Open Access

Shephard type problems for general \(L_{p}\)-centroid bodies

Journal of Inequalities and Applications20152015:287

https://doi.org/10.1186/s13660-015-0812-3

Received: 29 June 2015

Accepted: 1 September 2015

Published: 17 September 2015

Abstract

Lutwak and Zhang proposed the concept of \(L_{p}\)-centroid bodies. Then Feng and Wang gave the notion of general \(L_{p}\)-centroid bodies. In this article, based on the \(L_{p}\)-dual affine surface area, we address Shephard type problems for the general \(L_{p}\)-centroid bodies.

Keywords

Shephard problem \(L_{p}\)-dual affine surface areageneral \(L_{p}\)-centroid body

MSC

52A2052A4052A39

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 [37].
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, 1114]). For the further researches of asymmetric \(L_{p}\) Brunn-Minkowski theory, also see [10, 1527].

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 [2835]).

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

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 by the Natural Science Foundation of China (Grant No. 11371224).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, China Three Gorges University

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