Abstract
As an extension of the classical John ellipsoid and the \(L_{p}\)-John ellipsoids due to Lutwak–Yang–Zhang, this paper studies (p, q)-John ellipsoids. We consider an optimization problem about the (p, q)-mixed volumes, whose solution is uniquely existed for all \(0<p\le q\). The solution allows us to introduce the concept of (p, q)-John ellipsoids. As applications, we established an analog of the John’s inclusion theorem and Ball’s volume-ratio inequality for (p, q)-John ellipsoids. Moreover, the connection between the isotropy of measures and the characterization of (p, q)-John ellipsoids is demonstrated.
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1 Introduction
The concept of John ellipsoid, introduced by Fritz John [20], is extremely useful in convex geometry and Banach space geometry. For each convex body (compact convex set with nonempty interior) K in the n-dimensional Euclidean \({\mathbb {R}}^{n}\), its John ellipsoid JK is defined as the unique ellipsoid of maximal volume contained in K.
Two fundamental results concerning the John ellipsoid are John’s inclusion and Ball’s volume-ratio inequality. Let K be an origin-symmetric convex body K in \({\mathbb {R}}^{n}\). John’s inclusion shows that
As an application of John’s inclusion, the best upper bound of the Banach–Mazur distance is \(\sqrt{n}\), for an n-dimensional normed space to n-dimensional Euclidean space. Ball’s volume-ratio inequality states that
with equality if and only if K is a parallelotope. Here \(|\cdot |\) denotes n-dimensional volume and \(\omega _{n}=|B|=\pi ^{n/2}/\Gamma \left( 1+\frac{n}{2}\right) \) denotes the volume of the unit ball B in \({\mathbb {R}}^{n}\). The fact that there is equality in (1.2) only for parallelotopes was established by Barthe [3]. For more information about the John ellipsoid, one can refer to [1, 2, 12, 14, 15, 21, 22, 44] and the references within.
In 2005, Lutwak, Yang and Zhang [30] extend the John ellipsoid to \(L_{p}\) John ellipsoids, which is an important concept in the \(L_{p}\) Brunn–Minkowski theory initiated by Lutwak [27, 28]. During the last two decades, the \(L_{p}\) Brunn–Minkowski theory has achieved great developments and expanded rapidly, see, e.g., [4,5,6, 8, 9, 17,18,19, 24,25,26, 29, 31,32,34, 37, 38, 47,48,51]. Moreover, the Orlicz Brunn–Minkowski theory, as an extension of the \(L_{p}\) Brunn–Minkowski theory, emerged in [16, 35, 36]. In these papers, the fundamental notions of the \(L_{p}\) projection body and the \(L_{p}\) centroid body were extended to an Orlicz setting, see also [7, 53, 55]. For more information, please refer to the literature [11, 23, 39,40,41, 54, 56,57,58,59,60]. In particular, the classical John ellipsoid is extended to the \(L_{p}\) setting by Lutwak, Yang and Zhang [30] and to the Orlicz setting by Zou and Xiong [58].
Suppose \(p\in (0,\infty ]\) and K is a convex body in \({\mathbb {R}}^{n}\) with the origin in its interior. Among all origin-symmetric ellipsoids E, the unique ellipsoid that solves the constrained maximization problem
is called the \(L_{p}\) John ellipsoid [30] of K and denoted by \(E_{p}K\). Clearly, \(E_{p}B=B\). Here
is the normalized \(L_{p}\) mixed volume of K and E; \(S^{n-1}\) is the unit sphere in \({\mathbb {R}}^{n}\); \(h_{K}\) and \(h_{E}\) are the support functions (see Sect. 2) of K and E, respectively. In the case \(p=\infty \), we define
Therefore, when the John point of K, i.e., the center of JK, is at the origin, \(E_{\infty }K\) is precisely the classical John ellipsoid JK. In the case \(p=2\), the \(L_2\) John ellipsoid \(E_{2}K\) is the new ellipsoid \(\Gamma _{-2}K\) found by Lutwak, Yang and Zhang in [32], which is now called the LYZ ellipsoid and is in some sense dual to the Legendre ellipsoid of inertia in classical mechanics [42]. In the case \(p=1\), \(E_{1}K\) is the so-called Petty ellipsoid, see [13, 43]. The volume-normalized Petty ellipsoid is obtained by minimizing the surface area of K under \({\mathrm{SL}}(n)\)-transformations.
In general, the \(L_{p}\) John ellipsoid \(E_{p}K\) is not contained in K (except when \(p=\infty \)). However, when \(1\le p\le \infty \), it has \(|E_{p}K|\le |K|\). In reverse, for \(0<p\le \infty \), the \(L_{p}\) version of Ball’s volume-ratio inequality [30] states that
with equality if and only if K is a parallelotope.
By \(L_p\) dual curvature measures, Lutwak, Yang and Zhang [31] introduced the notion of \(L_{p}\) dual mixed volumes which unifies \(L_{p}\) mixed volumes of convex bodies in the \(L_{p}\) Brunn–Minkowski theory and dual mixed volumes of star bodies in the dual Brunn–Minkowski theory. Therefore, \(L_{p}\) dual mixed volumes become to be a core concept in convex geometry with unifying some contents of the \(L_{p}\) Brunn–Minkowski theory and the dual Brunn–Minkowski theory.
Let \({\mathcal {K}}^{n}_{o}\) denote the class of convex bodies in \({\mathbb {R}}^{n}\) that contain the origin in their interiors. And let \({\mathcal {S}}^{n}_{o}\) denote the set of star bodies (compact star-shaped set about the origin) in \({\mathbb {R}}^{n}\).
Suppose K is a convex body in \({\mathbb {R}}^{n}\). For each \(v\in {\mathbb {R}}^{n}\backslash \{o\}\), the hyperplane
is called the supporting hyperplane to K with outer normal v.
The spherical image (Gauss image) of \(\sigma \subset \partial K\) is defined by
Let \(\sigma _{K}\subset \partial K\) be the set consisting of boundary points \(x\in \partial K\), for which the set \(\varvec{\nu }_{K}(\{x\})\) contains more than a single element. It is well known that the spherical Lebesgue measure of \(\sigma _{K} \) is \({\mathcal {H}}^{n-1}(\sigma _{K})=0\) (see, e.g., [46, p. 84]). On precisely the functions
is called the spherical image map (Gauss map) of K and is continuous (see, e.g., [46, Lemma 2.2.12]). The set \(\partial K\backslash \sigma _{K}\) is usually abbreviated by \(\partial 'K\). Since \({\mathcal {H}}^{n-1}(\sigma _{K})=0\), the integrals over subsets of \(\partial 'K\) and \(\partial K\) are equal with respect to \({\mathcal {H}}^{n-1}\).
For \(\omega \subset S^{n-1}\), the radial Gauss image of \(\omega \) is denoted by
For a subset \(\eta \subset S^{n-1}\), the reverse radial Gauss image of \(\eta \) is denoted by
For \(K\in {\mathcal {K}}^{n}_{o}\), the radial map of K, \(r_{K}: S^{n-1}\rightarrow \partial K,\) is defined by
for \(u\in S^{n-1}\). Here, \(\rho _{K}(u)=\max \{\lambda >0: \lambda u\in K\}\) is the radial function of K for \(u\in S^{n-1}\). Note that \(r_{K}^{-1}: \partial K\rightarrow S^{n-1}\) is given by \(r_{K}^{-1}(x)=x/|x|\) for \(x\in \partial K\). Let \(\omega _{K}=\overline{\sigma _{K}}=r_{K}^{-1}(\sigma _{K})\). Observe that \(\omega _{K}\) has spherical Lebesgue measure 0, and the integrals over subsets of \(S^{n-1}\backslash \omega _{K}\) and \(S^{n-1}\) are equal with respect to the spherical Lebesgue measure.
The radial Gauss map of \(K\in {\mathcal {K}}^{n}_{o}\), \(\alpha _{K}: S^{n-1}\backslash \omega _{K}\rightarrow S^{n-1}\), is given by
Obviously, for any \(\lambda >0\) and any \(u\in S^{n-1}\),
For \(p,q\in {\mathbb {R}}\), \(K\in {\mathcal {K}}^{n}_{o}\), and \(Q\in {\mathcal {S}}^{n}_{o}\), the \(L_{p}\) dual curvature measures \({\widetilde{C}}_{p,q}(K,Q)\) are Borel measures on \(S^{n-1}\) given by
for each continuous function \(g: S^{n-1}\rightarrow {\mathbb {R}}\). For each Borel set \(\eta \subseteq S^{n-1}\), we have
It has shown that [31, Proposition 5.4] that the \(L_{p}\) surface area measure, the dual curvature measure and the integral measure are all special cases of the \(L_{p}\) dual curvature measure. In particular, for \(p,q\in {\mathbb {R}}\), and \(K\in {\mathcal {K}}^{n}_{o}\),
where \(S_{p}(K,\cdot )\) is the \(L_{p}\)-surface area measure of K.
Using \(L_{p}\) dual curvature measures, Lutwak, Yang and Zhang [31] introduced the concept of (p, q)-mixed volume volumes. For \(p,q\in {\mathbb {R}}\), and convex bodies \(K,L\in {\mathcal {K}}^{n}_{o}\), and a star body \(Q\in {\mathcal {S}}^{n}_{o}\), the (p, q)-mixed volume \({\widetilde{V}}_{p,q}(K,L,Q)\) is defined by
The concept of the (p, q)-mixed volume unifies the \(L_{p}\) mixed volume and the dual mixed volume in the sense that
In this paper we will consider the problem of minimizing total \(L_{p}\) dual curvature measures under \(\mathrm{SL}(n)\)-transformations. Let K be a smooth convex body in \({\mathbb {R}}^{n}\) with the origin in its interior, and let Q be a smooth star body in \({\mathbb {R}}^{n}\). For real number p, q, find
From (1.9) and [31, Proposition 7.3] (see also Lemma 2.3 of our paper), it follows that the original problem of minimizing total \(L_{p}\) dual curvature under \({\mathrm{SL}}(n)\)-transformations can be rewritten as
where the last minimum is taken over all origin-symmetric ellipsoids with volume \(\omega _n\). A \(\phi _{p,q}\in {\mathrm{SL}}(n)\) at which this minimum is attained defines an ellipsoid \({\bar{E}}_{p,q}(K,Q)\) which \(\phi _{p,q}\) maps into the unit ball B, i.e., \({\bar{E}}_{p,q}(K,Q)=\phi _{p,q}^{-1}B\). This ellipsoid is unique and will be called the volume-normalized (p, q)-John ellipsoid of K and Q. For \(p=\infty \), define
For \(r\in [0, +\infty )\), the normalized r-th dual area measure of \(K,Q\in {\mathcal {S}}^{n}_{o}\), \(\overline{{\widetilde{V}}}_{r}(K,Q; \cdot )\), is defined by
where \({\widetilde{V}}_{r}(K,Q)\) is the r-th dual mixed volume of \(K,Q\in {\mathcal {S}}^{n}_{o}\). Clearly, \({\mathrm{d}}\overline{{\widetilde{V}}}_{r}(K,Q; \cdot )\) is a probability measure on \(S^{n-1}\). In the case \(Q=K\), \({\mathrm{d}}\overline{{\widetilde{V}}}_{r}(K,K;u)={\mathrm{d}}\overline{{\widetilde{V}}}_{K}(u) =\frac{1}{n|K|} \rho ^{n}_{K}{\mathrm{d}}u,\text{ for }\ u\in S^{n-1},\) is the normalized dual area measure of \(K\in {\mathcal {S}}^{n}_{o}\). And for the cases \(r=0,n\), we have \({\mathrm{d}}\overline{{\widetilde{V}}}_{0}(K,Q; \cdot )={\mathrm{d}}\overline{{\widetilde{V}}}_{Q}(\cdot )\) and \({\mathrm{d}}\overline{{\widetilde{V}}}_{n}(K,Q; \cdot )={\mathrm{d}}\overline{{\widetilde{V}}}_{K}(\cdot )\).
In order to rewrite the formulation of our problem for the case \(p=\infty \), we next introduce a normalized version of (p, q)-dual mixed volumes. If \(K,L\in {\mathcal {K}}^{n}_{o}\), \(Q\in {\mathcal {S}}^{n}_{o}\) and \(q\ge p>0\) with \(r=q-p\ge 0\), then we define the normalized (p, q)-dual mixed volume by
In the case \(p=\infty \) (then \(q=\infty \)), define
Unless \(\frac{h_{L}(\alpha _{K}(u))\rho _{K}(u)}{h_{K}(\alpha _{K}(u))\rho _{Q}(u)}\) is constant on \({\mathrm{supp}}{\widetilde{V}}_{r}(K,Q; \cdot )\), it follows from (1.13) and Jensen’s inequality that
for \(0<p_{1}<p_{2}\le \infty , 0<q_{1}=p_{1}+r\le p_{2}+r=q_{2}\le \infty \), and
We shall require the fact that, for \(p_{0}\in (0,\infty ], q_{0}=p_{0}+r\in (0,\infty ]\) and \(r\in [0,\infty )\),
In fact, we have already proved a more general conclusion, see Theorem 3.1 in subsequent. By (1.14), we have
In the sequel, we use \({\mathcal {E}}^{n}\) to denote the class of origin-symmetric ellipsoids in \({\mathbb {R}}^{n}\).
Inspired by the constrained maximization problem (1.3) posed by Lutwak, Yang and Zhang [30], this paper will consider a (p, q)-version of the problem:
Optimization Problems 1.1
Let \(0<p\le q\) with \(q=p+r\), \(r\ge 0\). For \(K\in {\mathcal {K}}^{n}_{o}\) and \(Q\in {\mathcal {S}}^{n}_{o}\), find an ellipsoid, among all origin-symmetric ellipsoids, which solves the following constrained maximization problem:
An ellipsoid that solves the constrained maximization problem will be called a \(S_{p,q}\) solution for K and Q. The dual problem is
An ellipsoid that solves the dual problem will be called a \({\bar{S}}_{p.q}\) solution for K and Q.
We will prove in Sect. 4 there is a unique solution to the constrained maximization problem, which will be called the (p, q)-John ellipsoid \(E_{p,q}(K,Q)\) in Definition 4.6. The dual problem is equivalent to the problem of minimizing total \(L_{p}\) dual curvature measures under \(\mathrm{SL}(n)\)-transformations. The dual problem has a unique solution with volume \(\omega _n\), which differs by only a scale factor to the \(S_{p,q}\) solution. Therefore, it is called the normalized (p, q)-John ellipsoid \({\overline{E}}_{p,q}(K,Q)\).
In the case of \(Q=K\), \(E_{p,q}(K,Q)=E_{p}(K)\) is the \(L_{p}\) John ellipsoid studied by Lutwak, Yang and Zhang [30]. In the case that \(Q=B\) and \(p=n\), one also has \(E_{p,q}(K,Q)=E_{p}(K)\).
This paper is organized as follows. In Sect. 2 we recall some basic results in convex geometry. Section 3 proves the continuity of \({\widetilde{V}}_{p,q}\) and \(\overline{{\widetilde{V}}}_{p,q}\). We prove in Sect. 4 the existence, uniqueness and geometric characterization of the (p, q)-John ellipsoid which solves Problem 1.1. Using the continuity of \({\widetilde{V}}_{p,q}\) and \(\overline{{\widetilde{V}}}_{p,q}\), we study continuity of (p, q)-John ellipsoids in Sect. 5 In Sect. 6, we discuss generalizations of John’s inclusion for (p, q)-John ellipsoids. In the last section, the inequality for the volume ratio is established.
2 Preliminaries
For quick reference we recall some basic results of convex geometry. We refer the reader to [10, 46] for details.
The setting will be the n-dimensional Euclidean space \({\mathbb {R}}^{n}\). As usual \(x\cdot y\) denotes the standard inner product of x and y in \({\mathbb {R}}^{n}\). For \(x\in {\mathbb {R}}^{n}\), let \(|x|=\sqrt{x\cdot x}\) be the Euclidean norm of x. For \(x\in {\mathbb {R}}^{n}\backslash \{o\}\), we use both \({\bar{x}}\) and \(\langle x\rangle \) to denote x/|x|.
In addition to its denoting absolute value, without confusion we will use \(|\cdot |\) to denote the standard Euclidean norm on \({\mathbb {R}}^{n}\), often to denote n-dimensional volume, and on occasion to denote the absolute value of the determinant of an \(n\times n\) matrix.
For \(K\in {\mathcal {K}}^{n}_{o}\), its support function, \(h_{K}: {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}\) is defined by \(h_{K}(x)=\max \{x\cdot y: y\in K\}\), for \(x\in {\mathbb {R}}^{n}\). Obviously, for real \(\lambda >0\),
More generally, for \(\phi \in {\mathrm{GL}}(n)\) the image \(\phi K=\{\phi x: x\in K\}\) have that
where \(\phi ^{t}\) denotes the transpose of \(\phi \).
The Hausdorff distance between convex bodies K and L is given by
If \(K,L\in {\mathcal {K}}^{n}_{o}\), then for real \(p>0\), the \(L_{p}\)-mixed volume of K and L is defined by
If K contains the origin in its interior, then its polar body \(K^{*}\) is given by \(K^{*}=\{x\in {\mathbb {R}}^{n}: x\cdot y\le 1\ \ \text{ for } \text{ all }\ y\in K\}\). Obviously, for \(\phi \in {\mathrm{GL}}(n)\),
where \(\phi ^{-t}\) denotes the inverse of the transpose of \(\phi \).
A star body \(K\subset {\mathbb {R}}^{n}\) is a compact star-shaped set about the origin whose radial function \(\rho _{K}: {\mathbb {R}}^{n}\backslash \{o\}\rightarrow {\mathbb {R}}\), defined for \(x\in {\mathbb {R}}^{n}\backslash \{o\}\) by \(\rho _{Q}(x)=\max \{\lambda >0: \lambda x\in Q\}\), is continuous. We call two star bodies K and L in \({\mathcal {S}}^{n}_{o}\) are dilates (of one another) if \(\rho _{K}(u)/ \rho _{L}(u)\) is independent of \(u\in S^{n-1}\). If \(\lambda >0\), we have
More generally, for \(\phi \in {\mathrm{GL}}(n)\), the image \(\phi K=\{\phi x: x\in K\}\) of K have the property
for all \(x\in {\mathbb {R}}^{n}\backslash \{o\}\).
The radial distance between \(K,L\in {\mathcal {S}}^{n}_{o}\) is
The dual Brunn–Minkowski theory is a theory of dual mixed volumes of star bodies. For \(q\in {\mathbb {R}}\), the q-th dual mixed volume of \(K,Q\in {\mathcal {S}}^{n}_{o}\), is defined by (see [31])
where the integral is with respect to spherical Lebesgue measure. It is well know that for \(\phi \in {\mathrm{GL}}(n)\),
Dual Minkowski inequality can be expressed as follows: If \(0\le q\le n\) and \(K,Q\in {\mathcal {S}}^{n}_{o}\), then
with equality if and only if K and Q are dilates when \(0<q<n\).
If \(K\in {\mathcal {K}}^{n}_{o}\), then it is easy to see that the radial function and the support function of K are related by
From definitions of \({\widetilde{V}}_{p,q}\) and the radial Gauss map, the support function and the radial function imply that
Lemma 2.1
Let \(\lambda >0\), then
For \(\lambda >0\) and \(p\in (0,\infty ], q=p+r, r\in [0,\infty )\), based on the (1.13), (2.1) and (2.5), we can immediately obtain the results,
Lemma 2.2
Let \(\lambda >0\), then
We shall need the following fact.
Lemma 2.3
(cf. [31]) The (p, q)-mixed volume is \({\mathrm{SL}}(n)\)-invariant, in that for \(p,q\in {\mathbb {R}}\), and \(K,L\in {\mathcal {K}}^{n}_{o}\), with \(Q\in {\mathcal {S}}^{n}_{o}\),
for each \(\phi \in {\mathrm{SL}}(n)\).
Lemma 2.1, together with Lemma 2.3, shows that for \(\phi \in {\mathrm{GL}}(n)\),
We will also need the fact that for \(\phi \in {\mathrm{GL}}(n)\) and \(p\in (0,\infty ], q=p+r, r\in [0,\infty )\),
This follows immediately from (2.8) and (2.19) for all \(p\in (0,\infty ], q=p+r\) and \(r\in [0,\infty )\).
The following inequality for (p, q)-mixed volume is a generalization of the \(L_{p}\) Minkowski inequality for mixed volume (see [31]).
Lemma 2.4
Suppose p, q are such that \(1\le \frac{q}{n}\le p\). If \(K,L\in {\mathcal {K}}^{n}_{o}\) and \(Q\in {\mathcal {S}}^{n}_{o}\), then
with equality if and only if K, L, Q are dilates when \(1<\frac{q}{n}<p\), while only K and L need be dilates when \(q=n\) and \(p >1\), and K and L are homothets when \(q=n\) and \(p =1\).
We shall require the following definition.
Definition 2.5
(cf. [31]) Suppose \(p\in {\mathbb {R}}\). If \(\mu \) is a Borel measure on \(S^{n-1}\) and \(\phi \in {\mathrm{SL}}(n)\) then, \(\phi _{p}\dashv \mu \), the \(L_{p}\) image of \(\mu \) under \(\phi \), is a Borel measure such that
for each Borel \(f: S^{n-1}\rightarrow {\mathbb {R}}\).
Lemma 2.6
(cf. [31]) Suppose \(p\ne 0\) and \(q\ne 0\). Then for all \(Q\in {\mathcal {S}}^{n}_{o}\) and \(K,L\in {\mathcal {K}}^{n}_{o}\), and \(\phi \in {\mathrm{SL}}(n)\),
We also need the following lemma:
Lemma 2.7
(cf. [19]) Suppose \(K_{i}\in {\mathcal {K}}^{n}_{o}\) with \(\lim \limits _{i\rightarrow \infty }K_{i}=K_{0}\). Let \(\omega =\cup _{i=0}^{\infty }\omega _{K_{i}}\), be the set (of \({\mathcal {H}}^{n-1}\)-measure 0) off of which all of the \(\alpha _{K_{i}}\) are defined. Then if \(u_{i}\in S^{n-1}\backslash \omega \) are such that \(\lim \limits _{i\rightarrow \infty }u_{i}=u_{0}\in S^{n-1}\backslash \omega \), then \(\lim \limits _{i\rightarrow \infty }\alpha _{K_{i}}(u_{i})=\alpha _{K_{0}}(u_{0})\).
Let \(K\in {\mathcal {K}}^{n}_{o}\). The classical projection body \(\Pi K\) of K is given by (see [10])
We will use the concept of a \(L_{p}\)-projection body (see [28, 29, 45, 52]). For \(p\ge 1\), the \(L_{p}\)-projection body \(\Pi _{p}K\) is given by
where \(S_{p}(K,\cdot )\) is the \(L_{p}\)-surface area measure. Clearly, \(\Pi _1 K=\frac{1}{n}\Pi K\).
We shall use the concepts of (p, q)-mixed projection body and (p, q)-mixed polar projection body. For each \(K\in {\mathcal {K}}^{n}_{o}\) with a star body \(Q\in {\mathcal {S}}^{n}_{o}\), and \(p>0, q>0\), the (p, q)-mixed projection body, \(\Pi _{p,q}(K,Q)\), of K and Q is the origin-symmetric convex body whose support function is defined by
In particular, we have \(\Pi _{p,n}(K,B)=\Pi _{p,q}(K,K)=\Pi _{p}K\) for \(p>1\), and \(\Pi _{1,n}(K,B)=\Pi _{1,q}(K,K)=\Pi _{1}(K)=\frac{1}{n}\Pi K\).
If \(K\in {\mathcal {K}}^{n}_{o}\) and real \(p>0\), the star body \(\Gamma _{-p}K\) (called by \(L_{p}\)-polar projection body, see [30]) is defined as, for \(u\in S^{n-1}\) :
If \(K\in {\mathcal {K}}^{n}_{o}\) and \(Q\in {\mathcal {S}}^{n}_{o}\), and real \(p>0,q>0\) and \(q=p+r, r\in [0, +\infty )\), the star body \(\Gamma _{-p,-q}(K,Q)\) is defined by, for \(x\in {\mathbb {R}}^{n}\),
The star body \(\Gamma _{-p,-q}(K,Q)\) is called the (p, q)-mixed polar projection body of K and Q. It is easy to know that \(\Gamma _{-p,-q}(K,K)=\Gamma _{-p}K\).
Note that for \(q\ge p\ge 1\), the body \(\Gamma _{-p,-q}(K,Q)\) is a convex body. Define \(\Gamma _{-\infty ,-\infty }(K,Q)\) by
For real \(p >0,q=p+r, r\in [0, +\infty )\), and using(1.5), we can rewrite (2.25) as
for \(u\in S^{n-1}\). Thus, from (2.25) and (2.26),
3 The Continuity of \({\widetilde{V}}_{p,q}\) and \(\overline{{\widetilde{V}}}_{p,q}\)
In this section, we consider the continuity of \({\widetilde{V}}_{p,q}\) and \(\overline{{\widetilde{V}}}_{p,q}\).
Theorem 3.1
Suppose \(K,K_{i},L,L_{j}\in {\mathcal {K}}^{n}_{o}, Q,Q_{k}\in {\mathcal {S}}^{n}_{o}\) and \(p_{l},p, q_{m},q\in (0,\infty ]\), where \(i,j,k,l,m\in {\mathbb {N}}\). Let \(r\in [0,+\infty )\). If \(K_{i}\rightarrow K,L_{j}\rightarrow L,Q_{k}\rightarrow Q, p_{l}\rightarrow p\), and \(q_{m}\rightarrow q\) as \(i,j,k,l,m\rightarrow \infty \), then
and
Proof
Let
where
and
We first claim \(0<c_{m}\le c_{M}<\infty \). Since \(K_{i}\rightarrow K,L_{j}\rightarrow L\) and \(Q_{k}\rightarrow Q, p_{l}\rightarrow p\) as \(i,j,k\rightarrow \infty \), we have \(h_{K_{i}}\rightarrow h_{K}\), \(h_{L_{j}}\rightarrow h_{L}\) and \(h_{L_{k}}\rightarrow h_{L}\) uniformly on \(S^{n-1}\), respectively. From \(K,K_{i},L,L_{j}\in {\mathcal {K}}^{n}_{o}, Q,Q_{k}\in {\mathcal {S}}^{n}_{o}\), it follows that there exists an \(N_{0}\in N\), such that for all \(i,j,k>N_{0}\) and \(u\in S^{n-1}\),
For brevity, we write
where
and
Then we have \(0<a_{m}\le a_{M}<\infty \), and
Thus, by the definitions of \(c_{m}\) and \(c_{M}\), it yields
Next, we prove
For any \(\varepsilon >0\), three observations are in order. Firstly, let \(f(t)=t^p,f_{l}(t)=t^{p_{l}},l=1,2,\cdots ,\) defined on \([c_{m},c_{M}]\), then the sequence of \(\{f_{l}\}\) converges uniformly to f on \([c_{m},c_{M}]\). And let \(g(t)=t^p,g_{m}(t)=t^{p_{m}},m=1,2,\cdots ,\) defined on \([c_{m},c_{M}]\), then the sequence of \(\{g_{m}\}\) converges uniformly to g on \([c_{m},c_{M}]\). For all \(u\in S^{n-1}\),
there exists an \(N_{1}\in {\mathbb {N}}\), such that for all \(l,m\ge N_{1}\),
independently of i and j and uniformly on \(u\in S^{n-1}\).
Secondly, since \(K_{i}\rightarrow K,L_{j}\rightarrow L\) and \(Q_{k}\rightarrow Q, p_{l}\rightarrow p\) as \(i,j,k\rightarrow \infty \), and Lemma 2.7, there exists an \(N_{2}\in {\mathbb {N}}\) such that for all \(i,j,k>N_{2}\) and for all \(u\in S^{n-1}\),
Indeed, since functions f and g are all Lipschitzian on \([c_{m},c_{M}]\), there exist constants \(C_{1}, C_{2}>0\), such that for all \(u\in S^{n-1}\),
Thirdly, since the measure sequence \(\{\overline{{\widetilde{V}}}_{Q_{k}}\}\) weakly converges to \(\overline{{\widetilde{V}}}_{Q}\), there exists an \(N_{3}\in {\mathbb {N}}\), such that for all \(k\ge N_{3}\),
From (3.3), (3.4) and (3.5), it follows that for all \(i,j,k,l,m\ge \max \{N_{1},N_{2},N_{3}\}\),
Namely,
The first conclusion follows from the fact \(|Q_{k}|\rightarrow |Q|\) by sending k to infinity.
Finally, we proceed to prove
Fix \(\delta >0\). For \(0\le r<\infty \), we note that
The continuity of \(t^{\frac{1}{p}}\) on \([c_1,c_3]\) implies there exists an \(N_4>0\) such that for all \(l\ge N_4\),
holds independently of i, j and k.
From (1.11) and (3.1), it follows \(\lim \limits _{i,k\rightarrow \infty }{\widetilde{V}}_{r}(K_{i},Q_{k})={\widetilde{V}}_{r}(K,Q)\). Combining this with (3.1), the continuity of \(t^{\frac{1}{p}}\) on \([c_1,c_3]\) shows there exists an \(N_{5}>0\), such that for all \(i,j,k,l>N_{5}\),
In terms of (3.6) and (3.7), it follows that for \(i,j,k,l\ge \max \{N_{4},N_{5}\}\),
That is,
\(\square \)
4 (p, q)-John Ellipsoids
In this section, we focus on the main Problem 1.1 proposed in Sect. 1.
Optimization Problems. Let \(0<p\le q\) with \(q=p+r\), \(r\ge 0\). For \(K\in {\mathcal {K}}^{n}_{o}\) and \(Q\in {\mathcal {S}}^{n}_{o}\), find an ellipsoid, among all origin-symmetric ellipsoids, which solves the following constrained maximization problem:
An ellipsoid that solves the constrained maximization problem will be called a \(S_{p,q}\) solution for K and Q. The dual problem is
An ellipsoid that solves the dual problem will be called a \({\bar{S}}_{p.q}\) solution for K and Q.
The following theorem gives the existence of Problem \(S_{p,q}\) when \(0<p\le q\), and proves its uniqueness when \(1\le p\le q\).
Theorem 4.1
For any \(0<p\le q\), there exists an ellipsoid which solves Problem \(S_{p,q}\). The solution is unique for \(1\le p\le q\).
Proof
For an ellipsoid \(E\in {\mathcal {E}}^{n}\) (the class of origin-symmetric ellipsoids in \({\mathbb {R}}^n\)), we use \(d_{E}\) to denote its maximal principal radius. There exists a \(v_{E}\in S^{n-1}\) such that \(d_{E}|v_{E}\cdot u|\le h_{E}(u),\ \text{ for } \text{ all }\ u\in S^{n-1}\). From definitions of the (p, q)-mixed projection body and the \(L_{p}\)-dual mixed volume, it yields
Let \({\mathcal {E}}_{p,q}=\left\{ E\in {\mathcal {E}}^{n}:\overline{{\widetilde{V}}}_{p,q}(K,E,Q)\le 1\right\} \). Then, the above inequality yields that
Thus, the set \({\mathcal {E}}_{p,q}\) is bounded in the metric space \(({\mathcal {E}}^{n},\delta _{H})\). Using Theorem 3.1, the functional \(\overline{{\widetilde{V}}}_{p,q}(K,\cdot ,Q)\) is continuous, then \({\mathcal {E}}_{p,q}\) is also closed. According to the Blaschke selection theorem, each maximizing sequence of ellipsoids for Problem \(S_{p,q}\) has a convergent subsequence whose limit is still in \({\mathcal {E}}_{p,q}\). Therefore, a solution to Problem \(S_{p,q}\) exists.
We next prove the uniqueness by contradiction. We assume that the ellipsoids \(E_{1}\) and \(E_{2}\) are two different solutions to Problem \(S_{p,q}\). Let \(E_{1}=T_{1}B\) and \(E_{2}=T_{2}B\), where \(T_{1},T_{2}\in {\mathrm{GL}}(n)\). Then \(\det (T_{1})=\det (T_{2})\) and \(\overline{{\widetilde{V}}}_{p,q}(K,E_{i},Q)\le 1\), for \(i=1,2\).
Since each symmetric matrices \(T\in {\mathrm{GL}}(n)\) could be represented in the form \(T=PQ\), where P is symmetric, positive definite and Q is orthogonal. Then we may assume that \(T_{1}\) and \(T_{2}\) are symmetric and positive definite. Then \(T_{1}\ne \lambda T_{2}\), for all \(\lambda >0\). The Minkowski inequality for positive definite matrices implies
Let \(E_{3}=\frac{1}{2}(T_{1}+T_{2})B\). Then we have
From (2.2) and the triangle inequality, one has for all \(u\in S^{n-1}\),
Now, from Definition (1.13), the monotonicity of \(f(t)=t^{p},~p\ge 1\), (4.4), and the convexity of \(f(t)=t^{p}\), it follows that
Then \(E_{3}\in {\mathcal {E}}_{p,q}\). That is, \(E_{3}\) satisfies the constraint \(\overline{{\widetilde{V}}}_{p,q}(K,E_{3},Q)\le 1\). Then, it will result in \(|E_{3}|\le |E_{1}|=|E_{2}|\), which contradicts (4.3). \(\square \)
Our main problems \(S_{p,q}\) and \({\overline{S}}_{p,q}\) are two equivalent description. The solutions to \(S_{p,q}\) and \({\bar{S}}_{p,q}\) differ by only a scale factor. To prove this conclusion, we need the next lemma.
Lemma 4.2
Let \(p,q>0, K\in {\mathcal {K}}^{n}_{o}\) and \(Q\in {\mathcal {S}}^{n}_{o}\). Then
and
Proof
We first prove that the ellipsoid \(E_1\) with \(\overline{{\widetilde{V}}}_{p,q}(K,E_{1},Q)<1\) cannot be the maximizer of \(\max \nolimits _{\{E\in {\mathcal {E}}^{n}:\overline{{\widetilde{V}}}_{p,q}(K,E,Q)\le 1\}}|E|\). In fact, for the ellipsoid \(\overline{{\widetilde{V}}}_{p,q}(K,E_{1},Q)^{-1}E_{1}\), its volume is greater than the volume of \(E_1\), i.e.,
And one has from (2.16),
as required.
We next prove (4.6). For any ellipsoid \(E_2\) with \(|E_2|>\omega _n\), the ellipsoid \(\left( \frac{\omega _n}{|E_2|}\right) ^{\frac{1}{n}}E_2\) satisfies \(\left| \left( \frac{\omega _n}{|E_2|}\right) ^{\frac{1}{n}}E_2\right| =\omega _n\). And from (2.16), it follows that
\(\square \)
Theorem 4.3
Suppose \(p,q>0\) and K is an origin-symmetric convex body in \({\mathbb {R}}^{n}\), and Q is a star body in \({\mathbb {R}}^{n}\) about the origin.
\({\mathrm{(1)}}\) If \(E_{M}\) is an origin-symmetric ellipsoid that is a \(S_{p,q}\) solution for K and Q, then
is a solution to Problem \({\bar{S}}_{p,q}\).
\({\mathrm{(2)}}\) If \(E_{m}\) is an origin-symmetric ellipsoid that is a \({\bar{S}}_{p,q}\) solution for K and Q, then
is a solution to Problem \(S_{p,q}\).
Proof
\({\mathrm{(1)}}\) Let \(E\in \{E\in {\mathcal {E}}^{n}: |E|\ge \omega _{n}\}\). It follows from (2.16) that
Then, from the assumption that \(E_M\) is a \(S_{p,q}\) solution, it follows
Therefore,
where the last equality uses the fact \(\overline{{\widetilde{V}}}_{p,q}(K,E_{M},Q)=1\) by (4.5). Added that \(\left( \frac{\omega _{n}}{|E_{M}|}\right) ^{\frac{1}{n}}E_{M}\in \{E\in {\mathcal {E}}^{n}: |E|\ge \omega _{n}\}\), it implies that the ellipsoid \(\left( \frac{\omega _{n}}{|E_{M}|}\right) ^{\frac{1}{n}}E_{M}\) is a solution to Problem \({\bar{S}}_{p,q}\).
\({\mathrm{(2)}}\) Let \(E\in \left\{ E\in {\mathcal {E}}^{n}:\overline{{\widetilde{V}}}_{p,q}(K,E,Q)\le 1\right\} \). Since \(E_m\) is an \({\overline{S}}_{p,q}\) solution, and \(\left( \frac{\omega _{n}}{|E|}\right) ^{\frac{1}{n}}E\in \{E\in {\mathcal {E}}^{n}: |E|=\omega _{n}\},\) it follows from (2.16) that
Using (4.6), we have \(|E_m|=\omega _n\). Then \(\overline{{\widetilde{V}}}_{p,q}(K,E_{m},Q)^{-1}|E_m|^{\frac{1}{n}} \ge \overline{{\widetilde{V}}}_{p,q}(K,E,Q)^{-1}|E|^{\frac{1}{n}}.\) Thus, it results in
Then the proof is completed by observing \(\overline{{\widetilde{V}}}_{p,q}\left( K,\overline{{\widetilde{V}}}_{p,q}(K,E_{m},Q)^{-1}E_{m},Q\right) =1\). \(\square \)
In Theorem 4.1, we proved the existence for all cases of \(0<p\le q\), and the uniqueness for the cases of \(1<p\le q\). In order to show the uniqueness of for all cases of \(0<p\le q\), we need the next lemma that shows that, without loss of generality, we may assume that the ellipsoid E is the unit ball B in \({\mathbb {R}}^{n}\).
Lemma 4.4
Suppose real \(p, q\ne 0, K\in {\mathcal {K}}^{n}_{o}\) and \(Q\in {\mathcal {S}}^{n}_{o}\). If \(\phi \in {\mathrm{GL}}(n)\), then
if and only if
Proof
In light of Lemma 2.1, it suffices to prove the statement for \({\mathrm{SL}}(n)\). In terms of (2.2), (2.4) and Lemma 2.3, we have, for all \(x\in {\mathbb {R}}^{n}\),
Then, using Definition 2.5, (4.10) is equivalent to, for all \(x\in {\mathbb {R}}^{n}\),
which by Lemma 2.6 is in turn equivalent to
\(\square \)
Now we show the existence and uniqueness of solution \(S_{p,q}\) and \({\bar{S}}_{p,q}\) for all cases \(0<p\le q\).
Theorem 4.5
Suppose that \(0<p\le q=p+r, r\in [0,\infty ), K\in {\mathcal {K}}^{n}_{o}\) and \(Q\in {\mathcal {S}}^{n}_{o}\). Then \(S_{p,q}\) as well as \({\bar{S}}_{p,q}\) has a unique solution. Moreover, an ellipsoid \(E\in {\mathcal {E}}^{n}\) solves \({\bar{S}}_{p,q}\) if and only if it satisfies
and an ellipsoid \(E\in {\mathcal {E}}^{n}\) solves \(S_{p,q}\) if and only if it satisfies
Proof
We first show that an ellipsoid \(E\in {\mathcal {E}}^{n}\) solves \({\bar{S}}_{p,q}\) if and only if it satisfies (4.11). Without loss of generality, we may assume \(E=B\) by using Lemma 4.4. Namely, we will show that B is a \({\bar{S}}_{p,q}\) solution for K and Q if and only if
Firstly, we show if \(B\in {\mathcal {E}}^{n}\) solves \({\bar{S}}_{p,q}\), then (4.13) holds. Indeed, suppose that \(T\in {\mathrm{SL}}(n)\). Choose \(\varepsilon _{0}>0\) sufficiently small so that for all \(\varepsilon \in (-\varepsilon _{0},\varepsilon _{0})\), \(I_n+\epsilon T\) is invertible, where \(I_n\) is identity matrix. For \(\varepsilon \in (-\varepsilon _{0},\varepsilon _{0})\), define \(T_{\varepsilon }\in {\mathrm{SL}}(n)\) by
Since \(|T_{\varepsilon }|=1\), the ellipsoid \(E_{\varepsilon }=T^{t}_{\varepsilon }B\) clearly has volume \(\omega _{n}\). The support function of \(E_{\varepsilon }\) is given by
Since \(E_0=B\) is a \({\bar{S}}_{p,q}\) solution, we have
and hence using (1.9), it is equivalent to
Note that
and
then (4.14) implies
Let \(T=x\otimes x\) for nonzero \(x\in {\mathbb {R}}^{n}\), where the notation \(x\otimes x\) represents the rank 1 linear operator on \({\mathbb {R}}^{n}\) that takes y to \((x\cdot y)x\). It immediately gives that \({\mathrm{tr}}(x\otimes x)=|x|^{2}\). Using the facts \({\mathrm{tr}}(x\otimes x)=|x|^{2}\) and \(u\cdot (x\otimes x)u=(u\cdot x)^{2}\), (4.15) is
Secondly, we show if
then B is a solution to Problem \({\bar{S}}_{p,q}\). Moreover, B is a unique \({\bar{S}}_{p,q}\) solution.
To prove that B is a \({\bar{S}}_{p,q}\) solution for K, Q, we show that for any ellipsoid E with \(|E|=\omega _n\), one has
with equality if and only if \(E=B\). It is equivalent to show that for any ellipsoid E with \(E=P^t B\), \(P\in {\mathrm{SL}}(n)\), one has
with equality if and only if \(Pu=1\) for all \(u\in S^{n-1}\). From Jensen’s inequality,
with equality if and only if there exists \(c>0\) such that \(|Pu|=c\) for all \(u\in {\mathrm{supp}} {\widetilde{C}}_{p,q}(K,Q,\cdot )\). Hence, we need show
We write P as \(P=O^tDO\), where \(D={\mathrm{diag}}(\lambda _1,\lambda _2,\cdots ,\lambda _n)\) is a diagonal matrix with eigenvalues \(\lambda _1,\lambda _2,\cdots ,\lambda _n\), and O is orthogonal.
From Definition 2.5 and Lemma 2.6, it follows that
Then by the concavity of the \(\log \) function and (4.16),
where \(u_i\) denotes \(u\cdot e_i\) for \(i=1,\cdots ,n\). Since \(|D|=1\), we have \(\sum \limits _{i=1}^n\log (\lambda _i)=\log (\prod \limits _{i=1}^n\lambda _i)=0\) Thus (4.19) holds. And then we have (4.16), namely B is a solution to Problem \({\bar{S}}_{p,q}\).
For the uniqueness of Problem \({\bar{S}}_{p,q}\), we only need consider the equality condition. Note that the strict concavity of \(\log \) function implies that equality in (4.16) holds only if \(u_{i_1},\cdots ,u_{i_N}\ne 0\) implies \(\lambda _{i_1}=\cdots =\lambda _{i_N}\), for \(u\in {\mathrm{supp}}{\widetilde{C}}_{p,q}(O K,O Q,\cdot )\). Thus \(|Du|=\lambda _i\) when \(u_i\ne 0\) for \(u\in {\mathrm{supp}}{\widetilde{C}}_{p,q}(O K,O Q,\cdot )\). Equality in (4.18) forces \(|Pu|=c\) for all \(u\in {\mathrm{supp}}{\widetilde{C}}_{p,q}(O K,O Q,\cdot )\). Since \({\mathrm{supp}}{\widetilde{C}}_{p,q}(O K,O Q,\cdot )\) is not contained in an \((n-1)\)-dimensional subspace of \({\mathbb {R}}^n\), we have \(\lambda _i=c\) for all i. Combining with \(|D|=\lambda _1\cdots \lambda _n=1\), we have \(\lambda _i=1\) for all i. Thus \(D=I_n\), and \(P=I_n\).
Note that Theorems 4.1 and 4.3 get the existence of the solution to Problems \(S_{p,q}\) and \({\bar{S}}_{p,q}\). And their uniqueness is proved from the above proof and Theorem 4.3.
Finally, we let the ellipsoid \(E\in {\mathcal {E}}^n\) solve Problem \(S_{p,q}\). Using Theorem 4.3, it is equivalent to that \(c_0E\) is a solution to Problem \({\bar{S}}_{p,q}\), where \(c_0=(\frac{\omega _n}{|E|})^{\frac{1}{n}}\). It holds if and only if (4.11) holds, i.e.,
This completes the result by noticing that \(\overline{{\widetilde{V}}}_{p,q}(K,E,Q)= \left( \frac{{\widetilde{V}}_{p,q}(K,E,Q)}{{\widetilde{V}}_{r}(K,Q)}\right) ^{\frac{1}{p}} =1\) from Lemma 4.2. \(\square \)
Let \(0<p\le q\le \infty \). Theorem 4.5 shows that problem (\(S_{p,q}\)) has a unique solution. In the case \(Q=K\), the \(S_{p,q}\) problem had been considered by Lutwak, Yang and Zhang in [30].
In the case \(p=\infty \), with the aid of (1.16), we may rephrase \((S_{\infty ,q})\) as: Among all origin-symmetric ellipsoids, find an ellipsoid which solves the following constrained maximization problem:
When \(Q=K\), the problem is the classical John-ellipsoid problem (see, e.g., Giannopoulos and Milman [12]).
In light of Theorem 4.1, Theorem 4.3 and Theorem 4.5, we introduce a family of ellipsoids, which is an extension of LYZ’s \(L_{p}\) John ellipsoids.
Definition 4.6
Let \(0<p\le q=p+r\le \infty , r\in [0,\infty )\). Suppose K is a convex body in \({\mathbb {R}}^{n}\) that contains the origin in its interior and Q is a star body (about the origin) in \({\mathbb {R}}^{n}\). Among all origin-symmetric ellipsoids, the unique ellipsoid that solves the constrained maximization problem
will be called the (p, q)-John ellipsoid of K and Q, and will be denoted by \(E_{p,q}(K,Q)\).
Among all origin-symmetric ellipsoids, the unique ellipsoid that solves the constrained minimization problem
will be called the normalized (p, q)-John ellipsoid of K and Q, and will be denoted by \({\bar{E}}_{p,q}(K,L)\).
Note that in the case \(Q=K\), \(E_{p,q}(K,K)=E_{p}(K)\) is the \(L_p\)-John ellipsoid. In the case that \(q=n\) and \(Q=B\), \(E_{p,n}(K,B)=E_{p}(K)\) is also the \(L_p\)-John ellipsoid. In the case that \(p=\infty \) and \(Q=K\), \(E_{\infty ,\infty }(K,K)=J(K)\) is also the classic John ellipsoid.
From Definition 4.6 and (2.20), we immediately obtain
Lemma 4.7
Suppose \(K\in {\mathcal {K}}^{n}_{o}\) and \(Q\in {\mathcal {S}}^{n}_{o}\), and \(0<p\le q\le \infty \). Then for \(\phi \in {\mathrm{GL}}(n)\),
From \(E_{p,q}(B,B)=E_pB=B\) and Lemma 4.7, we see that if \(E\in {\mathcal {E}}^{n}\), then
Note that if the John point of K is at the origin (e.g., if K is origin-symmetric), then
From (2.24), (4.12) of Theorem 4.5, we immediately obtain
Lemma 4.8
Suppose \(K\in {\mathcal {K}}^{n}_{o}, Q\in {\mathcal {S}}^{n}_{o}\) and \(2\le q\le \infty \). Then
A finite positive Borel measure \(\mu \) on \(S^{n-1}\) is said to be isotropic if (see [12])
for all \(v\in S^{n-1}\), where \(|\mu |\) denotes the total mass of \(\mu \). For nonzero \(x\in {\mathbb {R}}^{n}\), the notation \(x\otimes x\) represents the rank 1 linear operator on \({\mathbb {R}}^{n}\) that takes y to \((x\cdot y)x\). It immediately gives that \({\mathrm{tr}}x\otimes x=|x|^{2}\). Equivalently, \(\mu \) is isotropic if
From definition (1.6) and (1.9), we see that
Therefore, the condition (4.11) is equivalent to
Then an immediate consequence of Theorem 4.5 is
Corollary 4.9
Suppose \(K\in {\mathcal {K}}^{n}_{o}\) with \(Q\in {\mathcal {S}}^{n}_{o}\), and \(0<p\le q\in (0,\infty ]\). Then there exists a unique solution to the following constrained minimization problem:
Moreover, the identity operator \(I_{n}\) is the solution if and only if \(L_{p}\) dual curvature measures \({\widetilde{C}}_{p,q}(K,Q,\cdot )\) are isotropic on \(S^{n-1}\).
Corollary 4.10
Suppose \(K\in {\mathcal {K}}^{n}_{o}\) with \(Q\in {\mathcal {S}}^{n}_{o}\), and \(0<p\le q\in (0,\infty ]\).
-
(1)
There exists an \({\mathrm{SL}}(n)\) transformation T, such that \({\widetilde{C}}_{p,q}(TK,TQ,\cdot )\) is isotropic on \(S^{n-1}\).
-
(2)
If \(T_{1},T_{2}\in {\mathrm{SL}}(n)\) such that \({\widetilde{C}}_{p,q}(T_{1}K,T_{1}Q,\cdot )\), \({\widetilde{C}}_{p,q}(T_{2}K,T_{2}Q,\cdot )\) are both isotropic on \(S^{n-1}\), then there exists an orthogonal \(O\in {\mathrm{O}}(n)\) such that \(T_{2}=O T_{1}\).
5 Continuity of (p, q)-John Ellipsoids
In this section, we show that the family of (p, q)-John ellipsoids associated with a convex body and a star body in \({\mathbb {R}}^{n}\) is continuous in \(p\in (0,\infty ]\).
We assume that \(K\in {\mathcal {K}}^{n}_{o}\) and \(Q\in {\mathcal {S}}^{n}_{o}\) are two fixed bodies in this section.
Lemma 5.1
Suppose \(0<p\le q\le \infty \). If \(aB\subseteq K\subseteq bB\) and \(aB\subseteq Q\subseteq bB\) for \(a,b>0\), then
where
Proof
From (4.1) and the definition of \({\bar{E}}_{p,q}(K,Q)\), we have
Now, we estimate the value of \(\overline{{\widetilde{V}}}_{p,q}(K,B,Q)\). By the definition of \(\overline{{\widetilde{V}}}_{p,q}(K,L,Q)\), we have
Note that
By the definition of (p, q)-mixed projection body and (5.3), we have
Together with (5.1), (5.2) and (5.4), and note that \({\widetilde{V}}_{r}(K,Q)\le \frac{\omega _{n}b^{n+r}}{a^{r}}\), we have
Therefore,
Note that \(\lim \limits _{p\rightarrow \infty }(c_{n-2,p})^{\frac{1}{p}}=1\), then \({\bar{E}}_{\infty ,\infty }(K,L)\subseteq \frac{b}{a}B\). \(\square \)
From Definition 4.6, we recall that for each \(p\in (0,\infty ]\) and \(q=p+r, r\in (0,\infty )\), the ellipsoid \({\bar{E}}_{p,q}(K,Q)\) is the unique ellipsoid that satisfies
Lemma 5.2
If \(p,p_{0}\in (0,\infty ], q=p+r, r\in (0,\infty ),p\rightarrow p_{0}, q\rightarrow p_{0}+r=q_{0}, K\in {\mathcal {K}}^{n}_{o}\), and \(Q\in {\mathcal {S}}^{n}_{o}\), then
Proof
Using the Definition \({\bar{E}}_{p,q}(K,Q)\), Theorem 3.1, (5.5), and again the definition of \({\bar{E}}_{p,q}(K,Q)\), we have
\(\square \)
Lemma 5.3
Suppose that \(p,p_{0}\in (0,\infty ], q=p+r, r\in (0,\infty ),p\rightarrow p_{0}, q\rightarrow p_{0}+r=q_{0}\), and \(K\in {\mathcal {K}}^{n}_{o}, Q\in {\mathcal {S}}^{n}_{o}\). If \(aB\subseteq Q\subseteq K\subseteq bB\) or \(aB\subseteq K\subseteq Q\subseteq bB\), for \(a,b>0\), then
Proof
We argue by contradiction and assume the conclusion to be false. Lemma 5.1, the Blaschke selection theorem, and our assumption, give a sequence \(p_{i}\rightarrow p_{0}\), as \(i\rightarrow \infty \), such that \(\lim \limits _{i\rightarrow \infty }{\bar{E}}_{p_{i},q_{i}}(K,Q)=E'\ne {\bar{E}}_{p_{0},q_{0}}(K,Q)\). Since the solution to Problem \(({\bar{S}}_{p,q})\) is unique, and by the uniform convergence established in Theorem 3.1, we get
This contradicts to Lemma 5.2. \(\square \)
Since, by Theorem 4.3, \(E_{p,q}(K,Q)=\overline{{\widetilde{V}}}_{p,q}\left( K,{\bar{E}}_{p,q}(K,Q),Q\right) ^{-1}{\bar{E}}_{p,q}(K,Q)\), the above gives
Theorem 5.4
If \(p,p_{0}\in (0,\infty ], q=p+r, r\in (0,\infty ),p\rightarrow p_{0}, q\rightarrow p_{0}+r=q_{0}, K\in {\mathcal {K}}^{n}_{o}\) and \(Q\in {\mathcal {S}}^{n}_{o}\), then
6 Generalizations of John’s Inclusion
John’s inclusion states that if K is an origin-symmetric convex body in \({\mathbb {R}}^{n}\), then
\(L_{p}\) version of John’s inclusion is (see [30]): If K is a convex body in \({\mathbb {R}}^{n}\) that contains the origin in its interior, then
In this section, we shall prove a (p, q)-version of John’s inclusion.
From (1.4), (2.1), (2.5) and Definition (2.26), we see immediately that if \(\lambda >0\), then
Lemma 6.1
If \(p\in (0,\infty ], q=p+r, r\in [0,\infty )\) and \(K\in {\mathcal {K}}^{n}_{o}\), as well as \(Q\in {\mathcal {S}}^{n}_{o}\), then for \(\phi \in {\mathrm{GL}}(n)\)
Proof
From (6.2) it is sufficient to prove the formula when \(\phi \in {\mathrm{SL}}(n)\). For real \(p>0\), it follows from Definition (2.24), Lemma 2.6, Definition 2.5, Definition (2.24) again, and (2.6) that for \(u\in S^{n-1}\),
The \(p=\infty \) case is now a direct consequence of the real case and Definition (2.25). \(\square \)
Lemma 6.2
If \(K\in {\mathcal {K}}^{n}_{o}, Q\in {\mathcal {S}}^{n}_{o}, p\in (0,\infty ]\) and \(q=p+r, r\in [0,\infty )\), then
Proof
Lemmas 4.7 and 6.1 show that it suffices to prove the inclusions when \(E_{p,q}(K,Q)=B\). For \(0<p<2\), Definition (2.24) and Theorem 4.5 show that for each \(u\in S^{n-1}\),
This gives \(\Gamma _{-p,-q}(K,Q)\subseteq B=E_{p,q}(K,Q)\) when \(0<p<2\).
When \(2\le p<\infty \), the inequality is reversed. Thus \(E_{p,q}(K,Q)\subseteq \Gamma _{-p,-q}(K,Q)\) for \(2\le p<\infty \). The case \(p=\infty \) follows from the real case together with Theorem 5.4 and Definition (2.25). \(\square \)
Of course the case \(p =2\) of Lemma 6.2 is known from Lemma 4.8: \(E_{2,q}(K,Q)=\Gamma _{-2,-q}(K,Q)\).
Our general \(L_{p}\) version of John’s inclusion will be a corollary of
Theorem 6.3
If \(K\in {\mathcal {K}}^{n}_{o}, Q\in {\mathcal {S}}^{n}_{o}, p_i\in (0,\infty ], q_i=p_i+r, r\in [0,\infty )\), \(i=1,2\), then
Proof
Note that \(q_{i}=p_{i}+r, i=1,2\) and \(0\le r<\infty \). Lemmas 4.7 and 6.1 show that it suffices to prove the inclusions when \(E_{p_{2},q_{2}}(K,Q)\) is the unit ball B. Since \(E_{p_{2},q_{2}}(K,Q)=B\), Definition 4.6 gives
Suppose \(0<p_{2}\le 2\). Now Definition (2.26), Definition (1.6), Jensen’s inequality, Definition (1.6) again, (6.3), Jensen’s inequality again, (6.3) again, and finally Theorem 4.5 show that for each \(u\in S^{n-1}\),
Thus, \(n^{\frac{1}{2}-\frac{1}{p_{1}}}E_{p_{2},q_{2}}(K,Q)\subseteq \Gamma _{-p_{1},q_{1}}(K,Q)\).
When \(2\le p_{1}\le p_{2}<\infty \), the inequality above is reversed. Thus,
The case \(p=\infty \) follows from the real case together with Theorem 5.4 and Definition (2.25). \(\square \)
By taking \(p_{1} =p_{2}=p\) in Theorem 6.3 and combining the inclusions with those of Lemma 6.2 we get the general \(L_{p}\) version of John’s inclusion:
Corollary 6.4
If \(K\in {\mathcal {K}}^{n}_{o}, Q\in {\mathcal {S}}^{n}_{o}, p,q\in (0,\infty ]\) with \(p\le q\), then
7 Volume-Ratio Inequalities
We first established the following inequality.
Theorem 7.1
If \(K\in {\mathcal {K}}^{n}_{o}, Q\in {\mathcal {S}}^{n}_{o}\), \(r\in [0,\infty )\) and \(p_{1},p_{2},p_{1},p_{2}\in (0,+\infty ]\) with satisfying that \(p_1<p_2\), \(q_{1}=p_1+r\) and \(q_2=p_{2}+r\), then
Proof
From Definitions (1.10), together with Jensen’s inequality, it follows that for \(0<p_{1}\le p_{2}\le \infty \),
This together with Definition 4.6 immediately gives the desired result for real \(p_{2}\) and \(q_{2}\). For the case \(p_{2}=\infty , q_{2}=\infty \), the result follows from the real case and Theorem 5.4. \(\square \)
In general, the (p, q)-John ellipsoid \(E_{p,q}(K,Q)\) is not contained in K or Q. However when \(1\le \frac{q}{n}\le p\le q \le n+p\le \infty \), the volume of \(E_{p,q}(K,Q)\) can be dominated by volume of Q.
Theorem 7.2
If \(K\in {\mathcal {K}}^{n}_{o}, Q\in {\mathcal {S}}^{n}_{o}\) and \(1\le \frac{q}{n}\le p\le q \le n+p\le \infty \), then
with equality if and only if K, Q are origin-symmetric ellipsoids with dilates of each other when \(1\le \frac{q}{n}< p\), while K, Q are an ellipsoid with dilates of each other when \(p=1,q=n\).
Proof
First suppose \(p<\infty \). From Definition (1.9), Definition 4.6 and the \(L_{p}\)-Minkowski inequality (see Lemma 2.4), we have
with equality if and only if K, Q and \(E_{p,q}(K,Q)\) are dilates when \(1<\frac{q}{n}<p\), while K and \(E_{p,q}(K,Q)\) are dilates when \(q=n\) and \(p >1\), and K and \(E_{p,q}(K,Q)\) are homothetic when \(q=n, p=1\).
From the dual \(L_{p}\)-Minkowski inequality (2.9), we have
with equality if and only if K and Q are dilates for \(0<r=q-p<n\).
Together with (7.2) and (7.3), we immediately get
The condition of equality follows from ones in (7.2) and (7.3).
For \(p=\infty \) the results follows from the argument for the real case and Theorem 7.1. \(\square \)
When \(Q=K\), an immediate consequence of Theorem 7.2 is
Corollary 7.3
If \(K\in {\mathcal {K}}^{n}_{o}\) and \(1\le p\le \infty \), then
with equality for \(p >1\), if and only if K is an origin-symmetric ellipsoid, and equality for \(p=1\) if and only if K is an ellipsoid.
Note that this inequality is about \(L_{p}\) John ellipsoid proved by Lutwak, Yang and Zhang [30].
If \(p,q\in (0,\infty ]\), K is an origin-symmetric convex body in \({\mathbb {R}}^{n}\), and Q is a star body (about the origin) in \({\mathbb {R}}^{n}\), then K is said to be (p, q)-isotropic with respect to Q, if there exists a \(c >0\), such that
For \(Q=K\), then K is said to be \(L_{p}\) isotropic (see [30]).
Theorem 4.5 shows that K is (p, q)-isotropic with respect to Q if and only if there exists a \(\lambda >0\), such that
Theorem 7.4
If \(0\le r\le n\), K and Q are origin-symmetric convex body in \({\mathbb {R}}^{n}\), and K is \((1,1+r)\)-isotropic with respect to Q, then for \(u\in S^{n-1}\),
Proof
If inequality (7.5) holds for bodies K and Q, then it obviously holds for all \(\lambda K\) and \(\lambda Q\) with \(\lambda >0\). Thus for K that is \((1,1+r)\)-isotropic with respect to Q we may assume that \(E_{1,1+r}(K,Q)=B\). It is necessary to show that
Definition 4.6 combined with Definition (1.13) gives
From Definition (2.23), (7.6), Jensen’s inequality, (7.6) again, and finally Theorem 4.5, it follows
Then we have,
Note that \(0\le r\le n\), by using dual Minkowski inequality (2.9), we have
\(\square \)
In particular, by taking \(Q=K\) in (7.5), and \(h_{\Pi _{1,1+r}(K,Q)}(u)=\frac{1}{n}h_{\Pi (K)}(u)=\frac{1}{n}{\mathrm{vol}}_{n-1}(K|u^{\bot })\), we have (see [30])
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This work was supported by the National Natural Science Foundation of China (Grant No. 11561020) and Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2020JQ-236)
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Ma, T., Wu, D. & Feng, Y. (p, q)-John Ellipsoids. J Geom Anal 31, 9597–9632 (2021). https://doi.org/10.1007/s12220-021-00621-4
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DOI: https://doi.org/10.1007/s12220-021-00621-4