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.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors completed the paper, and read and approved the final manuscript.
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.
Anzeige
The zonoids are investigated by many authors (see [3‐5]). The zonoid \(\mathcal{Z}\) is a convex body with support function
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]
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
The Legendre ellipsoid and its polar (the Binet ellipsoid) are well-known concepts from classical mechanics. See [7‐9] for historical references.
Anzeige
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 \(x\in\mathbb{R}^{n}\),
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
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
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
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
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
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])
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])
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
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
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 \(u_{0}\in S^{n-1}\)
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
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,
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.
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
If the inclusion is an identity, then all of the chords ofKandLparallel touhave 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
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. □
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 \(V(K)=V(B_{K}) (V(L)=V(B_{L}))\).
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.
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
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
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. □
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).
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors completed the paper, and read and approved the final manuscript.