1 Introduction and main results

Let \(\mathcal{K}^{n}\) denote the set of convex bodies (compact, convex subsets with non-empty interiors) in Euclidean space \(\mathbb {R}^{n}\). For the set of convex bodies containing the origin in their interiors, the set of convex bodies whose centroid lie at the origin and the set of origin-symmetric convex bodies in \(\mathbb{R}^{n}\), we write \(\mathcal{K}^{n}_{o}\), \(\mathcal{K}^{n}_{c}\), and \(\mathcal{K}^{n}_{os}\), respectively. \(\mathcal{S}^{n}_{o}\), \(\mathcal{S}^{n}_{os}\), respectively, denote the set of star bodies (about the origin) and the set of origin-symmetric star bodies in \(\mathbb{R}^{n}\). Let \({S}^{n-1}\) denote the unit sphere in \(\mathbb{R}^{n}\), and \(V(K)\) denotes the n-dimensional volume of body K. For the standard unit ball B in \(\mathbb{R}^{n}\), denote \(\omega_{n}=V(B)\).

In 1997, Lutwak and Zhang [1] gave the concept of an \(L_{p}\)-centroid body as follows: For each compact star-shaped about the origin \(K \subset \mathbb{R}^{n} \), real \(p\geq1 \), the \(L_{p}\)-centroid body, \(\Gamma_{p}K\), of K is an origin-symmetric convex body whose support function is defined by

$$h^{p}_{\Gamma_{p}K}(u)= \frac{1}{c_{n,p}V(K)}\int_{K} |u\cdot x|^{p} \,dx $$

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

$$ c_{n,p} = \omega_{n+p}/ \omega_{2} \omega_{n}\omega_{p-1}. $$
(1.1)

Meanwhile, they [1] obtained the \(L_{p}\)-centroid affine inequality, which implies the well-known Blaschke-Santaló inequality. Hereafter, associating the \(L_{p}\)-centroid bodies with the \(L_{p}\)-projection bodies, Lutwak et al. [2] established the \(L_{p}\)-Busemann-Petty centroid inequality and the \(L_{p}\)-Petty projection inequality. For the studies of \(L_{p}\)-centroid bodies, also see [3–7].

In 2005, Ludwig [8] introduced a function \(\varphi_{\tau}: \mathbb {R}\rightarrow[0, +\infty)\) by

$$ \varphi_{\tau}(t)=|t|+\tau t $$
(1.2)

for \(\tau\in[-1,1]\). Further, in [8] the general \(L_{p}\)-moment bodies and the general \(L_{p}\)-projection bodies were defined by (1.2). In 2009, Haberl and Schuster [9] derived the general \(L_{p}\)-moment body (the general \(L_{p}\)-projection body) is an \(L_{p}\)-Minkowski combination of the asymmetric \(L_{p}\)-moment body (the asymmetric \(L_{p}\)-projection body) and established the general \(L_{p}\)-Busemann-Petty centroid inequality and the general \(L_{p}\)-Petty projection inequality.

Recently, motivated by Ludwig’s, and Haberl and Schuster’s work, Feng et al. [10] defined asymmetric \(L_{p}\)-centroid bodies as follows: For \(K \in \mathcal{S}^{n}_{o}\), \(p\geq1\), the asymmetric \(L_{p}\)-centroid body, \(\Gamma^{+}_{p}K\), of K is a convex body whose support function is defined by

$$h^{p}_{\Gamma^{+}_{p}K}(u)= \frac{2}{c_{n,p}V(K)}\int_{K}(u \cdot x)^{p}_{+}\,dx $$

for any \(u\in{S}^{n-1}\). Using polar coordinates in the above definition, we easily obtain, for any \(u\in{S}^{n-1}\),

$$ h^{p}_{\Gamma^{+}_{p}K}(u)= \frac{2}{c_{n,p}(n+p)V(K)}\int_{S^{n-1}}(u \cdot v)^{p}_{+} \rho_{K}(v)^{n+p}\,dv, $$
(1.3)

where \((u \cdot x)_{+}=\max\{u\cdot x, 0\}\), \(c_{n, p}\) satisfies (1.1) and the integration is with respect to Lebesgue measure on \(S^{n-1}\). Obviously, \(\Gamma^{+}_{p}B=B\). They also defined

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

By (1.2), Feng et al. [10] introduced the general \(L_{p}\)-centroid bodies: For \(K \in\mathcal{S}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1,1]\), the general \(L_{p}\)-centroid body, \(\Gamma^{\tau}_{p}K\), of K is a convex body whose support function is defined by

$$\begin{aligned} h^{p}_{\Gamma^{\tau}_{p}K}(u) =& \frac{2}{c_{n,p}(\tau)V(K)}\int_{K} \varphi_{\tau}(u \cdot x)^{p} \,dx \\ =& \frac{2}{c_{n,p}(\tau )(n+p)V(K)}\int_{S^{n-1}} \varphi_{\tau}(u \cdot v)^{p}\rho_{K}(v)^{n+p} \,dv, \end{aligned}$$
(1.4)

where

$$c_{n, p}(\tau)= c_{n,p}\bigl[(1+\tau)^{p} + (1- \tau)^{p}\bigr] $$

and \(c_{n, p}\) satisfies (1.1).

Obviously, \(\Gamma^{\tau}_{p}B=B\), and if \(\tau=0\), then \(\Gamma^{\tau}_{p}K=\Gamma_{p}K\).

From the definition of \(\Gamma^{\pm}_{p}K\) and (1.4), it follows that if \(K \in\mathcal{S}^{n}_{os}\), \(p\geq1\), and \(\tau\in[-1,1]\), then, for any \(u\in S^{n-1}\),

$$ h\bigl(\Gamma^{\tau}_{p}K, u\bigr)^{p} = f_{1}(\tau)h\bigl(\Gamma^{+}_{p}K, u\bigr)^{p}+f_{2}( \tau)h\bigl(\Gamma ^{-}_{p}K, u\bigr)^{p}, $$
(1.5)

where

$$ f_{1}(\tau)= \frac{(1+\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}}, \qquad f_{2}(\tau)= \frac{(1-\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}} . $$
(1.6)

From (1.6), we know that

$$\begin{aligned}& f_{1}(-\tau)=f_{2}(\tau), \qquad f_{2}(- \tau)=f_{1}(\tau), \end{aligned}$$
(1.7)
$$\begin{aligned}& f_{1}(\tau) + f_{2}(\tau) = 1. \end{aligned}$$
(1.8)

The general \(L_{p}\)-centroid bodies belong to a new and rapidly evolving asymmetric \(L_{p}\) Brunn-Minkowski theory that has its origins in the work of Ludwig, Haberl, and Schuster (see [8, 9, 11–14]). For the further researches of asymmetric \(L_{p}\) Brunn-Minkowski theory, also see [10, 15–27].

In 1996, Lutwak [28] introduced the concept of an \(L_{p}\)-affine surface area as follows: For \(K \in\mathcal{K}^{n}_{o} \) and \(p\geq1\), the \(L_{p}\)-affine surface area, \(\Omega_{p}(K)\), of K is defined by

$$n^{-\frac{p}{n}}\Omega_{p}(K)^{\frac{n+p}{n}}=\inf\bigl\{ nV_{p}\bigl(K,Q^{*}\bigr)V(Q)^{\frac{p}{n}}: Q \in\mathcal{S}^{n}_{o} \bigr\} , $$

where \(V_{p}(M,N)\) denotes the \(L_{p}\)-mixed volume of \(M,N\in\mathcal{K}^{n}_{o}\).

Further, Wang and Leng [29] defined ith \(L_{p}\)-mixed affine surface area, \(\Omega_{p,i}(K)\), of K (for \(i=0\), \(\Omega_{p,0}(K)\) is just the \(L_{p}\)-affine surface area \(\Omega_{p}(K)\)) and extended some of Lutwak’s results. Regarding the studies of an \(L_{p}\)-affine surface area, many results have been obtained (see [28–35]).

Associated with the \(L_{p}\)-dual mixed volumes, Wang and He [36] gave the notion of the \(L_{p}\)-dual affine surface area. For \(K \in \mathcal{S}^{n}_{o} \) and \(1\leq p< n\), the \(L_{p}\)-dual affine surface area, \(\widetilde{\Omega}_{-p}(K)\), of K is defined by

$$ n^{\frac{p}{n}}\widetilde{\Omega}_{-p}(K)^{\frac{n-p}{n}}= \inf\bigl\{ {n\widetilde{V}_{-p}\bigl(K,Q^{*}\bigr)V(Q)^{-\frac{p}{n}}:Q\in \mathcal{K}^{n}_{c} }\bigr\} , $$
(1.9)

where \(\widetilde{V}_{-p}(M, N)\) denotes the \(L_{p}\)-dual mixed volume of \(M, N\in\mathcal{S}^{n}_{o} \).

In 2014, Feng and Wang [37] improved definition (1.9) from \(Q\in \mathcal{K}^{n}_{c} \) to \(Q\in\mathcal{S}^{n}_{os}\) as follows: For \(K \in \mathcal{S}^{n}_{o} \) and \(1\leq p< n\), the \(L_{p}\)-dual affine surface area, \(\widetilde{\Omega}_{-p}(K)\), of K is defined by

$$ n^{\frac{p}{n}}\widetilde{\Omega}_{-p}(K)^{\frac{n-p}{n}}= \inf\bigl\{ {n\widetilde{V}_{-p}\bigl(K, Q^{*}\bigr)V(Q)^{-\frac{p}{n}}: Q\in\mathcal {S}^{n}_{os}}\bigr\} . $$
(1.10)

Let \(Z^{*}_{p}\) denote the set of polar of all \(L_{p}\)-projection bodies, then \(\mathcal{Z}^{*}_{p}\subseteq \mathcal{S}^{n}_{os}\). If \(Q\in Z^{*}_{p}\) in (1.10), write \(\widetilde{\Omega}^{o}_{-p}(K)\) by

$$ n^{\frac{p}{n}}\widetilde{\Omega}^{o}_{-p}(K) =\inf\bigl\{ n \widetilde {V}_{-p}\bigl(K,Q^{*}\bigr)V(Q)^{\frac{p}{n}}: Q\in \mathcal{Z}^{*}_{p}\bigr\} . $$
(1.11)

According to (1.10) and (1.11), Feng and Wang [37] studied the Shephard type problems for the \(L_{p}\)-centroid bodies. First, they gave an affirmative form of the Shephard type problems for the \(L_{p}\)-centroid bodies as follows.

Theorem 1.A

For \(K, L\in\mathcal{S}^{n}_{o}\), \(1\leq p< n\), if \(\Gamma_{p}K \subseteq\Gamma_{p}L \), then

$$\frac{\widetilde{\Omega}^{o}_{-p}(K)^{\frac{n-p}{n}}}{V(K)} \leq\frac {\widetilde{\Omega}^{o}_{-p}(L)^{\frac{n-p}{n}}}{V(L)}, $$

with equality if and only if \(\Gamma_{p}K = \Gamma_{p}L \).

Hereafter, combining with definition (1.10) of the \(L_{p}\)-dual affine surface area, the authors [37] gave an improved form of the Shephard type problems for the \(L_{p}\)-centroid bodies.

Theorem 1.B

For \(K\in \mathcal{S}_{o}^{n}\), \(L\in\mathcal{ S}_{os}^{n}\) and \(1\leq p< n\), if \(\Gamma_{p}K=\Gamma_{p}L\), then

$$\widetilde{\Omega}_{-p}(K)\leq\widetilde{\Omega}_{-p}(L), $$

with equality if and only if \(K=L\).

Finally, they [37] obtained a negative form of the Shephard type problems for the \(L_{p}\)-centroid bodies.

Theorem 1.C

For \(L\in \mathcal{S}_{o}^{n}\) and \(1\leq p< n\), if L is not origin-symmetric star body, then there exists \(K\in \mathcal{S}_{os}^{n}\), such that

$$\Gamma_{p} K\subset\Gamma_{p} L, $$

but

$$\widetilde{\Omega}_{-p}(K)>\widetilde{\Omega}_{-p}(L). $$

In this paper, associated with definition (1.10) of the \(L_{p}\)-dual affine surface area, we will research the Shephard type problems for the general \(L_{p}\)-centroid bodies. For convenience, we improve definition (1.11) as follows: Let \(Z^{\tau,*}_{p}\) denote the set of polar of all general \(L_{p}\)-projection bodies, for \(K\in\mathcal{S}^{n}_{o}\) and \(1\leq p< n\), the \(L_{p}\)-dual affine surface area, \(\widetilde{\Omega}^{\star}_{-p}(K)\), of K is given by

$$ n^{\frac{p}{n}}\widetilde{\Omega}^{\star}_{-p}(K)^{\frac{n-p}{n}}= \inf\bigl\{ n\widetilde{V}_{-p}\bigl(K,Q^{*}\bigr)V(Q)^{\frac{p}{n}}: Q \in\mathcal{Z}^{\tau ,*}_{p}\bigr\} . $$
(1.12)

From definition (1.12), we first give an affirmative form of the Shephard type problems for the general \(L_{p}\)-centroid bodies, i.e., a general form of Theorem 1.A is obtained.

Theorem 1.1

For \(K,L\in\mathcal{S}^{n}_{o}\), \(1\leq p< n\), and \(\tau\in[-1,1]\), if \(\Gamma^{\tau}_{p}K \subseteq\Gamma^{\tau}_{p}L\), then

$$ \frac{\widetilde{\Omega}^{\star}_{-p}(K)^{\frac{n-p}{n}}}{V(K)} \leq\frac {\widetilde{\Omega}^{\star}_{-p}(L)^{\frac{n-p}{n}}}{V(L)}, $$
(1.13)

with equality if and only if \(\Gamma^{\tau}_{p}K =\Gamma^{\tau}_{p}L\).

Next, corresponding to Theorem 1.B and combining with definition (1.10), we get an improved form of the Shephard type problems for the general \(L_{p}\)-centroid bodies.

Theorem 1.2

Let \(K \in\mathcal{S}^{n}_{o}\), \(L \in \mathcal{S}^{n}_{os}\), \(1 \leq p < n\), and \(\tau\in[-1,1]\), if \(\Gamma^{\tau}_{p}K = \Gamma^{\tau}_{p}L \), then

$$ \widetilde{\Omega}_{-p}(K)\leq\widetilde{\Omega}_{-p}(L), $$
(1.14)

with equality if and only if \(K = L\).

Further, we prove a general version of Theorem 1.C, that is, a negative form of the Shephard type problems for the general \(L_{p}\)-centroid bodies is given.

Theorem 1.3

For \(L\in \mathcal{S}_{o}^{n}\), \(1\leq p< n\), and \(\tau\in(-1, 1)\), if L is not origin-symmetric star body, then there exists \(K\in \mathcal{S}_{o}^{n}\) (for \(\tau=0\), \(K\in\mathcal{S}^{n}_{os}\)), such that

$$\Gamma^{\tau}_{p} K\subset\Gamma^{\tau}_{p} L, $$

but

$$\widetilde{\Omega}_{-p}(K)>\widetilde{\Omega}_{-p}(L). $$

Besides, corresponding to Theorem 1.C, we generalize the scope of negative solutions of the Shephard type problems for the \(L_{p}\)-centroid bodies from \(K\in\mathcal{S}^{n}_{os} \) to \(K\in\mathcal{S}^{n}_{o}\).

Theorem 1.4

For \(L\in \mathcal{S}_{o}^{n}\) and \(1\leq p< n\), if L is not origin-symmetric star body, then there exists \(K\in \mathcal{S}_{o}^{n}\), such that

$$\Gamma_{p} K\subset\Gamma_{p} L, $$

but

$$\widetilde{\Omega}_{-p}(K)>\widetilde{\Omega}_{-p}(L). $$

The proofs of Theorems 1.1-1.4 are completed in Section 4. In order to prove our results, we give two inequalities for the general \(L_{p}\)-harmonic Blaschke bodies in Section 3.

2 Preliminaries

2.1 Support function, radial function and polar

If \(K \in\mathcal{K}^{n} \), then its support function, \(h_{K}=h(K,\cdot): \mathbb{R}^{n} \rightarrow(-\infty,\infty)\), is defined by (see [38, 39])

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

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

If K is a compact star-shaped (about the origin) in \(\mathbb{R}^{n} \), then its radial function, \(\rho_{K}=\rho(K,\cdot):\mathbb{R}^{n}\backslash \{0\}\rightarrow[0,+\infty)\), is defined by (see [38, 39])

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

Given \(c > 0\), we can get, for any \(u \in{S}^{n-1}\),

$$ \rho(cK, u)=c\rho(K,u). $$
(2.1)

If \(\rho_{K}\) is positive and continuous, K will be called a star body (about 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 non-empty set in \(\mathbb{R}^{n}\), the polar set, \(E^{\ast}\), of E is defined by (see [38, 39])

$$ E^{*}=\bigl\{ x\in\mathbb{R}^{n}: x\cdot y \leq1,y\in E \bigr\} . $$
(2.2)

From (2.2), we easily see that if \(K\in{\mathcal{S}}^{n}_{o}\), then \(K^{*}\in {\mathcal{K}}^{n}_{o}\) (see [38]).

2.2 \(L_{p}\)-Dual mixed volumes

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 [28])

$$ \rho(\lambda\star K +_{-p}\mu\star L,\cdot)^{-p} = \lambda \rho(K,\cdot )^{-p} + \mu\rho(L,\cdot)^{-p}, $$
(2.3)

where the operation ‘\(+_{-p}\)’ is called \(L_{p}\)-harmonic radical addition and \(\lambda\star K \) denotes the \(L_{p}\)-harmonic radical scalar multiplication. From (2.1) and (2.3), we have \(\lambda\star K = \lambda^{-\frac{1}{p}}K\).

Associated with (2.3), Lutwak [28] introduced the notion of an \(L_{p}\)-dual mixed volume as follows: For \(K,L \in\mathcal{S}^{n}_{o}\), \(p\geq1\), and \(\varepsilon>0\), the \(L_{p}\)-dual mixed volume, \(\widetilde {V}_{-p}(K,L)\), of K and L is defined by (see [28])

$$\frac{n}{-p}\widetilde{V}_{-p}(K, L) = \lim_{\varepsilon\rightarrow 0^{+}}{ \frac {V(K+_{-p}\varepsilon\star L)-V(K)}{\varepsilon}}. $$

The definition above and Hospital’s rule give the following integral representation of an \(L_{p}\)-dual mixed volume (see [28]):

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

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

The \(L_{p}\)-dual Minkowski inequality can be stated as follows (see [28]).

Theorem 2.A

If \(K,L\in\mathcal{S}^{n}_{o}\), \({p \geq 1}\), then

$$ \widetilde{V}_{-p}(K,L)\geq V(K)^{\frac{n+p}{n}}V(L)^{-\frac{p}{n}}, $$
(2.4)

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

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

The general \(L_{p}\)-projection body was introduced by Ludwig (see [8]). For \(K\in\mathcal{K}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1,1]\), the general \(L_{p}\)-projection body, \(\Pi^{\tau}_{p}K\in\mathcal{K}^{n}_{o}\), of K is given by

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

where \(\varphi_{\tau}\) satisfies (1.2) and

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

with \(\alpha_{n,p}=1/c_{n,p}(n+p)\omega_{n}\).

3 General \(L_{p}\)-harmonic Blaschke bodies

In order to prove our results, we require the notions of \(L_{p}\)-harmonic Blaschke combinations and general \(L_{p}\)-harmonic Blaschke bodies.

For \(K,L\in\mathcal{S}^{n}_{o}\), \(p \geq1\), and \(\lambda, \mu\geq0\) (not both zero), the \(L_{p}\)-harmonic Blaschke combination, \(\lambda * K\, \hat{+}_{p} \, \mu * L \in\mathcal{S}^{n}_{o}\), of K and L is defined by (see [37])

$$ \frac{\rho(\lambda * K\, \hat{+}_{p}\, \mu * L,\cdot)^{n+p}}{V(\lambda * K\, \hat{+}_{p}\, \mu * L)}= \lambda\frac{\rho(K,\cdot)^{n+p}}{V(K)}+\mu \frac{\rho(L,\cdot)^{n+p}}{V(L)}, $$
(3.1)

where the operation ‘\(\hat{+}_{p}\)’ is called \(L_{p}\)-harmonic Blaschke addition and \(\lambda * K\) denotes \(L_{p}\)-harmonic Blaschke scalar multiplication. From (2.1) and (3.1), we know \(\lambda * K=\lambda^{\frac {1}{p}}K\).

Let \(\lambda=\mu=\frac{1}{2}\) and \(L=-K\) in (3.1), then the \(L_{p}\)-harmonic Blaschke body, \(\widehat{\nabla}_{p}K\), of \(K\in\mathcal {S}^{n}_{o}\) is given by (see [37])

$$\widehat{\nabla}_{p}K=\frac{1}{2}*K\, \hat{+}_{p}\, \frac{1}{2}*(-K). $$

According to (3.1), Feng and Wang [15] defined general \(L_{p}\)-harmonic Blaschke bodies as follows: For \(K\in\mathcal{S}^{n}_{o}\), \(p\geq1\), and \(\tau\in[-1,1]\), the general \(L_{p}\)-harmonic Blaschke body, \(\widehat{\nabla}^{\tau}_{p} K = f_{1}(\tau )\circ K \, \hat{+}_{p}\, f_{2}(\tau)\circ(-K)\), of K is defined by

$$ \frac{\rho(\widehat{\nabla}^{\tau}_{p}K,\cdot)^{n+p}}{V(\widehat{\nabla }^{\tau}_{p}K)} = f_{1}(\tau)\frac{\rho(K,\cdot)^{n+p}}{V(K)} + f_{2}(\tau)\frac {\rho(-K,\cdot)^{n+p}}{V(-K)}, $$
(3.2)

where \(f_{1}(\tau)\), \(f_{2}(\tau)\) satisfy (1.6).

Obviously, if \(\tau=0 \), then \(\widehat{\nabla}^{\tau}_{p}K = \widehat {\nabla}_{p}K\). In addition, if \(\tau=\pm1\), then we write \(\widehat {\nabla}^{\tau}_{p}(K)=\widehat{\nabla}^{\pm}_{p}K\), and \(\widehat{\nabla }^{+}_{p}K=K\), \(\widehat{\nabla}^{-}_{p}K=-K\).

For the \(L_{p}\)-harmonic Blaschke combination (3.1), Feng and Wang [37] proved the following fact.

Theorem 3.A

If \(K,L\in\mathcal{S}^{n}_{o}\), \(p \geq 1\), \(\lambda,\mu\geq0\) (not both zero), then

$$ {V}(\lambda\ast K \, \hat{+}_{p}\, \mu\ast L)^{\frac{p}{n}} \geq \lambda V(K)^{\frac{p}{n}} + \mu V(L)^{\frac{p}{n}}, $$
(3.3)

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

From Theorem 3.A, we easily get the following.

Corollary 3.1

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

$$ V\bigl(\widehat{\nabla}^{\tau}_{p}K\bigr) \geq V(K). $$
(3.4)

For \(\tau\in(-1,1)\), equality holds if and only if K is origin-symmetric. For \(\tau=\pm1\), (3.4) is identic.

Proof

For \(\tau\in(-1,1)\), taking \(\lambda=f_{1}(\tau)\), \(\mu =f_{2}(\tau)\), and \(L=-K\) in (3.3), then by (1.8) we immediately get inequality (3.4). According to the equality condition of inequality (3.3), we see that equality holds in inequality (3.4) if and only if K and −K are dilates, i.e., K is origin-symmetric.

For \(\tau=\pm1\), by \(\widehat{\nabla}^{+}_{p}K=K\) and \(\widehat{\nabla }^{-}_{p}K=-K\), we know that (3.4) is identic. □

Further, according to the \(L_{p}\)-harmonic Blaschke combination (3.1) and definition (1.10) of the \(L_{p}\)-dual affine surface area, Feng and Wang [37] gave the following result.

Theorem 3.B

If \(K,L \in\mathcal{S}^{n}_{o}\), \(\lambda ,\mu\geq0\) (not both zero) and \(1\leq p< n\), then

$$ \frac{ \widetilde{\Omega}_{-p}(\lambda\ast K \, \hat{+}_{p}\, \mu\ast L)^{\frac{n-p}{n}}}{V(\lambda\ast K \, \hat{+}_{p}\, \mu\ast L )} \geq \lambda\frac{\widetilde{\Omega}_{-p}(K)^{\frac{n-p}{n}}}{V(K)} + \mu \frac{\widetilde{\Omega}_{-p}(L)^{\frac{n-p}{n}}}{V(L)}, $$
(3.5)

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

Corollary 3.2

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

$$ \widetilde{\Omega}_{-p}\bigl(\widehat{\nabla}^{\tau}_{p}K \bigr) \geq\widetilde{\Omega }_{-p}(K). $$
(3.6)

For \(\tau\in(-1,1)\), equality holds if and only if K is origin-symmetric. For \(\tau=\pm1\), (3.6) is identic.

Proof

For \(\tau\in(-1,1)\), let \(\lambda=f_{1}(\tau)\), \(\mu =f_{2}(\tau)\), and \(L=-K\) in (3.5), we obtain

$$ \frac{\widetilde{\Omega}(\widehat{\nabla}^{\tau}_{p}K)^{\frac {n-p}{n}}}{V(\widehat{\nabla}^{\tau}_{p}K)} \geq f_{1}(\tau)\frac{\widetilde {\Omega}_{-p}(K)^{\frac{n-p}{n}}}{V(K)} + f_{2}(\tau)\frac{\widetilde {\Omega}_{-p}(-K)^{\frac{n-p}{n}}}{V(-K)}. $$
(3.7)

For any \(Q\in\mathcal{S}^{n}_{os}\), using \(\rho_{Q^{*}}(u) = \rho _{-Q^{*}}(u)\), for any \(u\in S^{n-1}\), we get

$$ \widetilde{V}_{-p}\bigl(-K, Q^{*}\bigr) = \widetilde{V}_{-p} \bigl(K, Q^{*}\bigr). $$
(3.8)

Associated with (1.10) and (3.8), we have

$$ \widetilde{\Omega}_{-p}(-K) = \widetilde{\Omega}_{-p}(K). $$
(3.9)

Thus by (3.7), (3.9), and (1.8), we know

$$\biggl(\frac{\widetilde{\Omega}_{-p}(\widehat{\nabla}^{\tau}_{p}K)}{\widetilde{\Omega}_{-p}(K)} \biggr)^{\frac{n-p}{n}} \geq\frac {V(\widehat{\nabla}^{\tau}_{p}K)}{V(K)}. $$

This and inequality (3.4) yield inequality (3.6).

From the equality conditions of inequalities (3.4) and (3.5), we see that equality holds in (3.6) if and only if K is origin-symmetric.

For \(\tau=\pm1\), obviously, (3.6) is identic. □

4 Proofs of theorems

In this section, we complete the proofs of Theorems 1.1-1.4. In the proof of Theorem 1.1, we require a lemma as follows.

Lemma 4.1

([10])

If \(K\in\mathcal{S}^{n}_{o}\), \(p\geq1\), \(\tau\in[-1,1]\), then, for any \(Q\in\mathcal{K}^{n}_{o}\),

$$V_{p}\bigl(Q,\Gamma^{\tau}_{p}K\bigr)= \frac{\omega_{n}}{V(K)}\widetilde{V}_{-p}\bigl(K,\Pi ^{\tau,*}_{p}Q \bigr). $$

Proof of Theorem 1.1

Since \(\Gamma^{\tau}_{p}K\subseteq\Gamma^{\tau}_{p}L\), for any \(Q\in\mathcal{K}^{n}_{o}\),

$$ V_{p}\bigl(Q,\Gamma^{\tau}_{p}K\bigr) \leq V_{p}\bigl(Q,\Gamma^{\tau}_{p}L\bigr), $$
(4.1)

with equality if and only if \(\Gamma^{\tau}_{p}K=\Gamma^{\tau}_{p}L\).

Therefore, from (4.1) and Lemma 4.1, we have

$$ \frac{\widetilde{V}_{-p}(K,\Pi^{\tau,*}_{p}Q)}{V(K)}\leq\frac{\widetilde {V}_{-p}(L,\Pi^{\tau,*}_{p}Q)}{V(L)}. $$
(4.2)

Let \(M=\Pi^{\tau}_{p}Q\), then \(M\in Z^{\tau, \ast}_{p}\). From (1.11) and (4.2), we get

$$\begin{aligned} \frac{n^{\frac{p}{n}}\widetilde{\Omega}^{\star}_{-p}{(K)}^{\frac{n-p}{n}}}{V(K)} =&\inf \biggl\{ \frac{n\widetilde{V}_{-p}(K,M^{*})}{V(K)}V(M)^{-\frac {p}{n}}:M\in \mathcal{Z}^{\tau,*}_{p} \biggr\} \\ \leq&\inf \biggl\{ \frac {n\widetilde{V}_{-p}(L,M^{*})}{V(L)}V(M)^{-\frac{p}{n}}:M\in\mathcal {Z}^{\tau,*}_{p} \biggr\} \\ =&\frac{n^{\frac{p}{n}}\widetilde{\Omega}^{\star}_{-p}{(L)}^{\frac {n-p}{n}}}{V(L)}, \end{aligned}$$

i.e., (1.13) is obtained.

According to the equality condition of (4.1), we know that the equality holds in (1.13) if and only if \(\Gamma^{\tau}_{p}K=\Gamma^{\tau}_{p}L\). □

The proof of Theorem 1.2 requires the following lemmas.

Lemma 4.2

([37])

For \(K,L \in\mathcal {S}^{n}_{o}\), \(p\geq1\), if \(\Gamma_{p} K=\Gamma_{p} L\), then, for any \(Q\in \mathcal{S}^{n}_{os}\),

$$\frac{\widetilde{V}_{-p}(K,Q)}{V(K)}=\frac{\widetilde {V}_{-p}(L,Q)}{V(L)}. $$

Lemma 4.3

For \(K,L \in\mathcal{S}^{n}_{o}\), \(p\geq 1\), and \(\tau\in[-1,1]\), if \(\Gamma^{\tau}_{p} K = \Gamma^{\tau}_{p} L\), then, for any \(Q \in\mathcal{S}^{n}_{os}\),

$$ \frac{\widetilde{V}_{-p}(K,Q)}{V(K)}=\frac{\widetilde {V}_{-p}(L,Q)}{V(L)}. $$
(4.3)

Proof

Let \(\tau=0\) in (1.5), we have, for any \(u\in S^{n-1}\),

$$ h(\Gamma_{p}K, u)^{p}=\frac{1}{2}h\bigl( \Gamma^{+}_{p}K, u\bigr)^{p}+\frac{1}{2}h\bigl(\Gamma ^{-}_{p}K, u\bigr)^{p}. $$
(4.4)

On the other hand, by (1.5), (1.7), (1.8), and (4.4), we see that, for any \(u\in S^{n-1}\),

$$\begin{aligned}& \frac{1}{2}h\bigl(\Gamma^{\tau}_{p}K, u \bigr)^{p}+\frac{1}{2}h\bigl(\Gamma^{-\tau}_{p}K, u\bigr)^{p} \\& \quad =\frac{1}{2} \bigl[f_{1}(\tau)h_{\Gamma^{+}_{p}K}^{p}(u)+f_{2}( \tau )h_{\Gamma^{-}_{p}K}^{p}(u) \bigr] +\frac{1}{2} \bigl[f_{1}(-\tau)h_{\Gamma^{+}_{p}K}^{p}(u)+f_{2}(- \tau)h_{\Gamma ^{-}_{p}K}^{p}(u) \bigr] \\& \quad =\frac{1}{2} \bigl[f_{1}(\tau)h_{\Gamma^{+}_{p}K}^{p}(u)+f_{2}( \tau )h_{\Gamma^{-}_{p}K}^{p}(u) \bigr] +\frac{1}{2} \bigl[f_{2}(\tau)h_{\Gamma^{+}_{p}K}^{p}(u)+f_{1}( \tau)h_{\Gamma ^{-}_{p}K}^{p}(u) \bigr] \\& \quad =\frac{1}{2}h\bigl(\Gamma^{+}_{p}K, u\bigr)^{p}+ \frac{1}{2}h\bigl(\Gamma^{-}_{p}K, u\bigr)^{p}=h(\Gamma _{p}K, u)^{p}, \end{aligned}$$

i.e., for any \(u\in S^{n-1}\),

$$ h(\Gamma_{p}K, u)^{p}=\frac{1}{2}h\bigl( \Gamma^{\tau}_{p}K, u\bigr)^{p}+\frac{1}{2}h \bigl(\Gamma ^{-\tau}_{p}K, u\bigr)^{p}. $$
(4.5)

From this, if \(\Gamma^{\tau}_{p} K = \Gamma^{\tau}_{p} L \), then \(\Gamma ^{-\tau}_{p} K = \Gamma^{-\tau}_{p} L \). Thus by (4.5) we obtain \(\Gamma _{p}K=\Gamma_{p}L\). This combined with Lemma 4.2 gives (4.3). □

Proof of Theorem 1.2

According to (1.9), we know

$$ \frac{n^{\frac{p}{n}}\widetilde{\Omega}_{-p}(K)^{\frac{n-p}{n}}}{V(K)} = \inf \biggl\{ n\frac{\widetilde{V}_{-p}(K,Q^{*})}{V(K)}V(Q)^{-\frac {p}{n}} : Q\in \mathcal{S}^{n}_{os} \biggr\} . $$
(4.6)

Since \(\Gamma^{\tau}_{p}K=\Gamma^{\tau}_{p}L \), thus from Lemma 4.3, we get, for any \(Q\in\mathcal{S}^{n}_{os} \),

$$ \frac{\widetilde{V}_{-p}(K,Q^{*})}{V(K)}=\frac{\widetilde {V}_{-p}(L,Q^{*})}{V(L)}. $$
(4.7)

Thus from (4.6) and (4.7), we have

$$\frac{\widetilde{\Omega}_{-p}(K)^{\frac{n-p}{n}}}{V(K)} = \frac {\widetilde{\Omega}_{-p}(L)^{\frac{n-p}{n}}}{V(L)}, $$

i.e.,

$$ \biggl( \frac{\widetilde{\Omega}_{-p}(K)}{\widetilde{\Omega}_{-p}(L)} \biggr)^{\frac{n-p}{n}}=\frac{V(K)}{V(L)}. $$
(4.8)

But \(L\in\mathcal{S}^{n}_{os}\), thus taking \(Q=L\) in (4.3), and associated with inequality (2.4), we obtain

$$V(K)= \widetilde{V}_{-p}(K,L)\geq V(K)^{\frac{n+p}{n}}V(L)^{-\frac {p}{n}}, $$

i.e.,

$$V(K)\leq V(L). $$

This combined with (4.8), and noticing \(n>p\), leads to (1.14).

According to the equality condition of (2.4), we see that equality holds in (1.14) if and only if \(K=L \). □

Now we complete the proofs of Theorems 1.3 and 1.4. The following lemmas are required.

Lemma 4.4

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

$$ \Gamma^{+}_{p}\widehat{\nabla}^{\tau}_{p}K= \Gamma^{\tau}_{p}K $$
(4.9)

and

$$ \Gamma^{-}_{p}\widehat{\nabla}^{\tau}_{p}K= \Gamma^{-\tau}_{p}K. $$
(4.10)

Proof

From (1.3) and (3.2), we have, for all \(u\in S^{n-1}\),

$$\begin{aligned} h^{p}_{\Gamma^{+}_{p}\widehat{\nabla}^{\tau}_{p}K}(u) =&\frac {2}{c_{n,p}(n+p)V(\widehat{\nabla}^{\tau}_{p}K)}\int_{S^{n-1}} (u\cdot v)_{+}^{p}\rho_{\widehat{\nabla}^{\tau}_{p}K}(v)^{n+p}\,dv \\ =& \frac{2}{c_{n,p}(n+p)}\int_{S^{n-1}} (u\cdot v)_{+}^{p} \biggl[f_{1}(\tau)\frac{\rho_{K}(v)^{n+p}}{V(K)}+f_{2}(\tau) \frac{\rho _{-K}(v)^{n+p}}{V(-K)} \biggr]\,dv \\ =&f_{1}(\tau)h^{p}_{\Gamma^{+}_{p}K}(u)+f_{2}( \tau)h^{p}_{\Gamma^{+}_{p}(-K)}(u) \\ =&f_{1}(\tau)h^{p}_{\Gamma^{+}_{p}K}(u)+f_{2}( \tau)h^{p}_{\Gamma^{-}_{p}K}(u)=h^{p}_{\Gamma ^{\tau}_{p}K}(u). \end{aligned}$$

This immediately gives (4.9).

Similarly, we know that, for all \(u\in S^{n-1}\),

$$h^{p}_{\Gamma^{-}_{p}\widehat{\nabla}^{\tau}_{p}K}(u)=h^{p}_{\Gamma^{-\tau }_{p}K}(u). $$

This yields (4.10). □

Lemma 4.5

For \(L\in\mathcal{S}^{n}_{o}\), \(p\geq1\), and \(\tau\in(-1,1)\), if L is not origin-symmetric, then there exists \(K\in\mathcal{S}^{n}_{o}\) (for \(\tau =0\), \(K\in\mathcal{S}^{n}_{os}\)) such that

$$\Gamma^{+}_{p}K\subset\Gamma^{\tau}_{p}L,\qquad \Gamma^{-}_{p}K\subset\Gamma ^{-\tau}_{p}L, $$

but

$$\widetilde{\Omega}_{-p}(K)>\widetilde{\Omega}_{-p}(L). $$

Proof

Since L is not origin-symmetric and \(\tau\in(-1,1)\), thus by Corollary 3.2 we know \(\widetilde{\Omega}_{-p}(\widehat{\nabla }^{\tau}_{p}L)>\widetilde{\Omega}_{-p}(L)\). From this, choose \(\varepsilon>0\) such that \(1-\varepsilon>0\), and \(K=(1-\varepsilon)\widehat{\nabla}^{\tau}_{p}L\in\mathcal{S}^{n}_{o}\) (if \(\tau=0\) then \(K\in\mathcal{S}^{n}_{os}\)) satisfies

$$\widetilde{\Omega}_{-p}(K)=\widetilde{\Omega}_{-p}\bigl((1- \varepsilon )\widehat{\nabla}^{\tau}_{p}L\bigr)>\widetilde{ \Omega}_{-p}(L). $$

But by (4.9) and (4.10), and noticing that \(\Gamma^{\pm}_{p}(cM)=c\Gamma^{\pm}_{p}M\) (\(c>0\)), we, respectively, have

$$\Gamma^{+}_{p}K=\Gamma^{+}_{p}(1-\varepsilon)\widehat{ \nabla}^{\tau}_{p}L=(1-\varepsilon)\Gamma^{+}_{p} \widehat{\nabla}^{\tau}_{p}L =(1-\varepsilon) \Gamma^{\tau}_{p}L\subset\Gamma^{\tau}_{p}L $$

and

$$\Gamma^{-}_{p}K=\Gamma^{-}_{p}(1-\varepsilon)\widehat{ \nabla}^{\tau}_{p}L=(1-\varepsilon)\Gamma^{-}_{p} \widehat{\nabla}^{\tau}_{p}L =(1-\varepsilon) \Gamma^{-\tau}_{p}L\subset\Gamma^{-\tau}_{p}L. $$

 □

Proof of Theorem 1.3

Since L is not origin-symmetric and \(\tau \in(-1,1)\), thus by Lemma 4.5, there exists \(K\in\mathcal{S}^{n}_{o}\) such that

$$\Gamma^{+}_{p}K\subset\Gamma^{\tau}_{p}L ,\qquad \Gamma^{-}_{p}K\subset\Gamma ^{-\tau}_{p}L, $$

but

$$\widetilde{\Omega}_{-p}(K)>\widetilde{\Omega}_{-p}(L). $$

Because \(\tau\in(-1,1)\) is equivalent to \(-\tau\in(-1,1)\), we have \(\Gamma^{+}_{p}K\subset\Gamma^{\tau}_{p}L\), \(\Gamma^{-}_{p}K\subset\Gamma^{-\tau }_{p}L\) implying

$$\Gamma^{+}_{p}K\subset\Gamma^{\tau}_{p}L, \qquad \Gamma^{-}_{p}K\subset\Gamma ^{\tau}_{p}L. $$

From this together with (1.5) and (1.7), we obtain, for any \(u\in S^{n-1}\),

$$\begin{aligned} h\bigl(\Gamma^{\tau}_{p}K, u\bigr)^{p} =&f_{1}( \tau)h\bigl(\Gamma^{+}_{p}K, u\bigr)^{p}+f_{2}(\tau)h \bigl(\Gamma ^{-}_{p}K, u\bigr)^{p} \\ < &f_{1}(\tau)h\bigl(\Gamma^{\tau}_{p}L, u \bigr)^{p}+f_{2}(\tau)h\bigl(\Gamma^{\tau}_{p}L, u\bigr)^{p}=h\bigl(\Gamma^{\tau}_{p}L, u \bigr)^{p}, \end{aligned}$$

i.e., \(\Gamma^{\tau}_{p}K\subset\Gamma^{\tau}_{p}L\). □

Lemma 4.6

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

$$ \Gamma_{p}\bigl(\widehat{\nabla}^{\tau}_{p}K\bigr)= \Gamma_{p}K. $$
(4.11)

Proof

From (4.4), (4.9), (4.10), and (4.5), we have, for all \(u\in S^{n-1}\),

$$\begin{aligned} h^{p}_{\Gamma_{p}\widehat{\nabla}^{\tau}_{p}K}(u) =&\frac{1}{2} h^{p}_{\Gamma ^{+}_{p}\widehat{\nabla}^{\tau}_{p}K}(u)+ \frac{1}{2} h^{p}_{\Gamma^{-}_{p}\widehat{\nabla}^{\tau}_{p}K}(u) \\ =&\frac{1}{2} h^{p}_{\Gamma^{\tau}_{p}K}(u)+\frac{1}{2} h^{p}_{\Gamma^{-\tau}_{p}K}(u)=h^{p}_{\Gamma_{p}K}(u). \end{aligned}$$

So (4.11) is obtained. □

Proof of Theorem 1.4

Since L is not origin-symmetric, for \(\tau\in(-1, 1)\), by Corollary 3.2 we know

$$\widetilde{\Omega}_{-p}\bigl(\widehat{\nabla}^{\tau}_{p}L \bigr)>\widetilde{\Omega}_{-p}(L). $$

Choose \(\varepsilon> 0\), such that \(1-\varepsilon>0\) and

$$\widetilde{\Omega}_{-p}\bigl((1-\varepsilon)\widehat{ \nabla}^{\tau}_{p}L\bigr)>\widetilde{\Omega}_{-p}(L). $$

Let \(K=(1-\varepsilon)\widehat{\nabla}^{\tau}_{p} L\), thus \(K\in\mathcal {S}^{n}_{o}\) (if \(\tau=0\) then \(K\in\mathcal{S}^{n}_{os}\)) and \(\widetilde {\Omega}_{-p}(K)>\widetilde{\Omega}_{-p}(L)\).

But from Lemma 4.6 and \(\Gamma_{p}(cM)=c\Gamma_{p}M\) (\(c>0\)), we can get

$$\Gamma_{p}K = \Gamma_{p}(1-\varepsilon)\widehat{ \nabla}^{\tau}_{p}L = (1-\varepsilon)\Gamma_{p} \widehat{\nabla}^{\tau}_{p}L = (1-\varepsilon)\Gamma _{p}L\subset\Gamma_{p}L. $$

 □