Abstract
When \(p>0\), the dual \(L_p\) John ellipsoids provide a unified treatment for the Löwner ellipsoid and the Legendre ellipsoid associated with a convex body. When \(p<0\), very little relevance to known ellipsoids has been found for the dual \(L_p\) John ellipsoid so far. In this paper we investigate a sharp dual \(L_p\) John ellipsoid problem associated with origin-symmetric convex bodies when \(p\le -n-1\). The solution unifies the classic John ellipsoid and the classic Petty ellipsoid.
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1 Introduction
The classic John ellipsoid associated with a convex body \(K\subset \mathbb {R}^n\) is the ellipsoid of maximal volume contained within K. John (1948) proved that each convex body in \(\mathbb {R}^n\) contains a unique ellipsoid of maximal volume.
In their seminal work, Lutwak et al. (2005) showed that the Löwner–John ellipsoid, as well as the Petty ellipsoid (Petty 1961) and the Lutwak–Yang–Zhang ellipsoid (Lutwak et al. 2000b), are special solutions to the optimization problem minimizing the total \(L_p\) curvature of a convex body that contains the origin in its interior. The \(L_p\) John (Löwner–John) ellipsoid is defined to be the solution of minimizing the \(L_p\) surface area of a convex body in \(\mathbb {R}^n\). In fact, the \(L_p\) John (Löwner–John) ellipsoid theorem and its applications form an important ingredient of the \(L_p\) Brunn–Minkowski theory founded by Lutwak and his colleagues; see, e.g. Lutwak (1993, 1996) and Lutwak et al. (2000a, 2004). The very starting point of the \(L_p\) Brunn–Minkowski theory is the Minkowski–Firey \(L_p\) combination of convex bodies. If K, L are convex bodies that contain the origin in their interiors, then for \(p\ge 1\) and \(\varepsilon >0\), the \(L_p\) combination \(K+_p\varepsilon \cdot L\) of K and L can be determined by
where \(h_M\) is the support function of a compact set \(M\subset \mathbb {R}^n\) defined by \(h_M(x)=\max \{x\cdot y: y\in M\}\). If K, L are convex bodies that contain the origin in their interiors, then the \(L_p\) mixed volume \(V_p(K,L)\) of K and L can be expressed by (see Sect. 2 for details):
where \(S(K,\cdot )\) is the classic surface area measure of K. The measure \(h_K^{1-p}dS(K,\cdot )\) is known as the \(L_p\) surface area measure of K and usually is denoted by \(dS_p(K,\cdot )\); see Lutwak (1993) or next section for details. The \(L_p\) John (Löwner–John) ellipsoid was defined in Lutwak et al. (2005) as the solution to the following minimizing problem:
Indeed, in order to include the limiting case \(p=\infty \), a volume-normalizing treatment was adopted to the above problem in Lutwak et al. (2005). Extensions of John’s inclusion and Ball’s volume ratio inequality are also demonstrated therein. We refer to Böröczky et al. (2015), Giannopoulos and Milman (2000), Giannopoulos and Papadimitrakis (1999), Gruber (2011), Pisier (1989) and Schuster and Weberndorfer (2012) for a part of related research on affine positions of convex bodies and their applications.
The dual \(L_p\) John ellipsoid problem was studied in Bastero and Romance (2004) and Yu et al. (2007), which deals with optimization problems for the dual \(L_p\) mixed volumes of convex bodies. For a real number \(p\in \mathbb {R}\), the dual \(L_p\) mixed volume \(\widetilde{V}_{-p}(K,L)\) of convex bodies K and L that contain the origin in their interiors is defined as
where \(\rho _M\) is the radial function of star-shaped M (about the origin) given by \(\rho _M(u)=\max \{t>0: tu\in M\}\).
Bastero and Romance (2004) studied the following optimization problems for real p.
When \(p>0\), a volume-normalized technique for (1.2) was adopted in Yu et al. (2007) to achieve a unified treatment for the Löwner ellipsoid and the Legendre ellipsoid of the body K. However, when \(p<0\), the dual \(L_p\) John ellipsoids has been found very little relevance to known ellipsoids so far.
The main aim of this paper is to pose and solve a sharp version of the dual \(L_p\) John ellipsoid problem associated with origin-symmetric convex bodies when \(p\le -n-1\). The advantage of this sharp dual \(L_p\) John ellipsoid problem exists in that it provides a unified treatment for the John ellipsoid and the Petty ellipsoid of the involved symmetric convex body.
We shall begin with the normalized \(L_q\) Blaschke combination of two origin-symmetric convex bodies K and L. For \(q\ge 1\) and \(\varepsilon >0\), the volume-normalized \(L_q\) Blaschke combination \(K\overline{\#}_q\varepsilon \circ L\) of K and L is defined to be an origin-symmetric convex body:
We shall also need the notion of \(L_q\) polar projection body \(\Gamma _{-q}K\) of a convex body K that contains the origin in its interior: if \(q>0\), then
A useful property of \(\Gamma _{-q}K\) is the following
For \(q\ge 1\), we shall introduce a new affine invariant associated with two origin-symmetric convex bodies K and L as
It will be shown that
The optimization problem we are interested in this paper is the following
By using (1.3) and (1.4), one can prove that
where \(C_{n,q}\) is a constant depending on n and \(q\ge 1\) (see Proposition 3.5 of this paper for details). Thus, from (1.6) and the basic fact that \(\widetilde{V}_{-q}(B_2^n,\phi K)=\widetilde{V}_{n+q}(\phi K,B_2^n)\) we have
Therefore, by setting \(p=-n-q\) with \(q\ge 1\) (correspondingly, \(p\le -n-1\)) in the optimization problem (1.2), we see that up to a multiplication the minimization problem (1.5) is sharper than (1.2).
In order to unify some known important John type ellipsoid problems, we shall adopt the volume-normalized technique mentioned above. To this end, it would be more convenient to define a volume-normalized version of the affine invariant \(A_q(K,L)\) of convex bodies K and L:
We shall consider the following constrained optimization problem whose solution only differs by a scale factor from that of the problem (1.5):
As will be seen, when \(q=1\) the above minimizing problem reduces to Petty’s ellipsoid problem; and when \(q\rightarrow \infty \), it becomes the classic John ellipsoid problem for origin-symmetric bodies.
This paper is organized as follows. In Sect. 2 we collect some necessary basics from convex geometry so that the context is as self-contained as possible. We devote Sect. 3 to the aforementioned affine invariant and its normalization of two origin-symmetric convex bodies. In Sect. 4 we investigate the minimizing problem to achieve a sharp dual \(L_p\) John ellipsoid for \(p\le -n-1\). A characterization and uniqueness of the so-called sharp dual \(L_p\) John ellipsoid of an origin-symmetric convex body are established. The continuity of the dual \(L_p\) John ellipsoids and associated volume ratio inequalities are studied in Sects. 5 and 6, respectively.
2 Preliminaries
Excellent references for convex geometry are the books due to Schneider (2014), Gruber (2007) and Gardner (2006).
The setting for this paper is the n-dimensional Euclidean space, \(\mathbb {R}^n\). We shall write \(x\cdot y\) for the standard inner product of \(x,y\in \mathbb {R}^n\). Let \(B_2^n\) and \(S^{n-1}\) denote the standard Euclidean unit ball and the unit sphere in \(\mathbb {R}^n\).
For \(s\ne -2,-4,\ldots \), define the constant \(\omega _s\) by
where \(\Gamma (\cdot )\) is the Gamma function.
The most fundamental functional for convex body in \(\mathbb {R}^n\) is volume (Lebesgue measure), denoted by \(|\cdot |\). The volume of \(B_2^n\) is equal to \(\omega _n\); i.e., \(|B_2^n|=\omega _n\). Note that a convex body in \(\mathbb {R}^n\) in this paper is understood to be a compact, convex subset of \(\mathbb {R}^n\) with nonempty interior.
Let K be a convex body in \(\mathbb {R}^n\) and \(\nu _K: \partial ' K\rightarrow S^{n-1}\) the Gauss map, where \(\partial 'K\) is the set of boundary points of K that have only one unit normal vector. It is worth noting that \(\partial K\backslash \partial ' K\) has \(\mathcal {H}^{n-1}\)-measure equal to zero. For each Borel set \(\omega \subseteq S^{n-1}\) the inverse spherical image\(\nu _K^{-1}(\omega )\) is defined as a subset of \(\partial 'K\) such that the outer normal of \(x\in \partial 'K\) belongs to \(\omega \). For a convex body K in \(\mathbb {R}^n\), the classic surface area measure of K is defined by
for each Borel set \(\omega \subseteq S^{n-1}\). That is to say, \(S_K(\omega )\) is the \((n-1)\)-dimensional Hausdorff measure of the set of all points on \(\partial 'K\).
The support function of a convex body K in \(\mathbb {R}^n\) is defined by
for \(x\in \mathbb {R}^n\backslash \{0\}\). If a convex body K in \(\mathbb {R}^n\) contains the origin in its interior, then its polar body \(K^*\) is defined by
A compact set K in \(\mathbb {R}^n\) is star-shaped about the origin if the intersection of K with each straight line through the origin is a line segment. Associated with a star-shaped set K in \(\mathbb {R}^n\) is the radial function \(\rho _K: \mathbb {R}^n\backslash \{0\}\rightarrow \mathbb {R}\) which is defined for \(x\ne 0\) by
If \(\rho _K\) is positive and continuous, K is called a star body. The class of star bodies in \(\mathbb {R}^n\) will be denoted by \(\mathcal {S}_o^n\).
It is easily seen that for \(K\in \mathcal {S}_o^n\) and \(A\in \mathrm {SL}(n)\),
Moreover, if K is a convex body in \(\mathbb {R}^n\) that contains the origin in its interior, then it follows that
If in addition K is origin-symmetric then
For real \(a,b\ge 0\) (not both zero), the Minkowski linear combination\(aK+bL\) of convex bodies K, L can be defined either by
or by
More generally, if K, L are convex bodies that contain the origin in their interiors, then for \(p>1\) the Minkowski–Firey\(L_p\)combination\(a\cdot K+_p b\cdot L\) can be defined by
Obviously, \(a\cdot K+_p b\cdot L\) is a convex body containing the origin in its interior.
Throughout, we denote by \(\mathcal {K}^n\), \(\mathcal {K}_o^n\), \(\mathcal {K}_e^n\) the set of convex bodies in \(\mathbb {R}^n\), the set of convex bodies in \(\mathbb {R}^n\) that contain the origin in their interiors, and the set of origin-symmetric convex bodies in \(\mathbb {R}^n\), respectively.
The Hausdorff metric \(\Vert h_K-h_L\Vert _\infty \) of \(K, L\in \mathcal {K}^n\) is defined by
If the body K contains the origin in its interior, then for each \(p>0\), one can define the \(L_p\) surface area measure of K by
If \(K, L\in \mathcal {K}_o^n\), then for \(p\ge 1\) the \(L_p\) mixed volume \(V_p(K,L)\) of K and L is defined by
Obviously, for each \(K\in \mathcal {K}_o^n\),
It was shown by Lutwak (1993) that
The classic Minkowski mixed volume inequality can be stated as: For convex bodies K, L in \(\mathbb {R}^n\),
with equality if and only if there exist \(x\in \mathbb {R}^n\) and \(\lambda >0\) such that \(K=x+\lambda L\).
The \(L_p\) Minkowski mixed volume inequality was established in Lutwak (1993) and can be expressed as: If \(K,L\in \mathcal {K}_o^n\) and \(p>1\), then
with equality if and only if K and L are dilates.
To provide a unified treatment of \(V_p(K,L)\) for all \(p\in [1,\infty )\), especially for the limiting case \(p=\infty \), it is more convenient (see, e.g. Lutwak et al. 2005) to introduce the volume-normalized\(L_p\)mixed volume\(\overline{V}_p(K,L)\) of K and L:
If \(p=\infty \), then
For \(0<p\le \infty \) and \(K\in \mathcal {K}_o^n\), the \(L_p\) John (Löwner–John) ellipsoid of K is the unique solution of the following constrained maximization problem (Lutwak et al. 2005):
where E takes all origin-centered ellipsoids in \(\mathbb {R}^n\).
The principal significance of the \(L_p\) John (Löwner–John) ellipsoid is that it not only solves the optimization problem minimizing the total \(L_p\) curvature of a convex body in \(\mathcal {K}_o^n\), but also unifies the classic Löwner–John ellipsoid (John 1948), the LYZ ellipsoid (Lutwak et al. 2000b), and the Petty ellipsoid (Petty 1961).
For star bodies \(K, L\in \mathcal {S}_o^n\), and \(\varepsilon >0\), the harmonic\(L_p\)radial combination\(K\widetilde{+}_p\varepsilon \cdot L\) is the star body defined by
The dual\(L_p\)mixed volume\(\widetilde{V}_{-p}(K,L)\) of star bodies K and L is defined as
It follows from the polar coordinate formula and the above definition that
To facilitate the formulation of the context for the case \(p=\infty \), it is helpful to normalize the dual \(L_p\) mixed volume with respect to the volume of K:
where
is the dual cone-volume probability measure of K.
For \(p=\infty \), define
Associated with given convex body \(K\in \mathcal {K}_o^n\) is a new family of ellipsoids which are dual to the family of \(L_p\) John (Löwner–John) ellipsoids; see Yu et al. (2007). Let \(0<p\le \infty \) and \(K\in \mathcal {K}_o^n\). Among all origin-centered ellipsoids, the unique ellipsoid solving the constrained maximization problem
is called the dual\(L_p\)John (Löwner–John) ellipsoid of K. Note that both the polar of the classic Löwner–John ellipsoid and the Legendre ellipsoid are special cases of the family of dual \(L_p\) John (Löwner–John) ellipsoids (see Bastero and Romance 2004 for a different way for its characterization).
3 A new mixed-volume like affine invariant
We start this section by presenting the solution to the even normalized \(L_p\) Minkowski problem (see Lutwak et al. 2004): for \(p\ge 1\), if \(\mu \) is an even Borel measure on \(S^{n-1}\) whose support is not contained in a great subsphere of \(S^{n-1}\), then there exists an origin-symmetric convex body K such that \(d\mu =\frac{dS_p(K,\cdot )}{|K|}\). Furthermore, the body K is unique when \(p>1\) and is unique up to a translation when \(p=1\).
Using the uniqueness of the solution of the even \(L_p\) Minkowski problem, we define the normalized\(L_p\)Blaschke combination\(a\circ K\overline{\#}_p b\circ L\), for \(p>1\) and \(a,b\in \mathbb {R}\) (not both zero), of \(K, L\in \mathcal {K}_e^n\) as follows:
Note that when \(p=1\) the \(L_1\) Blaschke combination \(\alpha \circ K\overline{\#}\beta \circ L\) was defined by Lutwak in Lutwak (1988, 1991) as follows: For \(\alpha , \beta \ge 0\) (not both zero), and \(K,L\in \mathcal {K}^n\),
However, by combining the solution to the aforementioned even normalized \(L_1\) Minkowski problem, we see that definition (3.2) also admits to a normalized version just as (3.1) when \(K, L\in \mathcal {K}^n_e\).
We also note that if \(p>n\) the symmetry assumption of K and L in definition (3.1) can be removed because the existence and uniqueness of the \(L_p\) Minkowski problem for general measure have been completely solved for \(p>n\) by Hug et al. (2005).
For a measure \(d\mu (u)\) on \(S^{n-1}\) and a real \(p>0\), the affine image \(d\mu ^{(p)}(\phi u)\) under the affine transform \(\phi \in \mathrm {GL}(n)\) was defined in Lutwak et al. (2005) as
for each \(f\in C(S^{n-1})\). Here \(\langle x\rangle =x/|x|\) for \(x\ne 0\).
It was also shown in Lutwak et al. (2005) that for \(K\in \mathcal {K}_o^n\) and \(p>0\), it follows for \(\phi \in \mathrm {SL}(n)\) that
From this result and the definition of the \(L_p\) Blaschke combination follows:
Lemma 3.1
Suppose \(K, L\in \mathcal {K}_e^n\), and \(\varepsilon \ge 0\). If \(p>1\) and \(\phi \in \mathrm {SL}(n)\), then
If \(p=1\), the above identity holds up to a translation.
Proof
We only prove the case where \(p>1\). First claim that for \(\phi \in \mathrm {GL}(n)\) and \(u\in S^{n-1}\),
Indeed, from (3.3), (3.1), and (3.3) again, we have
for each \(f\in C(S^{n-1})\).
By (3.1), (3.4), (3.6), and (3.4) again, we see that
for each \(u\in S^{n-1}\). The desired result (3.5) follows by the uniqueness of the solution to the normalized \(L_p\) Minkowski problem. \(\square \)
For a convex body \(K\in \mathcal {K}_o^n\), the cone measure\(V_K\) of K is defined as
Observing that
we can define the cone-volume probability measure\(\overline{V}_K\) of K by
For \(p>0\), a star body \(\Gamma _{-p}K\) of a convex body \(K\in \mathcal {K}_o^n\) is defined for \(u\in S^{n-1}\) by
Note that for \(p\ge 1\) the body \(\Gamma _{-p}K\) is a convex body. In particular, if \(p=2\), then \(\Gamma _{-2}K\) defines an ellipsoid, which is known as LYZ ellipsoid associated with the body K; see Lutwak et al. (2000b, 2005). For the limiting case \(p=\infty \), we define \(\Gamma _{-\infty }K\) by
Therefore, from (3.8)
Note that when K is origin-symmetric,
From definitions (3.8) and (3.9), Jensen’s inequality, and the continuity of \(\rho _{\Gamma _{-p}K}\) in \(p\in (0,\infty )\), we have
Lemma 3.2
If \(0<p<r\le \infty \) and \(K\in \mathcal {K}_o^n\), then
It was shown in Lutwak et al. (2005) that for \(\lambda >0\) and \(p\in (0,\infty ]\),
Moreover, for \(\phi \in \mathrm {GL}(n)\) and \(p\in (0,\infty ]\),
A direct calculation shows that \(\Gamma _{-p}B_2^n=c_{n,p}B_2^n\), where \(c_{n,p}\) is a constant given by
Thus, from (3.12) we see that if E is an origin-symmetric ellipsoid, then
Proposition 3.3
Suppose that \(1\le p<\infty \), and \(K,L\in \mathcal {K}_e^n\), then
Proof
From (2.2), (3.8), (3.1), (2.5), and (2.2) again, we obtain
\(\square \)
To be consistent with the notations used in Introduction, henceforth we shall replace p by q for additional notations.
For \(1\le q<\infty \) and \(K, L\in \mathcal {K}_e^n\), we define a mixed affine invariant \(A_q(K,L)\) of K and L as
From definition (3.15), Lemma 3.1, (3.8), and (3.12), it follows that, for \(\phi \in \mathrm {SL}(n)\),
Note that for \(q\ge 1\), \(|\Gamma _{-q}K|\) in definition (3.15) actually is a variant of the \(L_q\)integral affine surface area of \(K\in \mathcal {K}_o^n\); see Zhang (2007) for details.
Lemma 3.4
If \(1\le q<\infty \) and \(K, L\in \mathcal {K}_e^n\), then
Proof
Since the convergence as \(\varepsilon \rightarrow 0^+\) in
is uniform on \(S^{n-1}\), it follows that
\(\square \)
Proposition 3.5
If \(1\le q<\infty \), \(L\in \mathcal {K}_e^n\), and E is an origin-symmetric ellipsoid in \(\mathbb {R}^n\), then
Proof
From Lemma 3.4, (3.13), definition (2.13), Lemma 3.2, and the fact that \(\widetilde{V}_{-q}(E,L)=\widetilde{V}_{n+q}(L,E)\), we have
\(\square \)
In order to facilitate the formulation of our problem for the limiting case \(q=\infty \), it will be helpful to introduce a normalized mixed\(L_q\)quasi-integral affine surface area: If \(1\le q<\infty \), and \(K,L\in \mathcal {K}_e^n\), then one can define \(\overline{A}_q(K,L)\) by
For the case where \(q\rightarrow \infty \), from (3.10) we see that
From (3.11) and definitions (3.19)–(3.20) we see immediately that for \(\lambda >0\) and \(q\in [1,\infty ]\),
Let \(\omega \) be a Borel subset of \(S^{n-1}\) and \(\langle \phi ^{-1}\omega \rangle =\{\langle \phi ^{-1}u\rangle : u\in \omega \}\). Observe that for \(q\in [1,\infty ]\) and \(\phi \in \mathrm {SL}(n)\), it follows that
This together with Lemma 3.4 and (3.20) gives
for \(\phi \in \mathrm {SL}(n)\).
Therefore, from (3.21) and (3.22) we conclude that if \(q\in [1,\infty ]\) and then for \(\phi \in \mathrm {GL}(n)\),
4 An alternative approach to extended Löwner–John ellipsoids
We shall be interesting in minimizing mixed \(L_q\) quasi-integral affine surface area \(\overline{A}_q(B_2^n,L)\) of \(B_2^n\) and a body \(L\in \mathcal {K}_e^n\) under \(\mathrm {SL}(n)\)-transformations of L: For \(q\in [1,\infty ]\) and find
This minimization problem stems from (3.12) and the following dual \(L_q\) Minkowski mixed volume inequality: If \(1\le q\le \infty \), then
with equality if and only if \(\Gamma _{-q}L\) is a ball in \(\mathbb {R}^n\).
Note that in view of (3.22) one can rewrite the minimization problem (4.1) as
Hence, we can formulate the minimization problem (4.1) in the following two equivalent ways: Given a convex body \(L\in \mathcal {K}_e^n\), find an ellipsoid E, amongst all origin-symmetric ellipsoids, which solves the constrained maximization problem for \(1\le q\le \infty \)
The dual problem is to find E so that
It is easily seen that the solutions of \(LJ_q\) and \(\overline{LJ}_q\) only differ by a scale factor.
Lemma 4.1
Suppose \(q\in [1,\infty ]\), and \(L\in \mathcal {K}_e^n\). If E is an ellipsoid centered at the origin that is an \(\overline{LJ}_q\) solution for L, then
is an \(LJ_q\) solution for L. If \(E'\) is an ellipsoid centered at the origin that is an \(LJ_q\) solution for L, then
is an \(\overline{LJ}_q\) solution for L.
Note that the \(LJ_\infty \) problem is exactly the classic John ellipsoid problem for the body L:
Furthermore, if \(q=1\), solving the \(\overline{LJ}_q\) problem gives rise to the classic Petty ellipsoid for the origin-symmetric body L. Indeed, from (2.2) and Cauchy’s surface area formula, we have
where \(\Pi L\) is the polar body of \(\Gamma _{-1}L\), whose support function can be expressed as the \((n-1)\)-dimensional volume of the orthogonal projection of L onto the hyperplane perpendicular to a given direction; i.e., \(h_{\Pi L}(u)=|L|u^\perp |\) for \(u\in S^{n-1}\).
To prove the existence of the solution to \(\overline{LJ}_q\), the following result due to Bastero and Romance (2004) will be required.
Lemma 4.2
Let \(K, L\in \mathcal {K}_o^n\), then
Since \(\widetilde{V}_{-q}(\Gamma _{-q}B_2^n, \phi \Gamma _{-q}L)=\widetilde{V}_{n+q}(\phi \Gamma _{-q}L,\Gamma _{-q}B_2^n)\), from Lemmas 3.4 and 4.2, as well as the Blaschke selection theorem, we see that
Lemma 4.3
There exists a solution of the problem \(\overline{LJ}_q\).
Let \(\mu \) be a finite Borel measure on \(S^{n-1}\) and \(\phi \in \mathrm {GL}(n)\), it was defined in Böröczky et al. (2015) the image \(\phi \mu \) of \(\mu \) under \(\phi \) by
where \(\langle \phi \omega \rangle =\{\langle \phi u\rangle : u\in \omega \}\) and \(\langle \phi u\rangle =\phi u/|\phi u|\). Any \(\phi \mu \) with \(\phi \in \mathrm {GL}(n)\) will be called an affine image of \(\mu \). An important property of affine image of a measure is that its total mass is invariant under \(\mathrm {SL}(n)\) transformations; i.e.,
for each \(\phi \in \mathrm {SL}(n)\).
If \(\phi \mu \) is an affine image of a finite Borel measure \(\mu \) on \(S^{n-1}\), then for each continuous \(f: S^{n-1}\rightarrow \mathbb {R}\) and \(\phi \in \mathrm {GL}(n)\) we have
Theorem 4.4
If \(q\ge 1\), and \(L\in \mathcal {K}_e^n\), then \(LJ_q\), as well as \(\overline{LJ}_q\), has a unique solution. Further, an ellipsoid E solves \(\overline{LJ}_q\) if and only if it satisfies
for all \(x\in \mathbb {R}^n\). An ellipsoid E solves \(LJ_q\) if and only if it satisfies
for all \(x\in \mathbb {R}^n\).
Proof
Suppose \(E_o=\phi _o^{-1}B_2^n\) is an \(\overline{LJ}_q\) solution. Choosing \(\psi \in \mathrm {GL}(n)\) arbitrarily, then there exists \(\varepsilon _0>0\) such that for all \(\varepsilon \in (-\varepsilon _0, \varepsilon _0)\) one can define \(E_\varepsilon \) as
Thus,
with \(|E_\varepsilon |=|E_o|=\omega _n\). From (4.8), Lemma 3.4, (3.13), (2.3), and the fact that \(\rho _{E_\varepsilon }(u)=|\phi _{\varepsilon }u|^{-1}\), we obtain
Set \(d\nu (u)=\rho _{\Gamma _{-q}E_o}^{n+q}(u)\rho _{\Gamma _{-q} L}^{-q}(u)du\). Thus, from the above equation, (4.4), and (4.5), we see that
Let \(\psi =\psi _{ij}\) in the above equation, where \(\psi _{ij}e_k=\delta _{jk}e_i\), then we obtain
This, together with the facts that \(|\nu |/n=A_q(E_o,L)\) and that \(|\phi _ou|=\rho ^{-1}_{E_o}(u)\), and (4.5), gives
Equivalently,
This proves (4.6).
Conversely, we suppose that (4.6) holds and shall prove that if \(|E|=\omega _n\), then
with equality if and only if \(E=E_o\).
Let \(E=\phi ^{-1} B_2^n\) and \(E_o=\phi _o^{-1}B_2^n\) with \(\phi ,\phi _o\in \mathrm {SL}(n)\). Also let
On one hand,
On another hand,
Thus, to prove (4.9), it suffices to show that if (4.6) holds then
with equality if and only if \(E=E_o\).
For each \(\phi \in \mathrm {SL}(n)\), it is easily seen that \(\phi \phi _o^{-1}\) also belongs to \(\mathrm {SL}(n)\). There exist a positive definite \(P\in \mathrm {SL}(n)\) and an orthogonal matrix O such that \(\phi \phi _o^{-1}=OP\). We can reduce having to prove (4.12) to having to prove
for all positive definite \(P\in \mathrm {SL}(n)\). For each positive definite \(P\in \mathrm {SL}(n)\), there exists an orthogonal matrix O and a diagonal matrix \(D=\mathrm {diag}\{\lambda _1,\ldots ,\lambda _n\}\) such that \(\lambda _i>0, i=1,\ldots , n\); \(\prod _{i=1}^n\lambda _i=1\), and \(P=O^tDO\). Thus, from (4.5) we obtain
Observing that (4.6) is equivalent to
This together with the fact that O is orthogonal gives
Therefore, from the concavity of the power function \(|t|^{-\frac{n+q}{2}}\), the fact that D is diagonal, (4.15), the AM–GM inequality, and the fact that \(\hbox {det}\ D=1\), we have
This proves the inequality (4.13).
From the strict concavity of the power function and the equality conditions of the AM–GM inequality, we see that the equality in (4.13) holds only if \(D=I_n\) and hence \(P=I_n\). From the fact that \(\phi \phi _o^{-1}=OP\), we obtain that \(\phi =O\phi _o\). This, together with the opposite of the procedure deducing (4.15) from (4.14), shows that the equality in (4.13) holds only if \(E=E_o\). Meanwhile, if \(E=E_o\), then it is easily seen that the inequality (4.13) becomes an equality. That proves the uniqueness of the solution. \(\square \)
Corollary 4.5
The unit ball \(B_2^n\) solves \(\overline{LJ}_q\) if and only if
Meanwhile, \(B_2^n\) solves \(LJ_q\) if and only if
Definition 1
Suppose \(q\in [1,\infty ]\), and \(L\in \mathcal {K}_e^n\). Amongst all origin-symmetric ellipsoids, the unique ellipsoid that solves the constrained maximization problem
will be called the \(L_q\) Löwner–John ellipsoid of L and will be denoted by \(\widetilde{E}_qL\). Amongst all origin-symmetric ellipsoids, the unique ellipsoid that solves the constrained minimization problem
will be called the normalized \(L_q\) Löwner–John ellipsoid of L and will be denoted by \(\overline{E}_qL\).
From (3.23) we obtain
Lemma 4.6
If \(1\le q\le \infty \), and \(L\in \mathcal {K}_e^n\), then for \(\phi \in \mathrm {GL}(n)\),
It follows from (4.16) that \(\widetilde{E}_qB_2^n=B_2^n\). This together with Lemma 4.6 gives
for any origin-symmetric ellipsoid E.
5 Continuity of \(L_q\) Löwner–John ellipsoids
In this section we demonstrate that \(L_q\) Löwner–John ellipsoids associated with a convex body is continuous in \(q\in [1,\infty ]\). We first show that \(\overline{A}_q(E,L)\) is monotonically increasing in \(q\in [1,\infty ]\).
Lemma 5.1
Suppose \(L\in \mathcal {K}_e^n\) and E is an origin-symmetric ellipsoid in \(\mathbb {R}^n\). Then for \(1\le q<r\le \infty \),
Proof
If \(q<r<\infty \), then from definition (3.19), Lemma 3.2, Jensen’s inequality, and definition (3.19) again, it follows that
From definition (3.20) and a limiting process of the above, we see that (5.1) also holds true when \(r=\infty \). \(\square \)
Throughout, we denote by \(\mathcal {E}^n\) the class of origin-symmetric ellipsoids in \(\mathbb {R}^n\). For \(E\in \mathcal {E}^n\), let \(d_E\) denote its maximal principal radius. If \(E=TB_2^n\) with \(T\in \mathrm {GL}(n)\), then
Let \(\overline{E}_qL\) be the solution of \(\overline{LJ}_q\). Since \(\overline{E}_qL\) is the origin-symmetric ellipsoid such that \(\overline{A}_q(E,L)\) attains its minimum value under the constraint \(|E|=\omega _n\), from Lemma 4.2 we see that there exists \(R>0\) such that \(d_{\overline{E}_qL}\le R<\infty \). This shows that \(\overline{E}_qL\) belongs to the following compact set
Lemma 5.2
Suppose that \(q_o\in [1, \infty ]\) and \(L\in \mathcal {K}_e^n\). If \(\{q_j\}\subset [1,\infty )\) is a sequence such that \(q_j<q_{j+1}\) and \(q_j\rightarrow q_o\) as \(j\rightarrow \infty \), then the convergence
is uniform in \(E\in \mathcal {E}_R\).
Proof
On one hand, for \(E\in \mathcal {E}_R\), let \(\{E_i\}\subset \mathcal {E}_R\) be a sequences of ellipsoids converging to E; i.e., \(E_i\rightarrow E\) as \(i\rightarrow \infty \). But this is equivalent to \(\rho _{E_i}\rightarrow \rho _E\) uniformly on \(S^{n-1}\), thus \(\overline{A}_q(E_i,L)\rightarrow \overline{A}_q (E,L)\) as \(i\rightarrow \infty \), which shows that \(\overline{A}_q(\cdot ,L)\) is continuous over the compact set \(\mathcal {E}_R\).
On another hand, by Lemma 5.1, we have
Recalling that \(\{\overline{A}_{q_j}(\cdot ,L)\}\) is a sequence of continuous functionals and it converges to the continuous functional \(\overline{A}_{q_o}(\cdot ,L)\) as \(j\rightarrow \infty \), in view of (5.3) one can apply Dini’s theorem to get
as \(j\rightarrow \infty \). \(\square \)
Lemma 5.3
Suppose that \(q_o\in [1, \infty ]\) and \(L\in \mathcal {K}_e^n\). If \(\{q_j\}\subset [1,\infty )\) is a sequence such that \(q_j<q_{j+1}\) and \(q_j\rightarrow q_o\) as \(j\rightarrow \infty \), then
Proof
From the definition of \(\overline{E}_qL\), Lemma 5.2, the continuity of \(\overline{A}_q(K,E)\) in q, and again the definition of \(\overline{E}_qL\), we have
\(\square \)
Lemma 5.4
Suppose that \(q_o\in [1, \infty ]\) and \(L\in \mathcal {K}_e^n\). If \(\{q_j\}\subset [1,\infty )\) is a sequence such that \(q_j<q_{j+1}\) and \(q_j\rightarrow q_o\) as \(j\rightarrow \infty \), then
Proof
Assume that
Since the \(\overline{LJ}_q\) solution is unique, by the uniform convergence established in Lemma 5.2, we obtain
which contradicts Lemma 5.3. \(\square \)
Noting that by Lemma 4.1,
This, together with Lemmas 5.3 and 5.4, gives
Theorem 5.5
Suppose that \(q_o\in [1, \infty ]\) and \(L\in \mathcal {K}_e^n\). If \(\{q_j\}\subset [1,\infty )\) is a sequence such that \(q_j<q_{j+1}\) and \(q_j\rightarrow q_o\) as \(j\rightarrow \infty \), then
6 Volume ratio inequalities
Ball’s volume ratio inequality (Ball 1991; Lutwak et al. 2004) demonstrates the connection between the volumes of the John ellipsoid \(\widetilde{E}_\infty K\) and the body K:
Theorem 6.1
If \(K\in \mathcal {K}_e^n\), then
with equality if and only if K is a parallelotope.
To extend the classic volume ratio inequality, we first establish a comparison theorem for volumes of \(L_q\) Löwner–John ellipsoids.
Theorem 6.2
If \(L\in \mathcal {K}_e^n\) and \(1\le q<r\le \infty \), then
Proof
Lemma 5.1 and Definition 1 immediately gives the desired result for real \(r<\infty \). For \(r=\infty \), let \(\{q_j\}\) be an increasing sequence of integers such that \(q<q_1<q_2<\cdots \). Then, by combining the real case with Theorem 5.5, we see that
\(\square \)
Theorem 6.3
If \(L\in \mathcal {K}_e^n\) and \(1\le q\le \infty \), then
with equality if and only if \(\Gamma _{-q}L\) is an ellipsoid centered at the origin.
Proof
First suppose \(q<\infty \). From Definition 1 and Hölder’s inequality, we have
with equality for \(q\ge 1\) if and only if \(\Gamma _{-q}L\) is a dilate of \(\Gamma _{-q}\widetilde{E}_qL\). For \(q=\infty \) combine this argument with Theorem 6.2. \(\square \)
By combining Theorem 6.2 with Ball’s volume ratio inequality (6.1), we immediately obtain
Theorem 6.4
If \(L\in \mathcal {K}_e^n\) and \(1\le q\le \infty \), then
When \(q=2\), or \(q=\infty \), the equality holds if and only if L is a parallelotope.
Note that if L is the cube \([-1,1]^n\) and \(q=2\) or \(q=\infty \), then \(\widetilde{E}_qL=B_2^n\). This shows that for origin-symmetric parallelotopes there is indeed equality in the inequality of Theorem 6.4 when \(q=2\) or \(q=\infty \).
References
Ball, K.: Volume ratios and a reverse isoperimetric inequality. J. Lond. Math. Soc. (2) 44, 351–359 (1991)
Bastero, J., Romance, M.: Positions of convex bodies associated with extremal problems and isotropic measures. Adv. Math. 184, 64–88 (2004)
Böröczky, K., Lutwak, E., Yang, D., Zhang, G.: Affine images of isotropic measures. J. Differ. Geom. 99, 407–442 (2015)
Gardner, R.: Geometric Tomography, 2nd edn. Cambridge University Press, New York (2006)
Giannopoulos, A.A., Milman, V.D.: Extremal problems and isotropic positions of convex bodies. Isael J. Math. 117, 29–60 (2000)
Giannopoulos, A., Papadimitrakis, M.: Isotropic surface area measures. Mathematika 46, 1–13 (1999)
Gruber, P.M.: John and Löwner ellipsoids. Discrete Comput. Geom. 46, 776–788 (2011)
Gruber, P.M.: Convex and Discrete Geometry. Springer, Berlin (2007)
Hug, D., Lutwak, E., Yang, D., Zhang, G.: On the \(L_p\) Minkowski problem for polytopes. Discrete Comput. Geom. 33, 699–715 (2005)
John, F.: Extremum problems with inequalities as subsidiary conditions, in Studies and Essays, presented to R. Courant on his 60th birthday. Intercience, New York (1948)
Lutwak, E.: Intersection bodies and dual mixed volumes. Adv. Math. 71, 232–261 (1988)
Lutwak, E.: Extended affine surface area. Adv. Math. 85, 39–68 (1991)
Lutwak, E.: The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)
Lutwak, E.: The Brunn–Minkowski–Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)
Lutwak, E., Yang, D., Zhang, G.: \(L_p\) affine isoperimetric inequality. J. Differ. Geom. 56, 111–132 (2000)
Lutwak, E., Yang, D., Zhang, G.: A new ellipsoid associated with convex bodies. Duke Math. J. 104, 375–390 (2000)
Lutwak, E., Yang, D., Zhang, G.: Volume inequalities for subspaces of \(L_p\). J. Differ. Geom. 68, 159–184 (2004)
Lutwak, E., Yang, D., Zhang, G.: \(L_p\) John ellipsoids. Proc. Lond. Math. Soc. 90, 497–520 (2005)
Petty, C.M.: Surface area of a convex body under affine transformations. Proc. Am. Math. Soc. 12, 824–828 (1961)
Pisier, G.: The Volume of Convex Bodies and Banach Space Geometry. Cambridge University Press, Cambridge (1989)
Schuster, F.E., Weberndorfer, M.: Volume inequalities for asymmetric Wulff shapes. J. Differ. Geom. 92, 263–283 (2012)
Schneider, R.: Convex bodies: the Brunn -Minkowski theory. Encyclopedia of Mathematics and Its Applications, 2nd edn. Cambridge University Press, Cambridge (2014)
Yu, W., Leng, G., Wu, D.: Dual \(L_p\) John ellipsoids. Proc. Edinb. Math. Soc. 50, 737–753 (2007)
Zhang, G.: New affine isoperimetric inequalities. In: ICCM II, 239–267 (2007)
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Lv, S. A sharp dual \(L_{p}\) John ellipsoid problem for \(p\le -n-1\). Beitr Algebra Geom 60, 709–732 (2019). https://doi.org/10.1007/s13366-019-00444-z
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DOI: https://doi.org/10.1007/s13366-019-00444-z