origin in their interiors and the set of origin-symmetric star bodies in \(\mathbb{R}^{n}\), we write \({\mathcal {S}}^{n}_{\mathrm {o}}\) and \({\mathcal {S}}^{n}_{\mathrm{os}}\), respectively. 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 subset of \(\mathbb{R}^{n}\), then the polar set \(E^{\ast}\) of E is defined by (see [1])

$$ E^{\ast}=\{x: x\cdot y\leq1, y\in E \},\quad x\in \mathbb{R}^{n}. $$
(2.1)

From (2.1) it follows that if \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), then

$$ h_{K^{\ast}}=1/\rho_{K}\quad\text{and}\quad \rho_{K^{\ast}}=1/h_{K}. $$
(2.2)

2.2 \(L_{p}\)-Minkowski and \(L_{p}\)-harmonic radial combinations

For \(K, L\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), the \(L_{p}\)-Minkowski combination (also called the Firey \(L_{p}\)-combination) \(\lambda\cdot K+_{p}\mu\cdot L\in{\mathcal {K}}^{n}_{\mathrm {o}}\) of K and L is defined by (see [48, 49])

$$ h(\lambda\cdot K+_{p}\mu\cdot L, \cdotp)^{p} = \lambda h(K, \cdot)^{p} + \mu h(L, \cdot)^{p}, $$
(2.3)

where ‘⋅’ in \(\lambda\cdot K\) denotes the \(L_{p}\)-Minkowski scalar multiplication.

For \(K, L\in\mathcal{S}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), the \(L_{p}\)-harmonic radial combination \(\lambda\circ K\,\tilde{+}_{-p}\,\mu\circ L\in\mathcal{S}^{n}_{\mathrm {o}}\) of K and L is defined by (see [50])

$$ \rho(\lambda\circ K\,\tilde{+}_{-p}\,\mu\circ L, \cdot)^{-p} = \lambda\rho(K, \cdot)^{-p} + \mu\rho(L, \cdot)^{-p}. $$
(2.4)

From (2.2), (2.3), and (2.4) we easily see that if \(K, L\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), then

$$ (\lambda\cdot K+_{p}\mu\cdot L)^{\ast}=\lambda\circ K^{\ast}\,\tilde{+}_{-p}\,\mu\circ L^{\ast}. $$
(2.5)

2.3 \(L_{p}\)-mixed and dual mixed volumes

Lutwak [51] gave the definition of \(L_{p}\)-mixed volume associated with \(L_{p}\)-Minkowski combinations of convex bodies: For \(K,L\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(\varepsilon>0\), and \(p\geq1\), the \(L_{p}\)-mixed volume \(V_{p}(K,L)\) of K and L is defined by

$$\frac{n}{p} V_{p}(K, L) = \lim_{\varepsilon\longrightarrow0^{+}}{ \frac {V(K+_{p}\varepsilon\cdot L)-V(K)}{\varepsilon}}. $$

Corresponding to each \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), Lutwak [51] proved that, for each \(L\in{\mathcal {K}}^{n}_{\mathrm {o}}\),

$$ V_{p}(K, L)=\frac{1}{n} \int_{S^{n-1}}h^{p}_{L}(v)\,dS_{p}(K,v). $$
(2.6)

From (2.6) and (1.3) it follows immediately that, for each \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\),

$$ V_{p}(K, K)=V(K)=\frac{1}{n} \int_{S^{n-1}}h_{K}(v)\,dS(K,v). $$
(2.7)

The \(L_{p}\)-Minkowski inequality states the following (see [51]):

Theorem 2.A

If \(K, L\in\mathcal{K}^{n}_{\mathrm {o}}\), and \(p\geq1\), then

$$ V_{p}(K,L)\geq V(K)^{(n-p)/n}V(L)^{p/n} $$
(2.8)

with equality for \(p>1\) if and only if K and L are dilates and for \(p=1\) if and only if K and L are homothetic.

Haberl [35] (also see [52]) introduced the notion of \(L_{p}\)-dual mixed volume as follows. For \({K, L}\in{\mathcal {S}}_{\mathrm {o}}^{n}\), \(p > 0\), and \(\varepsilon> 0\), the \(L_{p}\)-dual mixed volume \(\widetilde{V}_{p}(K, L)\) of K and L is defined by

$$ \widetilde{V}_{p}(K, L)=\frac{1}{n} \int_{S^{n-1}}\rho _{K}^{n-p}(u) \rho_{L}^{p}(u)\,dS(u), $$
(2.9)

where the integration is with respect to spherical Lebesgue measure on \(S^{n-1}\).

2.4 \(L_{p}\)-dual affine surface areas

Based on the \(L_{p}\)-dual mixed volume, Wang, Yuan, and He [53] defined the notion of \(L_{p}\)-dual affine surface area. For \(K\in {\mathcal {S}}_{\mathrm {o}}^{n}\) and \(0< p< n\), the \(L_{p}\)-dual affine surface area \(\widetilde{\Omega}_{p}(K)\) of K is defined by

$$ n^{-\frac{p}{n}}\widetilde{\Omega}_{p}(K)^{\frac{n+p}{n}}= \sup\bigl\{ n\widetilde{V}_{p}\bigl(K,Q^{\ast}\bigr)V(Q)^{\frac{p}{n}}: Q \in{\mathcal {K}}_{c}^{n}\bigr\} , $$
(2.10)

where \({\mathcal {K}}_{c}^{n}\) denotes the set of convex bodies whose centroids lie at the origin in \(\mathbb{R}^{n}\).

We extend definition (2.10) from \(Q\in{\mathcal {K}}_{c}^{n}\) to \(Q\in {\mathcal {S}}_{\mathrm{os}}^{n}\) as follows: For \(K\in{\mathcal {S}}_{\mathrm {o}}^{n}\) and \(0< p< n\), the \(L_{p}\)-dual affine surface area \(\widetilde{\Omega }_{p}(K)\) of K is defined by

$$ n^{-\frac{p}{n}}\widetilde{\Omega}_{p}(K)^{\frac{n+p}{n}}= \sup\bigl\{ n\widetilde{V}_{p}\bigl(K,Q^{\ast}\bigr)V(Q)^{\frac{p}{n}}: Q \in{\mathcal {S}}_{\mathrm{os}}^{n}\bigr\} . $$
(2.11)

2.5 Quermassintegrals and dual quermassintegrals

For \(K\in\mathcal{K}^{n}\), \(i=0, 1, \ldots, n-1\), the quermassintegrals \(W_{i}(K)\) of K are given by (see [1, 49])

$$ W_{i}(K) = \frac{1}{n} \int_{S^{n-1}}h(K, u)\,dS_{i}(K, u), $$
(2.12)

where \(S_{i}(K, \cdot)\) (\(i=0, 1, \ldots, n-1\)) denotes the ith surface area measure of K, and \(S_{0}(K, \cdot)=S(K, \cdot)\). From (2.12) and (2.7) we easily see that \(W_{0}(K) = V(K)\).

For the \(L_{p}\)-Minkowski combination, Lutwak [51] proved the following Brunn-Minkowski inequality for quermassintegrals.

Theorem 2.B

If \(K,L\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), \(i=0, 1, \ldots, n-1\), and \(\lambda, \mu\geq0\) (not both zero), then

$$ W_{i}(\lambda\cdot K+_{p}\mu\cdot L)^{\frac{p}{n-i}} \geq \lambda W_{i}(K)^{\frac{p}{n-i}} + \mu W_{i}(L)^{\frac{p}{n-i}} $$
(2.13)

with equality for \(p=1\) if and only if K and L are homothetic and for \(p>1\) if and only if K and L are dilates.

For \(K\in\mathcal{S}^{n}_{\mathrm {o}}\) and any real i, the dual quermassintegrals \(\widetilde{W}_{i}(K)\) of K are defined by (see [54])

$$ \widetilde{W}_{i}(K) = \frac{1}{n} \int_{S^{n-1}}\rho(K, u)^{n-i}\,dS(u). $$
(2.14)

Obviously, (2.14) implies

$$\widetilde{W}_{0}(K) = V(K)=\frac{1}{n} \int_{S^{n-1}}\rho(K, u)^{n}\,dS(u). $$

Associated with the \(L_{p}\)-harmonic radial combinations of star bodies, Wang and Leng [55] established the following Brunn-Minkowski inequality for dual quermassintegrals.

Theorem 2.C

If \(K,L\in\mathcal{S}^{n}_{\mathrm {o}}\), \(p\geq1\), \(\lambda, \mu\geq0\) (not both zero), and real \(i\neq n\), then, for \(i< n\) or \(n< i< n+p\),

$$ \widetilde{W}_{i}(\lambda\circ K\,\tilde{+}_{-p}\,\mu\circ L)^{-\frac {p}{n-i}}\geq\lambda\widetilde{W}_{i}(K)^{-\frac{p}{n-i}}+ \mu \widetilde{W}_{i}(L)^{-\frac{p}{n-i}} $$
(2.15)

and, for \(i>n+p\),

$$ \widetilde{W}_{i}(\lambda\circ K\,\tilde{+}_{-p}\,\mu\circ L)^{-\frac {p}{n-i}}\leq\lambda\widetilde{W}_{i}(K)^{-\frac{p}{n-i}}+ \mu \widetilde{W}_{i}(L)^{-\frac{p}{n-i}}. $$
(2.16)

In each inequality, equality holds if and only if K and L are dilates. For \(i=n+p\), (2.15) and (2.16) become equalities.

3 Some properties of general \(L_{p}\)-projection bodies

In this section, we recall some basic properties of general \(L_{p}\)-projection bodies.

Theorem 3.1

If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then

$$ \Pi^{\tau}_{p}(-K)=\Pi^{-\tau}_{p}K=- \Pi^{\tau}_{p}K. $$
(3.1)

Proof

From (1.5) it follows that, for all \(u\in S^{n-1}\),

$$\begin{aligned} h^{p}_{-\Pi^{\tau}_{p}K}(u) =& h^{p}_{\Pi^{\tau}_{p}K}(-u) = \alpha_{n,p}(\tau) \int_{S^{n-1}}\varphi_{\tau}(-u\cdot v)^{p}\,dS_{p}(K, v) \\ =&\alpha_{n,p}(\tau) \int_{S^{n-1}}\varphi_{\tau}\bigl(u\cdot (-v) \bigr)^{p}\,dS_{p}(-K, -v) = h^{p}_{\Pi^{\tau}_{p}(-K)}(u). \end{aligned}$$

This gives

$$ \Pi^{\tau}_{p}(-K)=-\Pi^{\tau}_{p}K. $$
(3.2)

In addition, by (1.9) we have that

$$\begin{aligned}& f_{1}(\tau)+f_{2}(\tau)=1, \end{aligned}$$
(3.3)
$$\begin{aligned}& f_{1}(-\tau)=f_{2}(\tau), \qquad f_{2}(- \tau)=f_{1}(\tau). \end{aligned}$$
(3.4)

From (3.3) and (3.4), together with (1.6) and (1.8), we obtain

$$\begin{aligned} \Pi^{-\tau}_{p}K =& f_{1}(-\tau)\cdot \Pi^{+}_{p}K+_{p}f_{2}(-\tau)\cdot\Pi ^{-}_{p}K \\ =& f_{2}(\tau)\cdot\Pi^{-}_{p}(-K)+_{p}f_{1}( \tau)\cdot\Pi ^{+}_{p}(-K)= \Pi^{\tau}_{p}(-K). \end{aligned}$$
(3.5)

Obviously, (3.2) and (3.5) yield (3.1). □

Theorem 3.2

If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), \(\tau\in[-1, 1]\), and \(\tau\neq0\), then

$$\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K \quad\Longleftrightarrow\quad \Pi^{+}_{p}K=\Pi^{-}_{p}K. $$

Proof

From (1.8) and (3.4) it follows that, for \(K\in{\mathcal {K}}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\tau\in[-1,1]\),

$$\Pi^{-\tau}_{p}K=f_{2}(\tau)\cdot \Pi^{+}_{p}K+_{p}f_{1}(\tau)\cdot \Pi^{-}_{p}K, $$

that is,

$$ h^{p}_{\Pi^{-\tau}_{p}K}(u)=f_{2}(\tau)h^{p}_{\Pi^{+}_{p}K}(u)+f_{1}( \tau )h^{p}_{\Pi^{-}_{p}K}(u) $$
(3.6)

for all \(u\in S^{n-1}\). Therefore, by (3.3), (1.7), and (3.6), if \(\Pi^{+}_{p}K=\Pi^{-}_{p}K\), then

$$h^{p}_{\Pi^{\tau}_{p}K}(u)=h^{p}_{\Pi^{-\tau}_{p}K}(u) $$

for all \(u\in S^{n-1}\). This gives \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\).

Conversely, if \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\), then (1.7) and (3.6) yield

$$\bigl[f_{1}(\tau)-f_{2}(\tau)\bigr]h^{p}_{\Pi^{+}_{p}K}(u)= \bigl[f_{1}(\tau)-f_{2}(\tau )\bigr]h^{p}_{\Pi^{-}_{p}K}(u) $$

for all \(u\in S^{n-1}\). Since \(f_{1}(\tau)-f_{2}(\tau)\neq0\) when \(\tau\neq0\), we get \(\Pi^{+}_{p}K=\Pi^{-}_{p}K\). □

Haberl and Schuster [30] proved the following fact.

Theorem 3.A

If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), and p is not odd integer, then \(\Pi^{+}_{p}K=\Pi^{-}_{p}K\) if and only if \(K\in\mathcal{K}^{n}_{\mathrm{os}}\).

According to Theorems 3.A and 3.2, we get the following:

Theorem 3.3

If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), and p is not odd integer, then, for \(\tau\in[-1, 1]\) and \(\tau\neq0\), \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\) if and only if \(K\in\mathcal{K}^{n}_{\mathrm{os}}\).

Theorem 3.4

If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then

$$ \Pi^{\tau}_{p}K+_{p}\Pi^{-\tau}_{p}K= \Pi^{+}_{p}K+_{p}\Pi^{-}_{p}K. $$
(3.7)

Proof

From (1.7) and (3.6), using (3.3), we have that, for any \(u\in S^{n-1}\),

$$h\bigl(\Pi^{\tau}_{p}K, u\bigr)^{p}+ h\bigl( \Pi^{-\tau}_{p}K, u\bigr)^{p}= h\bigl( \Pi^{+}_{p}K, u\bigr)^{p}+ h\bigl( \Pi^{-}_{p}K, u\bigr)^{p}, $$

that is,

$$h\bigl(\Pi^{\tau}_{p}K+_{p}\Pi^{-\tau}_{p}K, u\bigr)^{p}= h\bigl(\Pi^{+}_{p}K+_{p}\Pi ^{-}_{p}K, u\bigr)^{p}. $$

This is the desired relation. □

From Theorem 3.4 we deduce the following:

Corollary 3.1

If \(K\in\mathcal{K}^{n}_{\mathrm {o}}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then

$$ \Pi_{p}K=\frac{1}{2}\cdot\Pi^{\tau}_{p}K+_{p} \frac{1}{2}\cdot\Pi ^{-\tau}_{p}K. $$
(3.8)

Proof

Taking \(\tau=0\) in (1.8) and combining with (1.9) yield

$$ \Pi_{p}K=\frac{1}{2}\cdot\Pi^{+}_{p}K+_{p} \frac{1}{2}\cdot\Pi^{-}_{p}K. $$
(3.9)

From (3.9) and (3.7) we immediately get (3.8). □

Theorem 3.5

If \(K, L\in\mathcal{K}^{n}_{\mathrm{os}}\), \(p\geq1\) is not an even integer, and \(\tau \in[-1, 1]\), then

$$\Pi^{\tau}_{p}K=\Pi^{\tau}_{p}L \quad\Longrightarrow\quad K=L. $$

The proof of Theorem 3.5 requires the following two lemmas.

Lemma 3.1

If \(K, L\in\mathcal{K}^{n}_{\mathrm {o}}\), and \(p\geq1\) is not an even integer, then \(\Pi _{p} K=\Pi_{p} L\) if and only if \(V_{p}(K, Q)=V_{p}(L, Q)\) for any \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\).

Proof

From (1.1) we know that, for any \(u\in S^{n-1}\),

$$\begin{aligned} h^{p}_{\Pi_{p}(-K)}(u) =&\alpha_{n,p} \int_{S^{n-1}} | u\cdot v|^{p}\,dS_{p}(-K, v) \\ =&\alpha_{n,p} \int_{S^{n-1}} \bigl| u\cdot(-v)\bigr|^{p}\,dS_{p}(K, -v)=h^{p}_{\Pi_{p}K}(u), \end{aligned}$$

which implies \(\Pi_{p}(-K)=\Pi_{p}K\). Thus, for any \(u\in S^{n-1}\),

$$\begin{aligned} h^{p}_{\Pi_{p}K}(u) =&\frac{1}{2}h^{p}_{\Pi_{p}K}(u)+ \frac{1}{2}h^{p}_{\Pi _{p}(-K)}(u) \\ =&\frac{1}{2}\alpha_{n,p} \int_{S^{n-1}} | u\cdot v|^{p} \bigl[dS_{p}(K, v)+dS_{p}(-K, v)\bigr]. \end{aligned}$$

Thus, if \(\Pi_{p} K=\Pi_{p} L\), then, for any \(u\in S^{n-1}\),

$$\int_{S^{n-1}} | u\cdot v|^{p} \bigl[dS_{p}(K, v)+dS_{p}(-K, v)-dS_{p}(L, v)-dS_{p}(-L, v) \bigr]=0. $$

Letting \(\mu(v)=S_{p}(K, v)+S_{p}(-K, v)-S_{p}(L, v)-S_{p}(-L, v)\), we have

$$ \int_{S^{n-1}} | u\cdot v|^{p} \,d\mu(v)=0. $$
(3.10)

Since \(\mu(v)\) is an even Borel measure on \(S^{n-1}\) and \(p\geq1\) is not an even integer, it follows from (3.10) that \(\mu(v)=0\) (see, e.g., [30]), that is,

$$ S_{p}(K, \cdot)+S_{p}(-K, \cdot)=S_{p}(L, \cdot)+S_{p}(-L, \cdot). $$
(3.11)

Since \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\), we have \(h_{Q}(-v)=h_{Q}(v)\) for all \(v\in S^{n-1}\). Therefore, by (2.6) we get

$$V_{p}(K, Q)=\frac{1}{n} \int_{S^{n-1}}h^{p}_{Q}(-v)\,dS_{p}(K, -v)=\frac{1}{n} \int_{S^{n-1}}h^{p}_{Q}(v)\,dS_{p}(-K,v). $$

This and (2.6) yield

$$V_{p}(K, Q)=\frac{1}{2n} \int_{S^{n-1}}h^{p}_{Q}(v)\bigl[dS_{p}(K, v)+dS_{p}(-K, v)\bigr] $$

for any \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\). By (3.11) we see that if \(\Pi_{p} K=\Pi_{p} L\), then \(V_{p}(K, Q)=V_{p}(L, Q)\) for any \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\).

Conversely, if \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\), let \(Q=[-u, u]\) (\(u\in S^{n-1}\)). Then \(h_{Q}(v)=| u\cdot v|\) for any \(v\in S^{n-1}\). This, together with (2.6), yields

$$\begin{aligned} V_{p}(K, Q) =&\frac{1}{n} \int_{S^{n-1}}h^{p}_{Q}(v)\,dS_{p}(K, v) \\ =&\frac{1}{n} \int _{S^{n-1}}| u\cdot v|^{p}\,dS_{p}(K, v)=\frac{1}{n\alpha _{n,p}}h^{p}(\Pi_{p}K, u). \end{aligned}$$

Hence, if \(V_{p}(K, Q)=V_{p}(L, Q)\) for any \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\), then \(\Pi_{p}K=\Pi_{p}L\). □

Lemma 3.2

If \(K, L\in\mathcal{K}^{n}_{\mathrm{os}}\) and \(p\geq1\) is not an even integer, then

$$\Pi_{p}K=\Pi_{p}L\quad\Longrightarrow\quad K=L. $$

Proof

By Lemma 3.1, if \(\Pi_{p} K=\Pi_{p} L\) and p is not an even integer, then, for any \(Q\in\mathcal{K}^{n}_{\mathrm{os}}\),

$$ V_{p}(K, Q)=V_{p}(L, Q). $$
(3.12)

Taking K for Q in (3.12) and using (2.7) and (2.8), we obtain \(V(K)\geq V(L)\) with equality for \(p>1\) if and only if K and L are dilates (for \(p=1\), if and only if K and L are homothetic). Similarly, taking L for Q in (3.12) yields \(V(K)\leq V(L)\), and equality holds for \(p>1\) if and only if K and L are dilates (for \(p=1\), if and only if K and L are homothetic). Therefore, \(V(K)=V(L)\), and K and L are dilates when \(p>1\) (K and L are homothetic when \(p=1\)). Since \(K, L\in\mathcal{K}^{n}_{\mathrm{os}}\), we have that, for \(p\geq1\), \(K=L\). □

Proof of Theorem 3.5

If \(K\in\mathcal{K}^{n}_{\mathrm{os}}\), then by (3.5) and Corollary 3.1 we have that

$$\Pi_{p}K=\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K. $$

Therefore, if \(K, L\in\mathcal{K}^{n}_{\mathrm{os}}\), then, for \(\tau\in[-1, 1]\),

$$\Pi^{\tau}_{p}K=\Pi^{\tau}_{p}L\quad\Longleftrightarrow\quad \Pi_{p}K=\Pi_{p}L. $$

This, together with Lemma 3.2, completes the proof of Theorem 3.5. □

4 Proofs of the main results

In this section, we will complete the proofs of Theorems 1.1-1.3.

Proof of Theorem 1.1

From (1.8) and (2.5) we have

$$ \Pi^{\tau,\ast}_{p}K=f_{1}(\tau)\circ \Pi^{+,\ast}_{p}K\,\tilde{+}_{-p}\,f_{2}(\tau) \circ\Pi^{-,\ast}_{p}K. $$
(4.1)

Hence, for \(i< n\) or \(n< i< n+p\), using (4.1) and (2.15), we have that

$$ \widetilde{W}_{i}\bigl(\Pi^{\tau,\ast}_{p}K \bigr)^{-\frac{p}{n-i}}\geq f_{1}(\tau )\widetilde{W}_{i} \bigl(\Pi^{+,\ast}_{p}K\bigr)^{-\frac{p}{n-i}}+ f_{2}( \tau)\widetilde{W}_{i}\bigl(\Pi^{-,\ast}_{p}K \bigr)^{-\frac{p}{n-i}}. $$
(4.2)

But (3.1) yields \(\Pi^{-}_{p}K=\Pi^{+}_{p}(-K)=-\Pi^{+}_{p}K\), which implies \(\widetilde{W}_{i}(\Pi^{+,\ast}_{p}K)=\widetilde{W}_{i}(\Pi ^{-,\ast}_{p}K)\). Hence, by (4.2) and (3.3) we obtain

$$ \widetilde{W}_{i}\bigl(\Pi^{\tau,\ast}_{p}K \bigr)^{-\frac{p}{n-i}}\geq \widetilde{W}_{i}\bigl(\Pi^{\pm,\ast}_{p}K \bigr)^{-\frac{p}{n-i}}. $$
(4.3)

Now, if \(i< n\), then

$$ \widetilde{W}_{i}\bigl(\Pi^{\tau,\ast}_{p}K\bigr)\leq \widetilde{W}_{i}\bigl(\Pi^{\pm,\ast}_{p}K\bigr). $$
(4.4)

Inequality (4.4) is just the right-hand side inequality of (1.11). If \(n< i< n+p\), then by (4.3) we get

$$ \widetilde{W}_{i}\bigl(\Pi^{\tau,\ast}_{p}K\bigr)\geq \widetilde{W}_{i}\bigl(\Pi^{\pm,\ast}_{p}K\bigr), $$
(4.5)

which gives the right-hand side inequality of (1.12).

For \(i>n+p\), using (4.1) and (2.16), we arrive at

$$\widetilde{W}_{i}\bigl(\Pi^{\tau,\ast}_{p}K \bigr)^{-\frac{p}{n-i}}\leq \widetilde{W}_{i}\bigl(\Pi^{\pm,\ast}_{p}K \bigr)^{-\frac{p}{n-i}}, $$

which yields (4.4).

According to the conditions of equality in (2.15) and (2.16), we have that equality holds in (4.4) and (4.5) if and only if \(\Pi^{+,\ast}_{p}K\) and \(\Pi^{-,\ast}_{p}K\) are dilates. From this, letting \(\Pi^{+,\ast}_{p}K=c\Pi^{-,\ast}_{p}K\) (\(c>0\)) and using that \(\widetilde{W}_{i}(\Pi^{+,\ast}_{p}K)=\widetilde{W}_{i}(\Pi^{-,\ast}_{p}K)\), it follows that \(c=1\), that is, \(\Pi^{+,\ast}_{p}K=\Pi^{-,\ast}_{p}K\). This means that \(\Pi^{+}_{p}K=\Pi^{-}_{p}K\). Hence, from Theorem 3.A we see that if K is not origin-symmetric and p is not an odd integer, then equality holds in the right-hand side inequalities of (1.11) and (1.12) if and only if \(\tau=\pm1\).

Now we prove the left-hand side inequalities of (1.11) and (1.12).

From (3.8) and (2.5) we have that

$$ \Pi^{\ast}_{p}K=\frac{1}{2}\circ\Pi^{\tau,\ast}_{p}K \,\tilde{+}_{-p}\,\frac{1}{2}\circ\Pi^{-\tau,\ast}_{p}K. $$
(4.6)

Using (3.1) and respectively combining with inequalities (2.15) and (2.16), we obtain the left-hand side inequalities of (1.11) and (1.12).

Moreover, by the conditions of equality in (2.15) and (2.16) we see that equality holds in the left-hand side inequalities of (1.11) and (1.12) if and only if \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\). This, together with Theorem 3.3, yields that if K is not origin-symmetric and p is not an odd integer, then equality holds in the left-hand side inequalities of (1.11) and (1.12) if and only if \(\tau=0\). □

Proof of Theorem 1.2

Using (1.8) and inequality (2.13), we have

$$W_{i}\bigl(\Pi^{\tau}_{p}K\bigr)^{\frac{p}{n-i}}\geq f_{1}(\tau)W_{i}\bigl(\Pi ^{+}_{p}K \bigr)^{\frac{p}{n-i}}+ f_{2}(\tau)W_{i}\bigl( \Pi^{-}_{p}K\bigr)^{\frac{p}{n-i}}, $$

which, combined with (3.3), yields

$$W_{i}\bigl(\Pi^{\tau}_{p}K\bigr)\geq W_{i}\bigl(\Pi^{\pm}_{p}K\bigr). $$

This gives the right-hand side inequality of (1.13).

According to the condition of equality in (2.13), we see that equality holds in the right-hand side inequality of (1.13) for \(p>1\) if and only if \(\Pi^{+}_{p}K\) and \(\Pi^{-}_{p}K\) are dilates (for \(p=1\), if and only if \(\Pi^{+}_{p}K\) and \(\Pi^{-}_{p}K\) are homothetic), which yields \(\Pi^{+}_{p}K=\Pi^{-}_{p}K\). Thus, from Theorem 3.A it follows that if K is not origin-symmetric and p is not an odd integer, then equality holds in the right-hand side inequality of (1.13) if and only if \(\tau=\pm1\).

Meanwhile, from (3.8) and inequality (2.13) we obtain

$$W_{i}(\Pi_{p}K)^{\frac{p}{n-i}}\geq\frac{1}{2}W_{i} \bigl(\Pi^{\tau }_{p}K\bigr)^{\frac{p}{n-i}}+ \frac{1}{2}W_{i}\bigl(\Pi^{-\tau}_{p}K \bigr)^{\frac{p}{n-i}}, $$

which, together with (3.1), yields

$$W_{i}(\Pi_{p}K)\geq W_{i}\bigl( \Pi^{\tau}_{p}K\bigr). $$

This is the left-hand side inequality of (1.13), where equality holds if and only if \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\). This, together with Theorem 3.3, shows that if K is not origin-symmetric and p is not an odd integer, then equality holds in the left-hand side inequality of (1.13) if and only if \(\tau=0\). □

The proof of Theorem 1.3 requires the following two lemmas.

Lemma 4.1

If \(K, L\in\mathcal{S}^{n}_{\mathrm {o}}\), \(0< q< n\), \(p\geq1\), and \(\lambda, \mu\geq 0\) (not both zero), then, for any \(Q\in\mathcal{S}^{n}_{\mathrm {o}}\),

$$ \widetilde{V}_{q}(\lambda\circ K\bar{+}_{-p}\mu\circ L, Q)^{-\frac{p}{n-q}}\geq\lambda\widetilde {V}_{q}(K, Q)^{-\frac{p}{n-q}}+ \mu\widetilde{V}_{q}(L, Q)^{-\frac {p}{n-q}} $$
(4.7)

with equality if and only if K and L are dilates.

Proof

Since \(0< q< n\) and \(p\geq1\), we have \(-p/(n-q)<0\). Hence, from (2.9), (2.4), and the Minkowski integral inequality (see [56]), we obtain that, for any \(Q\in\mathcal{S}^{n}_{\mathrm {o}}\),

$$\begin{aligned} \widetilde{V}_{q}(\lambda\circ K\bar{+}_{-p}\mu\circ L, Q)^{-\frac{p}{n-q}} =& \biggl[\frac{1}{n} \int _{S^{n-1}}\rho_{\lambda\circ K\bar{+}_{-p}\mu\circ L}^{n-q}(u) \rho_{Q}^{q}(u)\,dS(u) \biggr]^{-\frac {p}{n-q}} \\ =& \biggl[\frac {1}{n} \int_{S^{n-1}} \bigl(\rho_{\lambda\circ K\bar{+}_{-p}\mu\circ L}^{-p}(u) \bigr)^{-\frac{n-q}{p}}\rho _{Q}^{q}(u)\,dS(u) \biggr]^{-\frac{p}{n-q}} \\ =& \biggl[\frac{1}{n} \int_{S^{n-1}} \bigl(\lambda\rho_{K}^{-p}(u)+ \mu \rho_{L}^{-p}(u) \bigr)^{-\frac{n-q}{p}} \rho_{Q}^{q}(u) \,dS(u) \biggr]^{-\frac{p}{n-q}} \\ \geq& \biggl[\frac{1}{n} \int_{S^{n-1}}\lambda\rho _{K}^{n-q}(u) \rho_{Q}^{q}(u)\,dS(u) \biggr]^{-\frac{p}{n-q}} \\ &{}+ \biggl[\frac{1}{n} \int_{S^{n-1}}\mu\rho _{L}^{n-q}(u) \rho_{Q}^{q}(u)\,dS(u) \biggr]^{-\frac{p}{n-q}} \\ =& \lambda\widetilde{V}_{q}(K, Q)^{-\frac{p}{n-q}}+\mu \widetilde{V}_{q}(L, Q)^{-\frac{p}{n-q}}. \end{aligned}$$

Thus, inequality (4.7) is proven.

According to the equality condition of the Minkowski integral inequality, equality holds in (4.7) if and only if there exists a constant \(c>0\) such that

$$\frac{\rho_{K}(u)^{n-q}\rho_{Q}^{q}(u)}{\rho_{L}(u)^{n-q}\rho_{Q}^{q}(u)}=c $$

for any \(u\in S^{n-1}\), that is, K and L are dilates. □

Lemma 4.2

If \(K, L\in\mathcal{S}^{n}_{\mathrm {o}}\), \(0< q< n\), \(p\geq1\), and \(\lambda, \mu\geq 0\) (not both zero), then

$$ \widetilde{\Omega}_{q}(\lambda\circ K\bar{+}_{-p}\mu\circ L)^{-\frac{p(n+q)}{n(n-q)}}\geq\lambda \widetilde{\Omega}_{q}(K)^{-\frac{p(n+q)}{n(n-q)}} +\mu\widetilde{\Omega}_{q}(L)^{-\frac{p(n+q)}{n(n-q)}} $$
(4.8)

with equality if and only if K and L are dilates.

Proof

For a bounded function \(\varphi>0\), we have

$$ (\sup\varphi)^{-1}=\inf\varphi^{-1}. $$
(4.9)

Thus, by (2.11), (4.7), and (4.9), noticing that \(-\frac{p}{n-q}<0\) when \(0< q< n\) and \(p\geq1\), we have that

$$\begin{aligned}& \bigl[n^{-\frac{q}{n}}\widetilde{\Omega}_{q}(\lambda\circ K \bar{+}_{-p}\mu\circ L)^{\frac{n+q}{n}} \bigr]^{-\frac{p}{n-q}} \\& \quad = \bigl[\sup\bigl\{ n\widetilde{V}_{q}\bigl(\lambda\circ K \bar{+}_{-p}\mu\circ L, Q^{\ast}\bigr)V(Q)^{\frac{q}{n}}: Q\in{ \mathcal {S}}_{\mathrm{os}}^{n}\bigr\} \bigr]^{-\frac{p}{n-q}} \\& \quad =\inf \bigl\{ \bigl[n\widetilde{V}_{q}\bigl(\lambda\circ K \bar{+}_{-p}\mu\circ L, Q^{\ast}\bigr)V(Q)^{\frac{q}{n}} \bigr]^{-\frac {p}{n-q}}: Q\in{\mathcal {S}}_{\mathrm{os}}^{n} \bigr\} \\& \quad =\inf \bigl\{ \bigl[n\widetilde{V}_{q}\bigl(\lambda\circ K \bar{+}_{-p}\mu\circ L, Q^{\ast}\bigr) \bigr]^{-\frac{p}{n-q}} \bigl[V(Q)^{\frac{q}{n}} \bigr]^{-\frac{p}{n-q}}: Q\in{\mathcal {S}}_{\mathrm{os}}^{n} \bigr\} \\& \quad \geq\inf \bigl\{ \lambda \bigl[n\widetilde{V}_{q}\bigl(K, Q^{\ast}\bigr)V(Q)^{\frac{q}{n}} \bigr]^{-\frac {p}{n-q}}+\mu \bigl[n \widetilde{V}_{q}\bigl(L, Q^{\ast}\bigr)V(Q)^{\frac {q}{n}} \bigr]^{-\frac{p}{n-q}}: Q\in{\mathcal {S}}_{\mathrm{os}}^{n} \bigr\} \\& \quad \geq\lambda\inf \bigl\{ \bigl[n\widetilde{V}_{q}\bigl(K, Q^{\ast}\bigr)V(Q)^{\frac{q}{n}} \bigr]^{-\frac{p}{n-q}}: Q\in{\mathcal {S}}_{\mathrm{os}}^{n} \bigr\} \\& \qquad{} +\mu\inf \bigl\{ \bigl[n\widetilde{V}_{q}\bigl(L, Q^{\ast}\bigr)V(Q)^{\frac {q}{n}} \bigr]^{-\frac{p}{n-q}}: Q\in{\mathcal {S}}_{\mathrm{os}}^{n} \bigr\} \\& \quad =\lambda \bigl[\sup \bigl\{ n\widetilde{V}_{q}\bigl(K, Q^{\ast}\bigr)V(Q)^{\frac {q}{n}}: Q\in{\mathcal {S}}_{\mathrm{os}}^{n} \bigr\} \bigr]^{-\frac{p}{n-q}} \\& \qquad{} +\mu \bigl[\sup \bigl\{ n\widetilde{V}_{q}\bigl(L, Q^{\ast}\bigr)V(Q)^{\frac {q}{n}}: Q\in{\mathcal {S}}_{\mathrm{os}}^{n} \bigr\} \bigr]^{-\frac{p}{n-q}} \\& \quad =\lambda \bigl[n^{-\frac{q}{n}}\widetilde{\Omega}_{q}(K)^{\frac {n+q}{n}} \bigr]^{-\frac{p}{n-q}} +\mu \bigl[n^{-\frac{q}{n}}\widetilde{\Omega}_{q}(L)^{\frac {n+q}{n}} \bigr]^{-\frac{p}{n-q}}. \end{aligned}$$

This gives (4.8).

According to the equality condition of inequality (4.7), equality holds in inequality (4.8) if and only if K and L are dilates. □

Proof of Theorem 1.3

From (4.1) and (4.8) we have that, for \(0< q< n\) and \(p\geq1\),

$$\begin{aligned} \widetilde{\Omega}_{q}\bigl(\Pi^{\tau,\ast}_{p}K \bigr)^{-\frac {p(n+q)}{n(n-q)}} =&\widetilde{\Omega}_{q} \bigl(f_{1}( \tau)\circ\Pi^{+,\ast}_{p}K\,\tilde{+}_{-p}\,f_{2}( \tau)\circ \Pi^{-,\ast}_{p}K \bigr)^{-\frac{p(n+q)}{n(n-q)}} \\ \geq& f_{1}(\tau)\widetilde{\Omega}_{q}\bigl(\Pi ^{+,\ast}_{p}K\bigr)^{-\frac{p(n+q)}{n(n-q)}} +f_{2}(\tau) \widetilde{\Omega}_{q}\bigl(\Pi^{-,\ast}_{p}K \bigr)^{-\frac {p(n+q)}{n(n-q)}}. \end{aligned}$$
(4.10)

But (2.9) shows that, for any \(Q\in{\mathcal {S}}_{\mathrm{os}}^{n}\), \(\widetilde {V}_{q}(-K, Q)=\widetilde{V}_{q}(K, Q)\). This and (2.11) give \(\widetilde {\Omega}_{q}(-K)=\widetilde{\Omega}_{q}(K)\). From this we see that

$$ \widetilde{\Omega}_{q}\bigl(\Pi^{-,\ast}_{p}K\bigr)= \widetilde{\Omega}_{q}\bigl(-\Pi ^{+,\ast}_{p}K\bigr)= \widetilde{\Omega}_{q}\bigl(\Pi^{+,\ast}_{p}K\bigr). $$
(4.11)

This, together with (4.10) and (3.3), yields

$$\widetilde{\Omega}_{q}\bigl(\Pi^{\tau,\ast}_{p}K \bigr)^{-\frac{p(n+q)}{n(n-q)}} \geq\widetilde{\Omega}_{q}\bigl( \Pi^{\pm,\ast}_{p}K\bigr)^{-\frac{p(n+q)}{n(n-q)}}, $$

that is, for \(0< q< n\) and \(p\geq1\),

$$\widetilde{\Omega}_{q}\bigl(\Pi^{\tau,\ast}_{p}K\bigr) \leq\widetilde{\Omega}_{q}\bigl(\Pi^{\pm,\ast}_{p}K \bigr). $$

This is the right-hand side inequality (1.15).

According to the equality condition of inequality (4.8), equality holds in the right-hand side inequality of (1.15) if and only if \(\Pi ^{+,\ast}_{p}K\) and \(\Pi^{-,\ast}_{p}K\) are dilates. This and (4.11) give \(\Pi^{+,\ast}_{p}K=\Pi^{-,\ast}_{p}K\), that is, \(\Pi^{+}_{p}K=\Pi ^{-}_{p}K\). From this, by Theorem 3.A, it follows that if K is not origin-symmetric and p is not an odd integer, then equality holds in the right-hand side inequality of (1.15) if and only if \(\tau=\pm1\).

On the other hand, by (4.6) and inequality (4.8), noticing that

$$ \widetilde{\Omega}_{q}\bigl(\Pi^{-\tau,\ast}_{p}K\bigr)= \widetilde{\Omega}_{q}\bigl(-\Pi^{\tau,\ast}_{p}K\bigr) = \widetilde{\Omega}_{q}\bigl(\Pi^{\tau,\ast}_{p}K \bigr), $$
(4.12)

we obtain that, for \(0< q< n\), \(p\geq1\) and \(\tau\in[-1,1]\),

$$\widetilde{\Omega}_{q}\bigl(\Pi^{\ast}_{p}K\bigr) \leq\widetilde{\Omega}_{q}\bigl(\Pi ^{\tau,\ast}_{p}K \bigr). $$

This yields the left-hand side inequality of (1.15).

According to the equality condition of (4.8) and using (4.12), we know that equality holds in the left-hand side inequality of (1.15) if and only if \(\Pi^{\tau}_{p}K=\Pi^{-\tau}_{p}K\). This, combined with Theorem 3.3, implies that if K is not origin-symmetric and p is not an odd integer, then equality holds in the left-hand side inequality of (1.15) if and only if \(\tau=0\). □