Open Access

Affine inequalities for \(L_{p}\)-mixed mean zonoids

Journal of Inequalities and Applications20162016:223

https://doi.org/10.1186/s13660-016-1167-0

Received: 26 February 2016

Accepted: 7 September 2016

Published: 15 September 2016

Abstract

In this paper, we introduce the \(L_{p}\)-mixed mean zonoid of convex bodies K and L, and we prove some important properties for the \(L_{p}\)-mixed mean zonoid, such as monotonicity, \(\operatorname{GL}(n)\) covariance, and so on. We also establish new affine isoperimetric inequalities for the \(L_{p}\)-mixed mean zonoid.

Keywords

\(L_{p}\)-zonoid \(L_{p}\)-mixed mean zonoidSteiner symmetrizationaffine inequality

MSC

52A3052A40

1 Introduction

Let \(\mathcal{K}^{n}\) denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space \(\mathbb{R}^{n}\). For the set of convex bodies containing the origin in their interiors, we write \(\mathcal{K}^{n}_{o}\). \(\mathcal {K}^{n}_{s}\) denotes the class of o-symmetric members of \(\mathcal {K}^{n}_{o}\) (o denotes the origin in \(\mathbb{R}^{n}\)). Let \(S^{n-1}\) denote the unit sphere in Euclidean space \(\mathbb{R}^{n}\) and let \(V(K)\) denote the n-dimensional volume of a body K. For the standard unit ball B in \(\mathbb{R}^{n}\), we write \(\omega_{n} = V(B)\) for its volume.

If \(K\in\mathcal{K}^{n}\), then its support function, \(h_{K}=h(K, \cdot):\mathbb {R}^{n}\rightarrow(-\infty, \infty)\), is defined by (see [1, 2]) \(h(K,x)=\max\{x \cdot y: y\in K\}, x\in\mathbb{R}^{n}\), where \(x\cdot y\) denotes the standard inner product of x and y.

The zonoids are investigated by many authors (see [35]). The zonoid \(\mathcal{Z}\) is a convex body with support function
$$h_{\mathcal{Z}}(u)=\frac{1}{2} \int_{S^{n-1}}\bigl\vert \langle u,v\rangle\bigr\vert \, \mathrm{d}\mu(v) \quad\mbox{for all } u\in S^{n-1}, $$
where μ is some positive, even Borel measure on \(S^{n-1}\) and \(\langle x,y\rangle\) denotes the standard inner product of vectors x and y in \(\mathbb{R}^{n}\).
For \(K\in\mathcal{K}^{n}\), the mean zonoid, \(\bar{\mathcal{Z}} K\), was defined by Zhang [6]
$$ h_{\bar{\mathcal{Z}}K}(u)=\frac{1}{V(K)^{2}} \int_{K} \int_{K}\bigl\vert \bigl\langle u,(x-y)\bigr\rangle \bigr\vert \,\mathrm{d}x\,\mathrm{d}y \quad\mbox{for all } u\in S^{n-1}, $$
(1.1)
where \(V(K)\) is the volume of the body K.

Further, Zhang [6] proved the affine isoperimetric inequality \(V(\bar{\mathcal{Z}}K)\geq V(\bar{\mathcal{Z}}B_{K})\), where \(B_{K}\) is the n-ball with the same volume as K.

For each convex subset in \(\mathbb{R}^{n}\), it is well known that there is a unique ellipsoid with the following property: The moment of inertia of the ellipsoid and the moment of inertia of the convex set are the same about every 1-dimensional subspace of \(\mathbb{R}^{n}\). This ellipsoid is called the ellipsoid of inertia \(\Gamma_{2}K\) (also called the Legendre ellipsoid) of the convex set. Namely, between the convex body K and the ellipsoid of inertia \(\Gamma_{2}K\) we have
$$\int_{K}\bigl\vert \langle x,y\rangle\bigr\vert ^{2}\,\mathrm{d}x= \int_{\Gamma_{2}K}\bigl\vert \langle x,y\rangle\bigr\vert ^{2}\,\mathrm{d}x, \quad\forall y\in\mathbb{R}^{n}. $$
The Legendre ellipsoid and its polar (the Binet ellipsoid) are well-known concepts from classical mechanics. See [79] for historical references.
A non-negative finite Borel measure μ on the unit sphere \(S^{n-1}\) is said to be isotropic if it has the same moment of inertia about all lines through the origin or, equivalently, if, for all \(x\in\mathbb{R}^{n}\),
$$\vert x\vert ^{2}= \int_{S^{n-1}}\bigl\vert \langle x,u\rangle\bigr\vert ^{2}\,\mathrm{d}\mu(u), $$
where \(\vert \cdot \vert \) denotes the standard Euclidean norm on \(\mathbb{R}^{n}\).
Based on the background of mechanics properties, the notion of \(L_{p}\)-zonoids was given by Schneider and Weil [10]. For \(p\geq 1\), an \(L_{p}\)-zonoid was defined by
$$ h_{\mathcal{Z}_{p}K}(u)^{p}= \int_{S^{n-1}}\bigl\vert \langle u,v\rangle\bigr\vert ^{p}\,\mathrm{d}\mu(v) \quad\mbox{for all } u\in S^{n-1}, $$
(1.2)
where μ is some positive, even Borel measure on \(S^{n-1}\). We also refer to [4, 11].
Xi, Guo and Leng [12] considered an extension for a class of bodies \(\bar{\mathcal{Z}}_{p}K\) named \(L_{p}\)-mean zonoids as follows: For \(K\in\mathcal{K}^{n}\) and \(p\geq1\), the \(L_{p}\)-mean zonoid, \(\bar {\mathcal{Z}}_{p}K\), of K is defined by
$$ h_{\bar{\mathcal{Z}}_{p}K}(z)= \biggl(\frac{1}{V(K)^{2}} \int_{K} \int _{K}\bigl\vert \bigl\langle z,(x-y)\bigr\rangle \bigr\vert \,\mathrm{d}x\,\mathrm{d}y \biggr)^{\frac{1}{p}} \quad\mbox{for all } z\in \mathbb{R}^{n}/\{o\}. $$
(1.3)
For \(p=1\), the body \(\bar{\mathcal{Z}}K\) is the mean zonoid of K [6]. Xi et al. also showed that \(\bar{\mathcal{Z}}_{p}K\) is an \(L_{p}\)-zonoid, and established the following affine isoperimetric inequality: For \(K\in\mathcal{K}^{n}\) and \(p\geq1\),
$$ V(\bar{\mathcal{Z}}_{p}K)\geq C_{n,p}V(K), $$
(1.4)
with equality if and only if K is an ellipsoid. Here \(C_{n,p}\) is a constant depending on p and the dimension n.

The main purpose of this paper is to introduce the notion of \(L_{p}\)-mixed mean zonoids, which extends the \(L_{p}\)-mean zonoids by Xi, Guo and Leng [12].

Definition 1.1

For \(K,L\in\mathcal{K}^{n}\) and \(p\geq1\), \(L_{p}\)-mixed mean zonoids, \(\bar {\mathcal{Z}}_{p}(K,L)\), of K and L are defined by
$$ h_{\bar{\mathcal{Z}}_{p}(K,L)}(z)= \biggl(\frac{1}{V(K)V(L)} \int_{K} \int _{L}\bigl\vert \bigl\langle z,(x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y \biggr)^{1/p} \quad\mbox{for all } z\in\mathbb{R}^{n}\backslash\{o\}. $$
(1.5)

Notice that when \(K=L\), (1.5) is defined by Xi, Guo, and Leng in [12].

Let \(\omega_{p}=\pi^{p/2}/\Gamma(1+p/2)\) and
$$C(n,p)= \biggl(\frac{2^{n+p}(2n+p+1)\omega_{2n+p}\omega _{2n+p+1}}{(n+1)\omega_{2}^{2}\omega_{n}^{2}\omega_{n+1}\omega _{p-1}\omega_{n+p-1}} \biggr)^{n/p}. $$

For the \(L_{p}\)-mixed mean zonoids, our main result is to establish the more general affine inequality as follows.

Theorem 1.2

Let \(K,L\in\mathcal{K}^{n}_{o}\) and \(p\geq1\). If \(K\subseteq L\), then
$$ V\bigl(\bar{\mathcal{Z}}_{p}(K,L)\bigr)\geq C(n,p)V(K)^{\frac{n+p}{p}}V(L)^{-\frac{n}{p}}, $$
(1.6)
with equality if and only if \(K=L\) is an ellipsoid.

If \(L=K\), then the above inequality (1.6) reduces to the affine inequality (1.4).

An immediate consequence of Theorem 1.2 is the following.

Corollary 1.3

Let \(K,L\in\mathcal{K}^{n}_{o}\). If \(K\subseteq L\), then
$$ \biggl(\frac{V(\bar{\mathcal{Z}}_{1}(K,L))}{V(K)} \biggr)^{\frac {1}{n}}\geq \bigl(C(n,1) \bigr)^{\frac{1}{n}} \biggl(\frac{\widetilde {V}_{1}(L,K)}{V(L)} \biggr)^{n}, $$
(1.7)
with equality if and only if \(K=L\) is an ellipsoid.

2 Notation and preliminaries

We refer to the books Gardner [1] and Schneider [2] for some terminologies and notations as regards convex bodies.

The Hausdorff metric \(\delta_{H}(K,L)\) between sets \(K,L\in\mathcal {K}^{n}\) can be defined by
$$\delta_{H}(K,L)=\sup_{x\in S^{n-1}}\bigl\vert h(K,x)-h(L,x)\bigr\vert . $$
A set K is star-shaped (about \(x_{0}\in K\)) if there exists \(x_{0}\in K\), such that the line segment from \(x_{0}\) to any point \(x \in K\) is contained in K. If K is a compact star-shaped (about the origin) set, then its radial function \(\rho_{K}(x,z):\mathbb{R}^{n}\backslash\{x\}\rightarrow [0,\infty)\) with respect to x is defined by
$$ \rho_{K}(x,z)=\max\{c: x+cz\in Z\} \quad\mbox{for all } z\in \mathbb {R}^{n} \backslash\{x\}. $$
(2.1)

If \(\rho_{K}\) is positive and continuous, then K will be called a star body (about the origin), and \(\mathcal{S}^{n}\) denotes the set of star bodies in \(\mathbb{R}^{n}\). We will use \(\mathcal{S}^{n}_{o}\) to denote the subset of star bodies in \(\mathcal{S}^{n}\) containing the origin in their interiors. Two star bodies K and L are said to be dilates of one another if \(\rho_{K}(u)/\rho_{L}(u)\) is independent of \(u\in S^{n-1}\).

For \(K,L\in\mathcal{S}^{n}_{o}\), \(p>0\), and \(\lambda,\mu\geq0\) (not both zero), the \(L_{p}\)-radial combination, \(\lambda\circ K\widetilde{+}_{p}\mu\circ L\in\mathcal{S}^{n}_{o}\), is defined by
$$ \rho(\lambda\circ K\widetilde{+}_{p}\mu\circ L, \cdot)^{p}=\lambda\rho (K,\cdot)^{p}+\mu\rho(L, \cdot)^{p}. $$
The dual \(L_{p}\)-mixed volume \(\widetilde{V}_{p}(K,L)\) of \(K,L\) was defined by
$$ \widetilde{V}_{p}(K,L)=\frac{p}{n}\lim _{\varepsilon\rightarrow 0^{+}}\frac{V(K\widetilde{+}_{p}\varepsilon\circ L)-V(K)}{\varepsilon}. $$
(2.2)
The integral representation of \(\widetilde{V}_{p}(K,L)\) was proved by
$$\widetilde{V}_{p}(K,L)=\frac{1}{n} \int_{S^{n-1}}\rho_{K}(u)^{n-p}\rho _{L}(u)^{p}\,\mathrm{d}S(u). $$
The \(L_{p}\)-Minkowski inequality for the dual \(L_{p}\)-mixed volume is: If \(K,L\in\mathcal{S}^{n}_{o}\) and \(0< p< n\), then
$$ \widetilde{V}_{p}(K,L)\leq V(K)^{\frac{n-p}{n}}V(L)^{\frac{p}{n}}, $$
(2.3)
with equality if and only if K and L are dilates.

The difference body \(D(K,L)\) of K and L is defined by \(D(K,L)=K-L=\{x-y: x\in K, y\in L\}\). Particularly, \(DK=K-K=\{x-y: x\in K, y\in K\}\).

For a star-shaped K and \(p\geq1\), the \(L_{p}\)-centroid body of K, \(\Gamma_{p}K\) is the origin-symmetric convex body with the support function
$$ h_{\Gamma_{p}K}(u)^{p}=\frac{1}{V(K)} \int_{K}\bigl\vert \langle u,x\rangle \bigr\vert ^{p}\,\mathrm{d}x=\frac{1}{(n+p)V(K)} \int_{S^{n-1}}\bigl\vert \langle u,v\rangle \bigr\vert ^{p}\rho_{K}(v)^{n+p}\,\mathrm{d}v, $$
(2.4)
for all \(u\in S^{n-1}\).
For \(K, L\in\mathcal{K}^{n}\), \(p>-1\), and \(K\subseteq L\), the generalized radial pth mean body, \(R_{p}(K,L, \lambda_{n})\), is defined by (see [13, 14])
$$ \rho_{R_{p}(K,L, \lambda_{n})}(u)= \biggl(\frac{1}{V(K)} \int_{K}\rho _{L}(x,u)^{p}\,\mathrm{d}x \biggr)^{1/p}, $$
(2.5)
for all \(u\in S^{n-1}\), where \(\lambda_{n}\) is the n-dimensional Lebesgue measure in \(\mathbb{R}^{n}\).

Lemma 2.1

([13])

For \(K,L\in\mathcal{K}^{n}\) and \(x\in\mathbb{R}^{n}\), the parallel section function on \(\mathbb{R}^{n}\) is defined by \(A_{K,L}(x):=V(K\cap(L+x))\). Then \(g_{K,L}(x)=A_{K,L}(x)^{\frac{1}{n}}\) is concave on its support.

If \(K\subseteq L\) and \(p>0\), then for all \(u\in S^{n-1}\) (see [13, 14])
$$ \int_{K}\rho_{L}(x,u)^{p}\,\mathrm{d}x=p \int_{0}^{\infty }A_{K,L}(ru)r^{p-1}\, \mathrm{d}r=p \int_{0}^{\rho _{DK}(u)}A_{K,L}(ru)r^{p-1}\, \mathrm{d}r. $$
(2.6)
For \(p,q>0\), define the β-function by
$$\beta(p,q)= \int_{0}^{1}t^{p-1}(1-t)^{q-1}\, \mathrm{d}t=\frac{\Gamma(p)\Gamma (q)}{\Gamma(p+q)}. $$

Lemma 2.2

If \(\lambda>0\) and \(A_{K,L}(r,u):=V(K\cap(L+ru))\), then
$$ \int_{0}^{\infty}A_{K,L}(r,u)r^{\lambda}\, \mathrm{d}r\leq n^{\lambda+1}\beta (\lambda+1,n)V(K)^{-\lambda} \biggl( \int_{0}^{\infty}A_{K,L}(r,u)\,\mathrm{d}r \biggr)^{\lambda+1}. $$
(2.7)

Proof

If
$$F(\lambda)= \biggl(\frac{1}{\beta(\lambda+1,n)} \int_{0}^{\infty}\frac {A_{K,L}(r,u)}{A_{K,L}(0,u)}r^{\lambda}\, \mathrm{d}r \biggr)^{\frac{1}{\lambda+1}}, $$
then \(F(\lambda)\) is a decreasing function on \((-1,+\infty)\). Particularly, if \(\lambda>0\), then \(F(\lambda)\leq F(0)\) with equality if and only if
$$1- \biggl(\frac{A_{K,L}(r,u)}{A_{K,L}(0,u)} \biggr)^{\frac{1}{n-1}}=\frac{r}{F(0)}. $$
Then
$$ \int_{0}^{\infty}A_{K,L}(r,u)r^{\lambda}\, \mathrm{d}r\leq n^{\lambda+1}\beta (\lambda+1,n)V(K)^{-\lambda} \biggl( \int_{0}^{\infty}A_{K,L}(r,u)\,\mathrm{d}r \biggr)^{\lambda+1}, $$
(2.8)
with equality if and only if \(A_{K,L}(r,u)=V(K) (1-\frac{rV(K)}{n\int_{K}\rho_{L}(x,u)\,\mathrm{d}x} )^{n-1}\). □

3 \(L_{p}\)-Mixed mean zonoids

Suppose \(K,L\in\mathcal{K}^{n}\) and \(p\geq1\). Define \(\bar{\mathcal {Z}}_{\infty}(K,L)\) by
$$h_{\bar{\mathcal{Z}}_{\infty}(K,L)}(u)=\max_{x\in K,y\in L}\bigl\vert \bigl\langle u,(x-y)\bigr\rangle \bigr\vert \quad \mbox{for all } u\in S^{n-1}. $$
Since \(\bar{\mathcal{Z}}_{\infty}(K,L)=D(K,L)\), it follows from Jensen’s inequality that
$$\bar{\mathcal{Z}}_{p}(K,L)\subseteq\bar{\mathcal{Z}}_{q}(K,L) \subseteq D(K,L) \quad\mbox{for } 1\leq p\leq q. $$

Property 3.1

Let \(K,L\in\mathcal{K}^{n}\) with \(K\subseteq L\). If \(p\geq1\), then
$$ \bar{\mathcal{Z}}_{p}(K,L)= \biggl(\frac{V(R_{n+p}(K,L,\lambda _{n}))}{(n+p)V(L)} \biggr)^{1/p}\Gamma_{p}\bigl(R_{n+p}(K,L, \lambda_{n})\bigr). $$
(3.1)

Proof

From (1.5), (2.1), the Fubini theorem, (2.4), and (2.5), passing to spherical coordinates we have
$$\begin{aligned} h_{\bar{\mathcal{Z}}_{p}(K,L)}(z) =& \biggl(\frac{1}{V(K)V(L)} \int_{K} \int _{L}\bigl\vert \bigl\langle z,(x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y \biggr)^{1/p} \\ =& \biggl(\frac{1}{V(K)V(L)} \int_{K} \int_{S^{n-1}} \int_{0}^{\rho _{L}(y,v)}\bigl\vert \langle z,v\rangle\bigr\vert ^{p}r^{n+p-1}\,\mathrm{d}r\,\mathrm{d}v\,\mathrm{d}y \biggr)^{1/p} \\ =& \biggl(\frac{1}{(n+p)V(K)V(L)} \int_{S^{n-1}}\bigl\vert \langle z,v\rangle \bigr\vert ^{p} \int_{K}\rho_{L}(y,v)^{n+p}\,\mathrm{d}y\, \mathrm{d}v \biggr)^{1/p} \end{aligned}$$
(3.2)
$$\begin{aligned} =& \biggl(\frac{1}{(n+p)V(L)} \int_{S^{n-1}}\bigl\vert \langle z,v\rangle\bigr\vert ^{p}\rho _{R_{n+p}(K,L,\lambda_{n})}(v)^{n+p}\,\mathrm{d}v \biggr)^{1/p} \\ =& \biggl(\frac{V(R_{n+p}(K,L,\lambda_{n}))}{(n+p)V(L)} \biggr)^{1/p}h_{\Gamma_{p}(R_{n+p}(K,L,\lambda_{n}))}(z). \end{aligned}$$
(3.3)
Combining with (3.3), we have
$$ \bar{\mathcal{Z}}_{p}(K,L)= \biggl(\frac{V(R_{n+p}(K,L,\lambda _{n}))}{(n+p)V(L)} \biggr)^{1/p}\Gamma_{p}\bigl(R_{n+p}(K,L,\lambda _{n})\bigr). $$
 □
Together (2.6) with (3.2), if \(K\subseteq L\), then
$$ h_{\bar{\mathcal{Z}}_{p}(K,L)}(z)= \biggl(\frac{1}{V(K)V(L)} \int _{S^{n-1}}\bigl\vert \langle z,u\rangle\bigr\vert ^{p} \int_{0}^{\infty }A_{K,L}(ru)r^{n+p-1}\, \mathrm{d}r\,\mathrm{d}u \biggr)^{1/p}. $$
(3.4)
Let
$$C_{K,L}(n,p)= \biggl(\frac{n^{n+p}(n+p)\beta(n+p,n)V(R_{1}(K,L,\lambda _{n}))}{V(L)} \biggr)^{1/p}. $$

Property 3.2

Let \(K,L\in\mathcal{K}^{n}\) and \(p\geq1\). If \(K\subseteq L\),then
$$\bar{\mathcal{Z}}_{p}(K,L)\subseteq C_{K,L}(n,p)\Gamma _{p}\bigl(R_{1}(K,L,\lambda_{n})\bigr). $$

Proof

By (3.4), (2.7), (2.6), (2.5), and (2.4), we have
$$\begin{aligned} h_{\bar{\mathcal{Z}}_{p}(K,L)}(u) =& \biggl(\frac{1}{V(K)V(L)} \int _{S^{n-1}}\bigl\vert \langle u,v\rangle\bigr\vert ^{p} \int_{0}^{\infty }A_{K,L}(ru)r^{n+p-1}\, \mathrm{d}r\,\mathrm{d}v \biggr)^{1/p}\\ \leq& \biggl(\frac{n^{n+p}\beta(n+p,n)}{V(K)^{n+p}V(L)} \int _{S^{n-1}}\bigl\vert \langle u,v\rangle\bigr\vert ^{p} \biggl( \int_{0}^{\infty }A_{K,L}(r,v)\,\mathrm{d}r \biggr)^{n+p} \biggr)^{1/p}\\ =& \biggl(\frac{n^{n+p}\beta(n+p,n)}{V(L)} \int_{S^{n-1}}\bigl\vert \langle u,v\rangle\bigr\vert ^{p} \biggl(\frac{1}{V(K)} \int_{K}\rho_{L}(x,v)\,\mathrm{d}x \biggr)^{n+p} \biggr)^{1/p}\\ =& \biggl(\frac{n^{n+p}\beta(n+p,n)}{V(L)} \int_{S^{n-1}}\bigl\vert \langle u,v\rangle\bigr\vert ^{p}\rho^{n+p}_{R_{1}(K,L,\lambda_{n})}(v)\,\mathrm{d}v \biggr)^{1/p}\\ =&C_{K,L}(n,p)h_{\Gamma(R_{1}(K,L,\lambda_{n}))}(u). \end{aligned}$$
This implies \(h_{\bar{\mathcal{Z}}_{p}(K,L)}(u)\leq C_{K,L}(n,p)h_{\Gamma _{p}(R_{1}(K,L,\lambda_{n}))}(u)\). □

The following property will be used to prove that \(\bar{\mathcal {Z}}_{p}: \mathcal{K}^{n}\times\mathcal{K}^{n}\rightarrow\mathcal {K}^{n}\) is continuous.

Property 3.3

If \(p\geq1, K_{i},L_{i}\in\mathcal{K}^{n}\) and \(K_{i}\rightarrow K\in\mathcal{K}^{n}, L_{i}\rightarrow L\in\mathcal{K}^{n}\), then
$$\bar{\mathcal{Z}}_{p}(K_{i},L_{i})\rightarrow \bar{\mathcal{Z}}_{p}(K,L). $$

Proof

Since \(K_{i}\rightarrow K\), \(\{K_{i}\}\) are uniformly bounded. Thus there is \(R_{K}>0\), such that \(K_{i}\subseteq R_{K}B^{n}\). Similarly, \(L_{i}\subseteq R_{L}B^{n}\) with \(R_{L}>0\). Taking (1.5) together with Minkowski’s inequality, it follows that for \(u_{0}\in S^{n-1}\)
$$\begin{aligned} &\bigl\vert h_{\bar{\mathcal{Z}}_{p}(K_{i},L_{i})}(u_{0})-h_{\bar{\mathcal {Z}}_{p}(K,L)}(u_{0}) \bigr\vert \\ &\quad = \biggl\vert \biggl(\frac{1}{V(K_{i})V(L_{i})} \int_{R_{K}B^{n}} \int _{R_{L}B^{n}}\textbf{1}_{K_{i}}(x)\textbf{1}_{L_{i}}(y) \bigl\vert \bigl\langle u_{0},(x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y \biggr)^{1/p} \\ &\qquad{} - \biggl(\frac{1}{V(K)V(L)} \int_{R_{K}B^{n}} \int _{R_{L}B^{n}}\textbf{1}_{K}(x)\textbf{1}_{L}(y) \bigl\vert \bigl\langle u_{0},(x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y \biggr)^{1/p} \biggr\vert \\ &\quad \leq \biggl(\frac{1}{V(K_{i})V(L_{i})} \int_{R_{K}B^{n}} \int _{R_{L}B^{n}}\bigl\vert \textbf{1}_{K_{i}}(x) \textbf{1}_{L_{i}}(y)-\textbf {1}_{K}(x)\textbf{1}_{L}(y) \bigr\vert \bigl\vert \bigl\langle u_{0},(x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y \biggr)^{1/p}\\ &\qquad{} + \biggl\vert \biggl( \biggl(\frac{1}{V(K_{i})V(L_{i})}-\frac {1}{V(K)V(L)} \biggr) \int_{R_{K}B^{n}} \int_{R_{L}B^{n}}\textbf{1}_{K}(x)\textbf{1}_{L}(y) \bigl\vert \bigl\langle u_{0},(x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y \biggr)^{1/p} \biggr\vert . \end{aligned}$$
This means \(h_{\bar{\mathcal{Z}}_{p}(K_{i},L_{i})}(u_{0})\rightarrow h_{\bar{\mathcal{Z}}_{p}(K,L)}(u_{0})\), which is the desired result. □

The following property will prove that \(\bar{\mathcal{Z}}_{p}: \mathcal {K}^{n}\times\mathcal{K}^{n}\rightarrow\mathcal{K}^{n}\) is \(\operatorname{GL}(n)\) covariant.

Property 3.4

If \(p\geq1\), \(K\in\mathcal{K}^{n}\) and \(T\in\operatorname{GL}(n)\), then
$$\bar{\mathcal{Z}}_{p}(T K,T L)=T\bigl(\bar{\mathcal{Z}}_{p}(K,L) \bigr). $$

Proof

Combining (1.5) with the substitution \(x=Tx_{1}, y=Ty_{1}\), we obtain
$$\begin{aligned} h_{\bar{\mathcal{Z}}_{p}(T K,T L)}(z) =& \biggl(\frac{1}{V(T K)V(T L)} \int _{TK} \int_{TL}\bigl\vert \bigl\langle z,(x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y \biggr)^{1/p}\\ =& \biggl(\frac{1}{V(T K)V(T L)}\vert T\vert ^{2} \int_{K} \int_{L}\bigl\vert \bigl\langle z,(Tx-T y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x_{1}\,\mathrm{d}y_{1} \biggr)^{1/p}\\ =& \biggl(\frac{1}{V(K)V(L)} \int_{K} \int_{L}\bigl\vert \bigl\langle T^{t}z,(x_{1}-y_{1}) \bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x_{1}\, \mathrm{d}y_{1} \biggr)^{1/p}\\ =&h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(T^{t}z\bigr)\\ =&h_{T\bar{\mathcal{Z}}_{p}(K,L)}(z). \end{aligned}$$
Namely, \(\bar{\mathcal{Z}}_{p}(T K,T L)=T(\bar{\mathcal{Z}}_{p}(K,L))\). □

4 Proof of main result

If \(u\in S^{n-1}\), then we denote by \(u^{\perp}\) the \((n-1)\)-dimensional subspace orthogonal to u, by \(l_{u}\) the line through o parallel to u, and by \(l_{u}(x)\) the line through the point x parallel to u. We denote by \(K_{u}\) the image of the orthogonal projection of K onto \(u^{\perp}\) for a convex body K. Let \(\overline{l}_{u}(K; y'): K_{u}\rightarrow\mathbb{R}\) and \(\underline{l}_{u}(K; y'): K_{u}\rightarrow\mathbb{R}\) for the overgraph and undergraph functions of K in the direction u; namely,
$$K=\bigl\{ y'+tu: -\underline{l}_{u}\bigl(K; y'\bigr)\leq t\leq\overline{l}_{u}\bigl(K;y' \bigr)\mbox{ for } y'\in K_{u}\bigr\} . $$
Thus, the overgraph and undergraph functions of the Steiner symmetrical \(S_{u}\) of \(K\in\mathcal{K}^{n}\) in direction u are defined by
$$\overline{l}_{u}\bigl(S_{u}K; y'\bigr)= \underline{l}_{u}\bigl(S_{u}K; y'\bigr)= \frac {1}{2} \bigl(\overline{l}_{u}\bigl(K; y' \bigr)+\underline{l}_{u}\bigl(K; y'\bigr) \bigr). $$

For \(y'\in K_{u}\), \(m_{y'}=m_{y'}(u)\) denotes \(m_{y'}(u)=\frac{1}{2} (\overline{l}_{u}(K; y')-\underline{l}_{u}(K; y') )\). Let the midpoint of the chord \(K\cap l_{u}(y')\) be \(y'+m_{y'}(u)u\), note that \(l_{u}(y')\) is the line through \(y'\) parallel to u, and let the length \(\vert K\cap l_{u}(y')\vert \) of this chord be \(\sigma_{y'}=\sigma_{y'}(u)\). For \(x=(x',s)\in\mathbb{R}^{n-1}\times\mathbb{R}\), we write \(h_{K}(x',s)\) throughout this section.

Lemma 4.1

([15])

If \(K\in\mathcal{K}^{n}_{o}\), \(u\in S^{n-1}\) and \(y'\in\operatorname{relint}K_{u}\), then
$$\begin{aligned}& \overline{l}_{u}\bigl(K; y'\bigr)=\min _{x'\in u^{\perp}}\bigl\{ h_{K}\bigl(x',1\bigr)- \bigl\langle x',y'\bigr\rangle \bigr\} , \end{aligned}$$
(4.1)
$$\begin{aligned}& \underline{l}_{u}\bigl(K; y'\bigr)=\min _{x'\in u^{\perp}}\bigl\{ h_{K}\bigl(x',-1\bigr)- \bigl\langle x',y'\bigr\rangle \bigr\} . \end{aligned}$$
(4.2)

Lemma 4.2

If \(K\in\mathcal{K}^{n}, p\geq1\), \(u\in S^{n-1}\), and \(z'_{1},z'_{2}\in u^{\perp}\), then
$$\begin{aligned}& h_{\bar{\mathcal{Z}}_{p}(S_{u}K,S_{u}L)} \biggl(\frac {z'_{1}+z'_{2}}{2},1 \biggr)\leq \frac{1}{2}h_{\bar{\mathcal {Z}}_{p}(K,L)}\bigl(z'_{1},1\bigr)+ \frac{1}{2}h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{2},-1\bigr), \end{aligned}$$
(4.3)
$$\begin{aligned}& h_{\bar{\mathcal{Z}}_{p}(S_{u}K,S_{u}L)} \biggl(\frac {z'_{1}+z'_{2}}{2},-1 \biggr)\leq \frac{1}{2}h_{\bar{\mathcal {Z}}_{p}(K,L)}\bigl(z'_{1},1\bigr)+ \frac{1}{2}h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{2},-1\bigr). \end{aligned}$$
(4.4)
Equality in (4.3) or (4.4) implies that all of the chords of K and L parallel to u have midpoints that lie in a hyperplane, respectively.

Proof

We only prove (4.3). Inequality (4.4) can be established in the same way. It follows from the definition of the \(L_{p}\)-mixed mean zonoid that
$$\begin{aligned} &h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{1},1\bigr)\\ &\quad = \biggl(\frac{1}{V(K)V(L)} \int_{K} \int_{L}\bigl\vert \bigl\langle \bigl(z'_{1},1 \bigr),(x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y \biggr)^{1/p}\\ &\quad = \biggl(\frac{1}{V(K)V(L)}\\ &\qquad{}\times \int_{K_{u}} \int_{m_{y'}-\frac{\sigma _{y'}}{2}}^{m_{y'}+\frac{\sigma_{y'}}{2}} \int_{L_{u}} \int_{m_{x'}-\frac {\sigma_{x'}}{2}}^{m_{x'}+\frac{\sigma_{x'}}{2}} \bigl\vert \bigl\langle \bigl(z'_{1},1\bigr),\bigl(\bigl(x',s_{1} \bigr)-\bigl(y',s_{2}\bigr)\bigr)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}s_{1}\,\mathrm{d}x'\, \mathrm{d}s_{2}\,\mathrm{d}y' \biggr)^{1/p}\\ &\quad = \biggl(\frac{1}{V(K)V(L)} \int_{K_{u}} \int_{m_{y'}-\frac{\sigma _{y'}}{2}}^{m_{y'}+\frac{\sigma_{y'}}{2}} \int_{L_{u}} \int_{m_{x'}-\frac {\sigma_{x'}}{2}}^{m_{x'}+\frac{\sigma_{x'}}{2}} \bigl\vert \bigl\langle z'_{1},\bigl(x'-y'\bigr) \bigr\rangle +s_{1}-s_{2}\bigr\vert ^{p}\, \mathrm{d}s_{1}\,\mathrm{d}x'\,\mathrm{d}s_{2} \,\mathrm{d}y' \biggr)^{1/p}\\ &\quad = \biggl(\frac{1}{V(K)V(L)}\\ &\qquad{}\times\int_{K_{u}} \int_{-\frac{\sigma _{y'}}{2}}^{\frac{\sigma_{y'}}{2}} \int_{L_{u}} \int_{-\frac{\sigma _{x'}}{2}}^{\frac{\sigma_{x'}}{2}} \bigl\vert \bigl\langle z'_{1},\bigl(x'-y'\bigr) \bigr\rangle +t_{1}-t_{2}+m_{x'}-m_{y'} \bigr\vert ^{p}\,\mathrm{d}t_{1}\,\mathrm{d}x' \,\mathrm{d}t_{2}\,\mathrm{d}y' \biggr)^{1/p}\\ &\quad = \biggl(\frac{1}{V(S_{u}K)V(S_{u}L)} \int_{S_{u}K} \int _{S_{u}L}\bigl\vert \bigl\langle z'_{1}, \bigl(x'-y'\bigr)\bigr\rangle +t_{1}-t_{2}+m_{x'}-m_{y'} \bigr\vert ^{p}\,\mathrm{d}t_{1}\,\mathrm{d}x' \,\mathrm{d}t_{2}\,\mathrm{d}y' \biggr)^{1/p}, \end{aligned}$$
by \(t_{1}=-m_{x'}+s_{1}, t_{2}=-m_{y'}+s_{2}\).
$$\begin{aligned} &h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{2},-1\bigr)\\ &\quad = \biggl(\frac{1}{V(K)V(L)} \int_{K} \int_{L}\bigl\vert \bigl\langle \bigl(z'_{2},-1 \bigr),(x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y \biggr)^{1/p}\\ &\quad = \biggl(\frac{1}{V(K)V(L)}\\ &\qquad{}\times\int_{K_{u}} \int_{m_{y'}-\frac{\sigma _{y'}}{2}}^{m_{y'}+\frac{\sigma_{y'}}{2}} \int_{L_{u}} \int_{m_{x'}-\frac {\sigma_{x'}}{2}}^{m_{x'}+\frac{\sigma_{x'}}{2}} \bigl\vert \bigl\langle \bigl(z'_{2},-1\bigr),\bigl(\bigl(x',s_{1} \bigr)-\bigl(y',s_{2}\bigr)\bigr)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}s_{1}\,\mathrm{d}x'\, \mathrm{d}s_{2}\,\mathrm{d}y' \biggr)^{1/p}\\ &\quad = \biggl(\frac{1}{V(K)V(L)} \int_{K_{u}} \int_{m_{y'}-\frac{\sigma _{y'}}{2}}^{m_{y'}+\frac{\sigma_{y'}}{2}} \int_{L_{u}} \int_{m_{x'}-\frac {\sigma_{x'}}{2}}^{m_{x'}+\frac{\sigma_{x'}}{2}} \bigl\vert \bigl\langle z'_{2},\bigl(x'-y'\bigr)\bigr\rangle -s_{1}+s_{2}\bigr\vert ^{p}\, \mathrm{d}s_{1}\,\mathrm{d}x'\,\mathrm{d}s_{2} \,\mathrm{d}y' \biggr)^{1/p}\\ &\quad = \biggl(\frac{1}{V(K)V(L)}\\ &\qquad{}\times\int_{K_{u}} \int_{-\frac{\sigma _{y'}}{2}}^{\frac{\sigma_{y'}}{2}} \int_{L_{u}} \int_{-\frac{\sigma _{x'}}{2}}^{\frac{\sigma_{x'}}{2}} \bigl\vert \bigl\langle z'_{2},\bigl(x'-y'\bigr)\bigr\rangle +t_{1}-t_{2}-m_{x'}+m_{y'}\bigr\vert ^{p}\,\mathrm{d}t_{1}\,\mathrm{d}x'\, \mathrm{d}t_{2}\,\mathrm{d}y' \biggr)^{1/p}\\ &\quad = \biggl(\frac{1}{V(S_{u}K)V(S_{u}L)} \int_{S_{u}K} \int _{S_{u}L}\bigl\vert \bigl\langle z'_{2}, \bigl(x'-y'\bigr)\bigr\rangle +t_{1}-t_{2}-m_{x'}+m_{y'} \bigr\vert ^{p}\,\mathrm{d}t_{1}\,\mathrm{d}x' \,\mathrm{d}t_{2}\,\mathrm{d}y' \biggr)^{1/p}. \end{aligned}$$
Let \(t_{1}=m_{x'}-s_{1}, t_{2}=m_{y'}-s_{2}\). Thus, combining with Minkowski’s inequality we have
$$\begin{aligned} &2h_{\bar{\mathcal{Z}}_{p}(S_{u}K,S_{u}L)} \biggl(\frac {z'_{1}+z'_{2}}{2},1 \biggr)\\ &\quad =2 \biggl(\frac{1}{V(S_{u}K)V(S_{u}L)} \int_{S_{u}K} \int _{S_{u}L} \biggl\vert \biggl\langle \biggl( \frac{z'_{1}+z'_{2}}{2},1 \biggr),(x-y) \biggr\rangle \biggr\vert ^{p}\, \mathrm{d}x\,\mathrm{d}y \biggr)^{1/p}\\ &\quad = \biggl(\frac{1}{V(S_{u}K)V(S_{u}L)} \int_{S_{u}K} \int _{S_{u}L}\bigl\vert \bigl\langle \bigl(z'_{1}+z'_{2} \bigr),\bigl(x'-y'\bigr)\bigr\rangle +2t_{1}-2t_{2} \bigr\vert ^{p}\,\mathrm{d}t_{1}\,\mathrm{d}x' \,\mathrm{d}t_{2}\,\mathrm{d}y' \biggr)^{1/p}\\ &\quad \leq \biggl(\frac{1}{V(S_{u}K)V(S_{u}L)} \int_{S_{u}K} \int _{S_{u}L}\bigl\vert \bigl\langle z'_{1}, \bigl(x'-y'\bigr)\bigr\rangle +t_{1}-t_{2}+m_{x'}-m_{y'} \bigr\vert ^{p}\,\mathrm{d}t_{1}\,\mathrm{d}x' \,\mathrm{d}t_{2}\,\mathrm{d}y' \biggr)^{1/p}\\ &\qquad{} + \biggl(\frac{1}{V(S_{u}K)V(S_{u}L)}\\ &\qquad{}\times\int_{S_{u}K} \int _{S_{u}L}\bigl\vert \bigl\langle z'_{2}, \bigl(x'-y'\bigr)\bigr\rangle +t_{1}-t_{2}-m_{x'}+m_{y'} \bigr\vert ^{p}\,\mathrm{d}t_{1}\,\mathrm{d}x' \,\mathrm{d}t_{2}\,\mathrm{d}y' \biggr)^{1/p}\\ &\quad =h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{1},1\bigr)+h_{\bar{\mathcal {Z}}_{p}(K,L)} \bigl(z'_{2},-1\bigr). \end{aligned}$$
From the condition of inequality in Minkowski’s inequality, we know that equality in (4.3) or (4.4) holds if and only if for \(\lambda\geq0\), we have
$$\bigl\langle z'_{1},\bigl(x'-y' \bigr)\bigr\rangle +t_{1}-t_{2}+m_{x'}-m_{y'}= \lambda \bigl(\bigl\langle z'_{2},\bigl(x'-y' \bigr)\bigr\rangle +t_{1}-t_{2}-m_{x'}+m_{y'} \bigr), $$
for all \((x'_{1},t_{1})\in K, (y'_{1},t_{2})\in L\). This is equivalent to
$$ \bigl\langle \bigl(z'_{1}-\lambda z'_{2}\bigr), \bigl(x'-y' \bigr)\bigr\rangle +(1+\lambda ) (m_{x'}-m_{y'})=(\lambda-1) (t_{1}-t_{2}), $$
(4.5)
for all \((x'_{1},t_{1})\in K, (y'_{1},t_{2})\in L\).

We fix \(x',y'\). If change \(t_{1},t_{2}\) in (4.5) with \((x'_{1},t_{1})\in K, (y'_{1},t_{2})\in L\), then the left of (4.5) will not change; thus we obtain \(\lambda=1\). Namely, equality in (4.3) or (4.4) implies all of the chords of K and L parallel to u have midpoints that lie in a hyperplane, respectively. □

Lemma 4.3

If \(K,L\in\mathcal{K}^{n}, p\geq1\) and \(u\in S^{n-1}\), then
$$ \bar{\mathcal{Z}}_{p}(S_{u}K,S_{u}L) \subseteq S_{u}\bigl(\bar{\mathcal{Z}}_{p}(K,L)\bigr). $$
(4.6)
If the inclusion is an identity, then all of the chords of K and L parallel to u have midpoints that lie in a hyperplane, respectively.

Proof

Let \(y'\in\operatorname{relint}(\bar{Z}_{p}(K,L))_{u}\). Lemma 4.1 means that there exist \(z'_{1}=z'_{1}(y')\) and \(z'_{2}=z'_{2}(y')\) in \(u^{\perp}\) with
$$\begin{aligned}& \overline{l}_{u}\bigl(\bar{\mathcal{Z}}_{p}(K,L);y' \bigr)=h_{\bar{\mathcal {Z}}_{p}(K,L)}\bigl(z'_{1},1\bigr)-\bigl\langle z'_{1},y'\bigr\rangle ,\\& \underline{l}_{u}\bigl(\bar{\mathcal{Z}}_{p}(K,L);y' \bigr)=h_{\bar{\mathcal {Z}}_{p}(K,L)}\bigl(z'_{2},-1\bigr)-\bigl\langle z'_{2},y'\bigr\rangle . \end{aligned}$$
Combining (4.1), (4.2), (4.3), and (4.4), it follows that
$$\begin{aligned} \overline{l}_{u}\bigl(S_{u}\bigl(\bar{ \mathcal{Z}}_{p}(K,L)\bigr);y'\bigr) =&\frac {1}{2} \overline{l}_{u}\bigl(\bar{\mathcal{Z}}_{p}(K,L);y' \bigr)+\frac {1}{2}\underline{l}_{u}\bigl(\bar{ \mathcal{Z}}_{p}(K,L);y'\bigr) \\ =&\frac{1}{2} \bigl(h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{1},1 \bigr)-\bigl\langle z'_{1},y'\bigr\rangle \bigr)+\frac{1}{2} \bigl(h_{\bar{\mathcal {Z}}_{p}(K,L)}\bigl(z'_{2},-1 \bigr)-\bigl\langle z'_{2},y'\bigr\rangle \bigr) \\ =&\frac{1}{2}h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{1},1 \bigr)+\frac {1}{2}h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{2},-1 \bigr)- \biggl\langle \biggl(\frac {1}{2}z'_{1}+ \frac{1}{2}z'_{2} \biggr),y' \biggr\rangle \\ \geq& h_{\bar{\mathcal{Z}}_{p}(S_{u}(K,L))} \biggl(\frac {z'_{1}+z'_{2}}{2},1 \biggr)- \biggl\langle \biggl(\frac{1}{2}z'_{1}+\frac {1}{2}z'_{2} \biggr),y' \biggr\rangle \\ \geq& \min_{x'\in u^{\perp}} \bigl\{ h_{\bar{\mathcal {Z}}_{p}(S_{u}K,S_{u}L)} \bigl(x',1\bigr)-\bigl\langle x',y'\bigr\rangle \bigr\} \\ =& \overline{l}_{u}\bigl(\bar{\mathcal{Z}}_{p}(S_{u}K,S_{u}L); y'\bigr) \end{aligned}$$
and
$$\begin{aligned} \underline{l}_{u}\bigl(S_{u}\bigl(\bar{ \mathcal{Z}}_{p}(K,L)\bigr);y'\bigr) =&\frac {1}{2} \overline{l}_{u}\bigl(\bar{\mathcal{Z}}_{p}(K,L);y' \bigr)+\frac {1}{2}\underline{l}_{u}\bigl(\bar{ \mathcal{Z}}_{p}(K,L);y'\bigr) \\ =&\frac{1}{2} \bigl(h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{1},1 \bigr)-\bigl\langle z'_{1},y'\bigr\rangle \bigr)+\frac{1}{2} \bigl(h_{\bar{\mathcal {Z}}_{p}(K,L)}\bigl(z'_{2},-1 \bigr)-\bigl\langle z'_{2},y'\bigr\rangle \bigr) \\ =&\frac{1}{2}h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{1},1 \bigr)+\frac {1}{2}h_{\bar{\mathcal{Z}}_{p}(K,L)}\bigl(z'_{2},-1 \bigr)- \biggl\langle \biggl(\frac {1}{2}z'_{1}+ \frac{1}{2}z'_{2} \biggr),y' \biggr\rangle \\ \geq& h_{\bar{\mathcal{Z}}_{p}(S_{u}(K,L))} \biggl(\frac {z'_{1}+z'_{2}}{2},-1 \biggr)- \biggl\langle \biggl(\frac{1}{2}z'_{1}+\frac {1}{2}z'_{2} \biggr),y' \biggr\rangle \\ \geq& \min_{x'\in u^{\perp}} \bigl\{ h_{\bar{\mathcal {Z}}_{p}(S_{u}K,S_{u}L)} \bigl(x',-1\bigr)-\bigl\langle x',y' \bigr\rangle \bigr\} \\ =& \underline{l}_{u}\bigl(\bar{\mathcal{Z}}_{p}(S_{u}K,S_{u}L); y'\bigr). \end{aligned}$$

Let the inclusion be an identity. Then equality in both (4.3) and (4.4) holds; thus all of the chords of K and L parallel to u have midpoints that lie in a hyperplane, respectively. □

Now, we show that \(\bar{Z}_{p}(K,L)\) contains the origin in its interior.

Lemma 4.4

Suppose that \(K,L\in\mathcal{K}^{n}, p\geq1\). Then there exists \(c_{0}>0\) such that
$$\int_{K} \int_{L}\bigl\vert \bigl\langle u, (x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y\geq c_{0}, $$
for all \(u\in S^{n-1}\).

Proof

Given \(u_{0}\in S^{n-1}\). Taking the Euclidean n-balls \(B_{1}\subseteq K\) and \(B_{2}\subseteq L\) such that \(x-y\) is not orthogonal to \(u_{0}\) for all \((x,y)\in B_{1}\times B_{2}\), then it follows from the continuity that the above result holds. □

Proof of Theorem 1.2

It follows from the standard Steiner symmetrization argument that there exists a sequence of directions \(\{u_{i}\}\) with the sequences of convex bodies \(\{K_{i}\}\) and \(\{L_{i}\}\), defined by
$$K_{i+1}=S_{u_{i}}K_{i}, \qquad K_{0}=K, $$
and
$$L_{i+1}=S_{u_{i}}L_{i},\qquad L_{0}=L, $$
converge to \(B_{K}\) and \(B_{L}\), respectively. Note that \(B_{K}\ (B_{L})\) is the n-ball, where \(V(K)=V(B_{K}) (V(L)=V(B_{L}))\).
By Property 3.3 and Lemma 4.3, we have
$$\begin{aligned} V\bigl(\bar{\mathcal{Z}}_{p}(K_{i},L_{i}) \bigr) =&V\bigl(\bar{\mathcal {Z}}_{p}(S_{u_{i-1}}K_{i-1},S_{u_{i-1}}L_{i-1}) \bigr) \\ \leq& V\bigl(\bar{\mathcal{Z}}_{p}(K_{i-1},L_{i-1}) \bigr)\leq \cdots \\ \leq& V\bigl(\bar{\mathcal{Z}}_{p}(K,L)\bigr). \end{aligned}$$
(4.7)
From Lemma 4.4, Lemma 4.3, (4.7), and the definitions of \(B_{K}\) and \(B_{L}\), we get
$$V\bigl(\bar{\mathcal{Z}}_{p}(K,L)\bigr)\geq V\bigl(\bar{ \mathcal {Z}}_{p}(B_{K},B_{L})\bigr). $$
Since K and L are the ellipsoids, it follows from Property 3.4 that
$$V\bigl(\bar{\mathcal{Z}}_{p}(K,L)\bigr)=V\bigl(\bar{ \mathcal{Z}}_{p}(B_{K},B_{L})\bigr). $$
Conversely, let \(V(\bar{\mathcal{Z}}_{p}(K,L))=V(\bar{\mathcal {Z}}_{p}(B_{K},B_{L}))\). Clearly, for all \(u\in S^{n-1}\) the inclusion in (4.6) is the identity. Thus we see that all of the chords of K and L parallel to u have midpoints that lie in a hyperplane, respectively, for all \(u\in S^{n-1}\), namely, K and L are ellipsoids.
This implies
$$ V\bigl(\bar{\mathcal{Z}}_{p}(K,L)\bigr)\geq V\bigl(\bar{ \mathcal{Z}}_{p}(B_{K},B_{L})\bigr), $$
(4.8)
where \(V(B_{K})=V(K)\) and \(V(B_{L})=V(L)\), and equality holds if and only if K and L are dilated ellipsoids having the same midpoints.
Furthermore, we know that \(h(\bar{\mathcal{Z}}_{p}(B_{K},B_{L}),u)\) is a constant independent of u. Thus \(\bar{\mathcal {Z}}_{p}(B_{K},B_{L})\) is an n-ball. Thus one has
$$ \biggl(\frac{V(\bar{\mathcal{Z}}_{p}(B_{K},B_{L}))}{\omega_{n}} \biggr)^{1/n}= \biggl( \frac{1}{V(B_{K})V(B_{L})} \int_{B_{K}} \int _{B_{L}}\bigl\vert \bigl\langle u,(x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}x\,\mathrm{d}y \biggr)^{1/p}. $$
(4.9)
The following formula is well known:
$$ \int_{S^{n-1}}\bigl\vert \bigl\langle u, (x-y)\bigr\rangle \bigr\vert ^{p}\,\mathrm{d}u=\frac{(n+p)\omega _{n+p}}{\omega_{2}\omega_{p-1}}\vert x-y\vert ^{p}. $$
(4.10)
Together (4.9) with (4.10), it follows that
$$ \biggl(\frac{V(\bar{\mathcal{Z}}_{p}(B_{K},B_{L}))}{\omega_{n}} \biggr)^{1/n}= \biggl( \frac{(n+p)\omega_{n+p}}{n\omega_{2}\omega_{p-1}\omega _{n}V(B_{K})V(B_{L})} \int_{B_{K}} \int_{B_{L}} \vert x-y\vert ^{p}\,\mathrm{d}x\, \mathrm{d}y \biggr)^{1/p}. $$
(4.11)
Suppose that \(r_{B_{K}}\) and \(r_{B_{L}}\) denote radii of the balls \(B_{K}\) and \(B_{L}\), respectively. Without loss of generality, let \(B_{K}\subseteq B_{L}\). For all \(u\in S^{n-1}\), it is obvious that \(\rho _{DB_{K}}(u)=2r_{B_{K}}\) and \(\rho_{DB_{L}}(u)=2r_{B_{L}}\). It follows from the spherical polar coordinates, (2.1), the Fubini theorem, and (2.6) that
$$\begin{aligned} \int_{B_{K}} \int_{B_{L}}\vert x-y\vert ^{p}\,\mathrm{d}x\, \mathrm{d}y =& \int_{B_{K}} \int _{S^{n-1}} \int_{0}^{\rho_{B_{L}}(y,u)}r^{n+p-1}\,\mathrm{d}r\, \mathrm{d}u\,\mathrm{d}y \\ =&\frac{1}{n+p} \int_{S^{n-1}} \int_{B_{K}}\rho_{B_{L}}(y,u)^{n+p}\,\mathrm{d}y \,\mathrm{d}u \\ =& \int_{S^{n-1}} \int_{0}^{\rho_{DB_{K}}(u)}V \bigl(B_{K}\cap (B_{L}+tu) \bigr)t^{n+p-1}\,\mathrm{d}t\,\mathrm{d}u \\ =& \int_{S^{n-1}} \int_{0}^{2r_{B_{K}}}V \bigl(B_{K} \cap(B_{L}+tu) \bigr)t^{n+p-1}\,\mathrm{d}t\,\mathrm{d}u. \end{aligned}$$
(4.12)
Since \(g_{B_{K},B_{L}}(tu)^{1/n}=V(B_{K}\cap(B_{L}+tu))^{1/n}\) is concave on \(DB_{K}\), it follows from Lemma 2.1 that
$$ V\bigl(B_{K}\cap(B_{L}+tu)\bigr)\geq V(B_{K}) \biggl(1-\frac{t}{2r_{B_{K}}} \biggr)^{n}\quad \mbox{for } 0\leq t\leq2r_{B_{K}}, $$
(4.13)
with equality if and only if \(B_{K}=B_{L}\).
Taking (4.12) together with (4.13), it follows that
$$\begin{aligned} \int_{B_{K}} \int_{B_{L}}\vert x-y\vert ^{p}\,\mathrm{d}x\, \mathrm{d}y& \geq n\omega_{n}(2 r_{B_{K}})^{n+p}V(K) \int _{0}^{1}y^{n+p-1}(1-y)^{n}\,\mathrm{d}y \\ & = n\omega_{n}(2 r_{B_{K}})^{n+p}\beta(n+p,n+1)V(K) \\ & = 2^{n+p}n\omega_{n}^{-\frac{p}{n}}\beta(n+p,n+1)V(K)^{\frac {2n+p}{n}}. \end{aligned}$$
(4.14)
Combining (4.11) with (4.14), we have
$$ \biggl(\frac{V(\bar{\mathcal{Z}}_{p}(B_{K},B_{L}))}{\omega_{n}} \biggr)^{1/n}\geq \biggl( \frac{2^{n+p}(n+p)\omega_{n+p}\beta(n+p,n+1)}{\omega _{2}\omega_{p-1}\omega_{n}^{\frac{n+p}{n}}} \biggr)^{1/p} \biggl(\frac {V(K)^{\frac{n+p}{n}}}{V(L)} \biggr)^{1/p}. $$
(4.15)
From (4.8) and (4.15), this yields
$$ V\bigl(\bar{\mathcal{Z}}_{p}(K,L)\bigr)\geq C(n,p)V(K)^{\frac{n+p}{p}}V(L)^{-\frac{n}{p}}, $$
(4.16)
with equality if and only if \(K=L\) is an ellipsoid. □

Proof of Corollary 1.3

Let \(p=1\). From the \(L_{p}\)-Minkowski inequality (2.3), we have
$$V(K)^{n-1}V(L)\geq\widetilde{V}_{1}(K,L)^{n}. $$
Exchange the order of K and L, then
$$V(L)^{n-1}V(K)\geq\widetilde{V}_{1}(L,K)^{n}, $$
namely,
$$ \frac{V(K)}{V(L)}\geq \biggl(\frac{\widetilde{V}_{1}(L,K)}{V(L)} \biggr)^{n}, $$
(4.17)
with equality if and only if K and L are dilates.
On the other hand, from inequality (1.6), we have
$$ \biggl(\frac{V(\bar{\mathcal{Z}_{1}}(K,L))}{V(K)} \biggr)^{\frac {1}{n}}\geq \bigl(C(n,1) \bigr)^{\frac{1}{n}}\frac{V(K)}{V(L)}, $$
(4.18)
with equality if and only if \(K=L\) is an ellipsoid.
Taking (4.17) together with (4.18), it follows that
$$\biggl(\frac{V(\bar{\mathcal{Z}}_{1}(K,L))}{V(K)} \biggr)^{\frac {1}{n}}\geq \bigl(C(n,1) \bigr)^{\frac{1}{n}} \biggl(\frac{\widetilde {V}_{1}(L,K)}{V(L)} \biggr)^{n}. $$

Together with the equality conditions of inequalities (4.17) and (4.18), we see with equality in (1.7) if and only if \(K=L\) is an ellipsoid. □

Declarations

Acknowledgements

The authors are indebted to the editors and the anonymous referees for many valuable suggestions and comments. This work is supported by the National Natural Science Foundations of China (Grant No. 11561020 and Grant No. 11161019), the Science and Technology Plan of Gansu Province (Grant No. 145RJZG227), the Young Foundation of Hexi University (Grant No. QN2015-02) and partly the National 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)
College of Mathematics and Statistics, Hexi University
(2)
College of Mathematics and Statistics, Northwest Normal University

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