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Extremum of geometric functionals involving general \(L_{p}\)-projection bodies

Journal of Inequalities and Applications20162016:135

https://doi.org/10.1186/s13660-016-1076-2

Received: 1 June 2015

Accepted: 29 April 2016

Published: 10 May 2016

Abstract

Following the discovery of general \(L_{p}\)-projection bodies by Ludwig, Haberl and Schuster determined the extremum of the volume of the polars of this family of \(L_{p}\)-projection bodies. In this paper, the result of Haberl and Schuster is extended to all dual quermassintegrals, and a dual counterpart for the quermassintegrals of general \(L_{p}\)-projection bodies is also obtained. Moreover, the extremum of the \(L_{q}\)-dual affine surface areas of polars of general \(L_{p}\)-projection bodies are determined.

Keywords

general \(L_{p}\)-projection bodyextremumquermassintegraldual quermassintegral \(L_{q}\)-dual affine surface area

MSC

52A4052A20

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 [112]).
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, 1421]).

Through the characterization of so-called \(L_{p}\)-Minkowski valuations, Ludwig [15] discovered (see also [2229] 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, 3047]).

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 \(\tau=0\). □

Declarations

Acknowledgements

The authors would like to sincerely thank the referees for very valuable and helpful comments and suggestions, which made the paper more accurate and readable. Research is supported 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. 2014PY065).

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, Yichang, P.R. China

References

  1. Gardner, RJ: Geometric Tomography, 2nd edn. Cambridge University Press, Cambridge (2006) View ArticleMATHGoogle Scholar
  2. Bolker, ED: A class of convex bodies. Trans. Am. Math. Soc. 145, 323-345 (1969) MathSciNetView ArticleMATHGoogle Scholar
  3. Bourgain, J, Lindenstrauss, J: Projection Bodies, Geometric Aspects of Functional Analysis. Lecture Notes in Math., vol. 1317, pp. 250-270. Springer, Berlin (1988) View ArticleGoogle Scholar
  4. Brannen, NS: Volumes of projection bodies. Mathematika 43, 255-264 (1996) MathSciNetView ArticleMATHGoogle Scholar
  5. Chakerian, GD, Lutwak, E: Bodies with similar projections. Trans. Am. Math. Soc. 349, 1811-1820 (1997) MathSciNetView ArticleMATHGoogle Scholar
  6. Leng, GS, Zhao, CJ, He, BW, Li, XY: Inequalities for polars of mixed projection bodies. Sci. China Ser. A 47(2), 175-186 (2004) MathSciNetView ArticleMATHGoogle Scholar
  7. Lutwak, E: Mixed projection inequalities. Trans. Am. Math. Soc. 287, 91-106 (1985) MathSciNetView ArticleMATHGoogle Scholar
  8. Lutwak, E: Inequalities for mixed projection bodies. Trans. Am. Math. Soc. 339, 901-916 (1993) MathSciNetView ArticleMATHGoogle Scholar
  9. Micheal, S: Petty’s projection inequality and Santaló’s affine isoperimetric inequality. Geom. Dedic. 57, 285-295 (1995) View ArticleMATHGoogle Scholar
  10. Petty, CM: Projection bodies. In: Proc. Coll. Convexity, Copenhagen, 1965, pp. 234-241. Københavns Univ. Math. Inst., Copenhagen (1967) Google Scholar
  11. Petty, CM: Isoperimetric problems. In: Proc. Conf. Convexity and Combinatorial Geometry, Norman, 1971, pp. 26-41. University of Oklahoma Press, Norman (1972) Google Scholar
  12. Zhang, GY: Restricted chord projection and affine inequalities. Geom. Dedic. 39, 213-222 (1991) MathSciNetView ArticleMATHGoogle Scholar
  13. Lutwak, E, Yang, D, Zhang, GY: \(L_{p}\) affine isoperimetric inequalities. J. Differ. Geom. 56, 111-132 (2000) MathSciNetMATHGoogle Scholar
  14. Ludwig, M: Projection bodies and valuations. Adv. Math. 172, 158-168 (2002) MathSciNetView ArticleMATHGoogle Scholar
  15. Ludwig, M: Minkowski valuations. Trans. Am. Math. Soc. 357, 4191-4213 (2005) MathSciNetView ArticleMATHGoogle Scholar
  16. Wang, WD, Lu, FH, Leng, GS: A type of monotonicity on the \(L_{p}\) centroid body and \(L_{p}\) projection body. Math. Inequal. Appl. 8(4), 735-742 (2005) MathSciNetMATHGoogle Scholar
  17. Wang, WD, Leng, GS: The Petty projection inequality for \(L_{p}\)-mixed projection bodies. Acta Math. Appl. Sin. 23(8), 1485-1494 (2007) MathSciNetView ArticleMATHGoogle Scholar
  18. Wang, WD, Leng, GS: On the \(L_{p}\)-versions of the Petty’s conjectured projection inequality and applications. Taiwan. J. Math. 12(5), 1067-1086 (2008) MathSciNetMATHGoogle Scholar
  19. Wang, WD, Leng, GS: Some affine isoperimetric inequalities associated with \(L_{p}\)-affine surface area. Houst. J. Math. 34(2), 443-453 (2008) MathSciNetMATHGoogle Scholar
  20. Lv, SJ, Leng, GS: The \(L_{p}\)-curvature images of convex bodies and \(L_{p}\)-projection bodies. Proc. Indian Acad. Sci. Math. Sci. 118, 413-424 (2008) MathSciNetView ArticleMATHGoogle Scholar
  21. Ryabogin, D, Zvavitch, A: The Fourier transform and Firey projections of convex bodies. Indiana Univ. Math. J. 53, 667-682 (2004) MathSciNetView ArticleMATHGoogle Scholar
  22. Abardia, J: Difference bodies in complex vector spaces. J. Funct. Anal. 263, 3588-3603 (2012) MathSciNetView ArticleMATHGoogle Scholar
  23. Abardia, J: Minkowski valuations in a 2-dimensional complex vector space. Int. Math. Res. Not. 2015, 1247-1262 (2015) MathSciNetMATHGoogle Scholar
  24. Abardia, J, Bernig, A: Projection bodies in complex vector spaces. Adv. Math. 227, 830-846 (2011) MathSciNetView ArticleMATHGoogle Scholar
  25. Haberl, C: Minkowski valuations intertwining with the special linear group. J. Eur. Math. Soc. 14, 1565-1597 (2012) MathSciNetView ArticleMATHGoogle Scholar
  26. Parapatits, L, Schuster, FE: The Steiner formula for Minkowski valuations. Adv. Math. 230, 978-994 (2012) MathSciNetView ArticleMATHGoogle Scholar
  27. Parapatits, L, Wannerer, T: On the inverse Klain map. Duke Math. J. 162, 1895-1922 (2013) MathSciNetView ArticleMATHGoogle Scholar
  28. Schuster, FE: Crofton measures and Minkowski valuations. Duke Math. J. 154, 1-30 (2010) MathSciNetView ArticleMATHGoogle Scholar
  29. Schuster, FE, Wannerer, T: Even Minkowski valuations. Am. J. Math. 137, 1651-1683 (2015) MathSciNetView ArticleMATHGoogle Scholar
  30. Haberl, C, Schuster, F: General \(L_{p}\)-affine isoperimetric inequalities. J. Differ. Geom. 83, 1-26 (2009) MathSciNetMATHGoogle Scholar
  31. Wang, WD, Wan, XY: Shephard type problems for general \(L_{p}\)-projection bodies. Taiwan. J. Math. 16(5), 1749-1762 (2012) MathSciNetMATHGoogle Scholar
  32. Wang, WD, Feng, YB: A general \(L_{p}\)-version of Petty’s affine projection inequality. Taiwan. J. Math. 17(2), 517-528 (2013) View ArticleMATHGoogle Scholar
  33. Feng, YB, Wang, WD: General \(L_{p}\)-harmonic Blaschke bodies. Proc. Indian Acad. Sci. Math. Sci. 124(1), 109-119 (2014) MathSciNetView ArticleMATHGoogle Scholar
  34. Feng, YB, Wang, WD, Lu, FH: Some inequalities on general \(L_{p}\)-centroid bodies. Math. Inequal. Appl. 18(1), 39-49 (2015) MathSciNetMATHGoogle Scholar
  35. Haberl, C: \(L_{p}\)-intersection bodies. Adv. Math. 217, 2599-2624 (2008) MathSciNetView ArticleMATHGoogle Scholar
  36. Haberl, C, Ludwig, M: A characterization of \(L_{p}\) intersection bodies. Int. Math. Res. Not. 2006, Article ID 10548 (2006) MathSciNetMATHGoogle Scholar
  37. Haberl, C, Schuster, FE: Asymmetric affine \(L_{p}\) Sobolev inequalities. J. Funct. Anal. 257, 641-658 (2009) MathSciNetView ArticleMATHGoogle Scholar
  38. Haberl, C, Schuster, FE, Xiao, J: An asymmetric affine Pólya-Szegö principle. Math. Ann. 352, 517-542 (2012) MathSciNetView ArticleMATHGoogle Scholar
  39. Parapatits, L: \(SL(n)\)-covariant \(L_{p}\)-Minkowski valuations. J. Lond. Math. Soc. 89, 397-414 (2014) MathSciNetView ArticleMATHGoogle Scholar
  40. Parapatits, L: \(SL(n)\)-contravariant \(L_{p}\)-Minkowski valuations. Trans. Am. Math. Soc. 366, 1195-1211 (2014) MathSciNetView ArticleMATHGoogle Scholar
  41. Schuster, FE, Wannerer, T: \(GL(n)\) contravariant Minkowski valuations. Trans. Am. Math. Soc. 364, 815-826 (2012) MathSciNetView ArticleMATHGoogle Scholar
  42. Schuster, FE, Weberndorfer, M: Volume inequalities for asymmetric Wulff shapes. J. Differ. Geom. 92, 263-283 (2012) MathSciNetMATHGoogle Scholar
  43. Wang, WD, Li, YN: Busemann-Petty problems for general \(L_{p}\)-intersection bodies. Acta Math. Sin. Engl. Ser. 31(5), 777-786 (2015) MathSciNetView ArticleMATHGoogle Scholar
  44. Wang, WD, Li, YN: General \(L_{p}\)-intersection bodies. Taiwan. J. Math. 19(4), 1247-1259 (2015) View ArticleGoogle Scholar
  45. Wang, WD, Ma, TY: Asymmetric \(L_{p}\)-difference bodies. Proc. Am. Math. Soc. 142(7), 2517-2527 (2014) View ArticleMATHGoogle Scholar
  46. Wannerer, T: \(GL(n)\) equivariant Minkowski valuations. Indiana Univ. Math. J. 60, 1655-1672 (2011) MathSciNetView ArticleMATHGoogle Scholar
  47. Weberndorfer, M: Shadow systems of asymmetric \(L_{p}\) zonotopes. Adv. Math. 240, 613-635 (2013) MathSciNetView ArticleMATHGoogle Scholar
  48. Firey, WJ: p-means of convex bodies. Math. Scand. 10, 17-24 (1962) MathSciNetMATHGoogle Scholar
  49. Schneider, R: Convex Bodies: The Brunn-Minkowski Theory, 2nd expanded edn. Cambridge University Press, Cambridge (2014) MATHGoogle Scholar
  50. Lutwak, E: The Brunn-Minkowski-Firey theory II: affine and geominimal surface areas. Adv. Math. 118, 244-294 (1996) MathSciNetView ArticleMATHGoogle Scholar
  51. Lutwak, E: The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131-150 (1993) MathSciNetMATHGoogle Scholar
  52. Grinberg, E, Zhang, GY: Convolutions transforms and convex bodies. Proc. Lond. Math. Soc. 78(3), 77-115 (1999) MathSciNetView ArticleMATHGoogle Scholar
  53. Wang, W, Yuan, J, He, BW: Large inequalities for \(L_{p}\)-dual affine surface area. Math. Inequal. Appl. 7, 34-45 (2008) Google Scholar
  54. Lutwak, E: Dual mixed volumes. Pac. J. Math. 58, 531-538 (1975) MathSciNetView ArticleMATHGoogle Scholar
  55. Wang, WD, Leng, GS: A correction to our paper ‘\(L_{p}\)-dual mixed quermassintegrals’. Indian J. Pure Appl. Math. 38(6), 609 (2007) MathSciNetGoogle Scholar
  56. Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1959) MATHGoogle Scholar

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