A sharp dual $L_{p}$ John ellipsoid problem for $p\le -n-1$

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.

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

$$\begin{aligned} h_{K+_p\varepsilon \cdot L}=(h_K^p+\varepsilon h_L^p)^{\frac{1}{p}}, \end{aligned}$$

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):

$$\begin{aligned} V_p(K,L)=\frac{p}{n}\lim _{\varepsilon \rightarrow 0^+}\frac{|K+_p\varepsilon \cdot L|-|K|}{\varepsilon }=\frac{1}{n}\int _{S^{n-1}}h_L(u)^ph_K^{1-p}(u)dS(K,u), \end{aligned}$$

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:

$$\begin{aligned} \min _{\phi \in \mathrm {SL}(n)}\int _{S^{n-1}}dS_p(K,u) =\min _{\phi \in \mathrm {SL}(n)}nV_p(\phi K,B_2^n). \end{aligned}$$

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

$$\begin{aligned} \widetilde{V}_{-p}(K,L)=\frac{1}{n}\int _{S^{n-1}}\rho _K(u)^{n+p}\rho _L(u)^{-p}du, \end{aligned}$$

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.

$$\begin{aligned}&\max _{\phi \in \mathrm {SL}(n)}\widetilde{V}_{-p}(\phi K,B_2^n)\qquad \mathrm {if}~~p\in (-n,0); \end{aligned}$$

(1.1)

$$\begin{aligned}&\min _{\phi \in \mathrm {SL}(n)}\widetilde{V}_{-p}(\phi K,B_2^n)\qquad \mathrm {if}~~p\notin [-n,0]. \end{aligned}$$

(1.2)

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:

$$\begin{aligned} \frac{S_q(K\overline{\#}_q\varepsilon \circ L,\cdot )}{|K\overline{\#}_q\varepsilon \circ L|}=\frac{S_q(K,\cdot )}{|K|}+\varepsilon \frac{S_q(L,\cdot )}{|L|}. \end{aligned}$$

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

$$\begin{aligned} \rho _{\Gamma _{-q}K}^{-q}(u)=\frac{1}{|K|}\int _{S^{n-1}}|u\cdot v|^qdS_q(K,v). \end{aligned}$$

A useful property of $\Gamma _{-q}K$ is the following

$$\begin{aligned} K\subseteq n^{\frac{1}{q}}\Gamma _{-q}K. \end{aligned}$$

(1.3)

For $q\ge 1$, we shall introduce a new affine invariant associated with two origin-symmetric convex bodies K and L as

$$\begin{aligned} A_q(K,L)=\frac{-q}{n}\lim _{\varepsilon \rightarrow 0^+}\frac{|\Gamma _{-q} (K\overline{\#}_q\varepsilon \circ L)|-|\Gamma _{-q}K|}{\varepsilon }. \end{aligned}$$

It will be shown that

$$\begin{aligned} A_q(K,L)=\widetilde{V}_{-q}(\Gamma _{-q}K,\Gamma _{-q}L). \end{aligned}$$

(1.4)

The optimization problem we are interested in this paper is the following

$$\begin{aligned} \min _{\phi \in \mathrm {SL}(n)}A_q(B_2^n,\phi K). \end{aligned}$$

(1.5)

By using (1.3) and (1.4), one can prove that

$$\begin{aligned} {A}_q(B_2^n,\phi K)\le C_{n,q}\widetilde{V}_{-q}(B_2^n,\phi K), \end{aligned}$$

(1.6)

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

$$\begin{aligned} A_q(B_2^n,\phi K)\le C_{n,q}\widetilde{V}_{n+q}(\phi K,B_2^n)=C_{n,q}\widetilde{V}_{-(-n-q)}(\phi K,B_2^n). \end{aligned}$$

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:

$$\begin{aligned} \overline{A}_q(K,L)=n^{-\frac{1}{q}}c_{n,q}^{-1} \left( \frac{A_q(K,L)}{|\Gamma _{-q}K|}\right) ^{1/q}. \end{aligned}$$

(1.7)

We shall consider the following constrained optimization problem whose solution only differs by a scale factor from that of the problem (1.5):

$$\begin{aligned} \max _E\left( \frac{|E|}{\omega _n}\right) ^{1/n}\quad \text {subject}~~\text {to}~~\quad \overline{A}_q(E,K)\le 1. \end{aligned}$$

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

$$\begin{aligned} \omega _s=\frac{\pi ^{\frac{s}{2}}}{\Gamma \left( 1+\frac{s}{2}\right) }, \end{aligned}$$

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

$$\begin{aligned} S_K(\omega )=\mathcal {H}^{n-1}(\nu _K^{-1}(\omega )), \end{aligned}$$

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

$$\begin{aligned} h_K(x)=\max \{x\cdot y: y\in K\}, \end{aligned}$$

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

$$\begin{aligned} K^*=\{x\in \mathbb {R}^n: x\cdot y\le 1~~\text {for}~~\text {all}~~ y\in K\}. \end{aligned}$$

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

$$\begin{aligned} \rho _K(x)=\max \{\lambda \ge 0: \lambda x\in K\}. \end{aligned}$$

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 $A\in \mathrm {SL}(n)$,

$$\begin{aligned} \rho _{AK}(x)=\rho _K(A^{-1}x)\quad x\in \mathbb {R}^n\backslash \{0\}. \end{aligned}$$

(2.1)

Moreover, if K is a convex body in $\mathbb {R}^n$ that contains the origin in its interior, then it follows that

$$\begin{aligned} \rho _{K^*}=1/h_K \quad \mathrm {and}\quad h_{K^*}=1/\rho _K. \end{aligned}$$

(2.2)

If in addition K is origin-symmetric then

$$\begin{aligned} \rho _K(u)=\min \{h_K(v)/|u\cdot v|: v\in S^{n-1}\}. \end{aligned}$$

(2.3)

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

$$\begin{aligned} aK+bL=\{ax+by: x\in K, y\in L\}, \end{aligned}$$

or by

$$\begin{aligned} h_{aK+bL}=ah_K+bh_L. \end{aligned}$$

(2.4)

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

$$\begin{aligned} h_{a\cdot K+_p b\cdot L}^p=ah_K^p+bh_L^p. \end{aligned}$$

(2.5)

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

$$\begin{aligned} \Vert h_K-h_L\Vert _\infty =\max _{u\in S^{n-1}}|h_K(u)-h_L(u)|. \end{aligned}$$

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

$$\begin{aligned} dS_p(K,\cdot )=h_K^{1-p}dS(K,\cdot ). \end{aligned}$$

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

$$\begin{aligned} V_p(K,L)=\frac{p}{n}\lim _{\varepsilon \rightarrow 0^+}\frac{|K+_p\varepsilon \cdot L|-|K|}{\varepsilon }. \end{aligned}$$

Obviously, for each $K\in \mathcal {K}_o^n$,

$$\begin{aligned} V_p(K,K)=|K|. \end{aligned}$$

It was shown by Lutwak (1993) that

$$\begin{aligned} V_p(K,L)=\frac{1}{n}\int _{S^{n-1}}h_L(u)^pdS_p(K,u). \end{aligned}$$

(2.6)

The classic Minkowski mixed volume inequality can be stated as: For convex bodies K, L in $\mathbb {R}^n$,

$$\begin{aligned} V_1(K,L)^n\ge |K|^{n-1}|L|, \end{aligned}$$

(2.7)

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

$$\begin{aligned} V_p(K,L)^n\ge |K|^{n-p}|L|^p, \end{aligned}$$

(2.8)

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:

$$\begin{aligned} \overline{V}_p(K,L)=\left( \frac{V_p(K,L)}{|K|}\right) ^{1/p}. \end{aligned}$$

(2.9)

If $p=\infty$, then

$$\begin{aligned} \overline{V}_\infty (K,L)=\max \{h_K(u)/h_L(u): u\in \mathrm {supp}S(K,\cdot )\}. \end{aligned}$$

(2.10)

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):

$$\begin{aligned} \max _E|E|\quad \text {subject}~~\text {to}\quad \overline{V}_p(K,E)\le 1, \end{aligned}$$

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

$$\begin{aligned} \rho _{K\widetilde{+}_p\varepsilon \cdot L}(u)^{-p}=\rho _K(u)^{-p}+\varepsilon \rho _L(u)^{-p}. \end{aligned}$$

(2.11)

The dual$L_p$mixed volume$\widetilde{V}_{-p}(K,L)$ of star bodies K and L is defined as

$$\begin{aligned} \widetilde{V}_{-p}(K,L)=\frac{-p}{n}\lim _{\varepsilon \rightarrow 0^+} \frac{|K\widetilde{+}_p\varepsilon \cdot L|-|K|}{\varepsilon }. \end{aligned}$$

(2.12)

It follows from the polar coordinate formula and the above definition that

$$\begin{aligned} \widetilde{V}_{-p}(K,L)=\frac{1}{n}\int _{S^{n-1}}\rho _K(u)^{n+p}\rho _L(u)^{-p}du. \end{aligned}$$

(2.13)

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:

$$\begin{aligned} \widehat{V}_{-p}(K,L)=\left( \frac{\widetilde{V}_{-p}(K,L)}{|K|}\right) ^{1/p} =\left( \int _{S^{n-1}}\left( \frac{\rho _K(u)}{\rho _L(u)}\right) ^pd \widetilde{V}_K(u)\right) ^{1/p}, \end{aligned}$$

(2.14)

where

$$\begin{aligned} d\widetilde{V}_K(u)=\frac{\rho _K^n(u)}{n|K|}du \end{aligned}$$

is the dual cone-volume probability measure of K.

For $p=\infty$, define

$$\begin{aligned} \widehat{V}_{-\infty }(K,L)=\max \{\rho _K(u)/\rho _L(u): u\in S^{n-1}\}. \end{aligned}$$

(2.15)

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

$$\begin{aligned} \max _E\frac{1}{|E|}\quad \text {subject}~~\text {to}\quad \widehat{V}_{-p}(K,E)\le 1 \end{aligned}$$

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:

$$\begin{aligned} \frac{S_p(a\circ K\overline{\#}_pb\circ L, \cdot )}{|a\circ K\overline{\#}_pb\circ L|}=a\frac{S_p(K,\cdot )}{|K|}+b\frac{S_p(L,\cdot )}{|L|}. \end{aligned}$$

(3.1)

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 $K,L\in \mathcal {K}^n$,

$$\begin{aligned} S(\alpha \circ K\overline{\#}\beta \circ L,\cdot )=\alpha S(K,\cdot )+\beta S(L,\cdot ). \end{aligned}$$

(3.2)

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

$$\begin{aligned} \int _{S^{n-1}}f(u)d\mu ^{(p)}(\phi u)=\int _{S^{n-1}}|\phi ^{-1}u|^pf(\langle \phi ^{-1}u\rangle )d\mu (u), \end{aligned}$$

(3.3)

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

$$\begin{aligned} dS_p(\phi K,u)=dS_p^{(p)}(K,\phi ^tu). \end{aligned}$$

(3.4)

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

$$\begin{aligned} \phi \left( K\overline{\#}_p\varepsilon \circ L\right) =(\phi K)\overline{\#}_p\varepsilon \circ (\phi L). \end{aligned}$$

(3.5)

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 $u\in S^{n-1}$,

$$\begin{aligned} \frac{dS_p^{(p)}(K\overline{\#}_p\varepsilon \circ L,\phi ^tu)}{|K\overline{\#}_p\varepsilon \circ L|}=\frac{dS_p^{(p)}(K,\phi ^tu)}{|K|}+\varepsilon \frac{dS_p^{(p)}(L,\phi ^tu)}{|L|}. \end{aligned}$$

(3.6)

Indeed, from (3.3), (3.1), and (3.3) again, we have

$$\begin{aligned}&\int _{S^{n-1}}f(u)\frac{dS_p^{(p)}(K\overline{\#}_p \varepsilon \circ L,\phi ^tu)}{|K\overline{\#}_p\varepsilon \circ L|}\\&\quad =\int _{S^{n-1}}|\phi ^{-t}u|^pf(\langle \phi ^{-t}u\rangle ) \frac{dS_p(K\overline{\#}_p\varepsilon \circ L, u)}{|K\overline{\#}_p\varepsilon \circ L|}\\&\quad =\int _{S^{n-1}}|\phi ^{-t}u|^pf(\langle \phi ^{-t}u\rangle ) \frac{dS_p(K,u)}{|K|}+\varepsilon \int _{S^{n-1}}|\phi ^{-t}u|^pf (\langle \phi ^{-t}u\rangle )\frac{dS_p(L,u)}{|L|}\\&\quad =\int _{S^{n-1}}f(u)\frac{dS_p^{(p)}(K,\phi ^tu)}{|K|}+\varepsilon \int _{S^{n-1}}f(u)\frac{dS_p^{(p)}(L,\phi ^tu)}{|L|}, \end{aligned}$$

for each $f\in C(S^{n-1})$.

By (3.1), (3.4), (3.6), and (3.4) again, we see that

$$\begin{aligned} \frac{S_p((\phi K)\overline{\#}_p\varepsilon \circ (\phi L), u)}{|(\phi K)\overline{\#}_p\varepsilon \circ (\phi L)|}&=\frac{S_p(\phi K,u)}{|\phi K|}+\varepsilon \frac{S_p(\phi L,u)}{|\phi L|}\\&=\frac{S_p^{(p)}(K,\phi ^tu)}{|K|} +\varepsilon \frac{S_p^{(p)}(L,\phi ^tu)}{|L|}\\&=\frac{S_p^{(p)}(K\overline{\#}_p\varepsilon \circ L,\phi ^tu)}{|K\overline{\#}_p\varepsilon \circ L|}\\&=\frac{S_p(\phi (K\#_p\varepsilon \circ L),u)}{|\phi (K\#_p\varepsilon \circ L)|}, \end{aligned}$$

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

$$\begin{aligned} dV_K=\frac{1}{n}h_KdS_K. \end{aligned}$$

Observing that

$$\begin{aligned} |K|=\int _{S^{n-1}}dV_K(u), \end{aligned}$$

we can define the cone-volume probability measure$\overline{V}_K$ of K by

$$\begin{aligned} \overline{V}_{K}=\frac{dV_K}{|K|}. \end{aligned}$$

(3.7)

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

$$\begin{aligned} \rho ^{-p}_{\Gamma _{-p}K}(u)&=\frac{1}{|K|}\int _{S^{n-1}}|u\cdot v|^pdS_p(K,v)\nonumber \\&=n\int _{S^{n-1}}|u\cdot v|^ph_K(u)^{-p}d\overline{V}_K(v). \end{aligned}$$

(3.8)

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

$$\begin{aligned} \Gamma _{-\infty } K=\lim _{p\rightarrow \infty }\Gamma _{-p}K. \end{aligned}$$

Therefore, from (3.8)

$$\begin{aligned} \rho _{\Gamma _{-\infty }K}^{-1}(u)=\max \{|u\cdot v|/h_K(v): v\in \mathrm {supp}S_K\}. \end{aligned}$$

(3.9)

Note that when K is origin-symmetric,

$$\begin{aligned} \Gamma _{-\infty }K=K. \end{aligned}$$

(3.10)

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

$$\begin{aligned} n^{\frac{1}{p}}\Gamma _{-p}K\supseteq n^{\frac{1}{r}}\Gamma _{-r}K. \end{aligned}$$

It was shown in Lutwak et al. (2005) that for $\lambda >0$ and $p\in (0,\infty ]$,

$$\begin{aligned} \Gamma _{-p}\lambda K=\lambda \Gamma _{-p}K. \end{aligned}$$

(3.11)

Moreover, for $\phi \in \mathrm {GL}(n)$ and $p\in (0,\infty ]$,

$$\begin{aligned} \Gamma _{-p}\phi K=\phi \Gamma _{-p}K. \end{aligned}$$

(3.12)

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

$$\begin{aligned} c^{-p}_{n,p}=\frac{2\Gamma (\frac{p+1}{2}) \Gamma (1+\frac{n}{2})}{\sqrt{\pi }\Gamma (\frac{n+p}{2})}. \end{aligned}$$

Thus, from (3.12) we see that if E is an origin-symmetric ellipsoid, then

$$\begin{aligned} \Gamma _{-p}E=c_{n,p}E. \end{aligned}$$

(3.13)

Proposition 3.3

Suppose that $1\le p<\infty$, and $K,L\in \mathcal {K}_e^n$, then

$$\begin{aligned} \Gamma _{-p}(K\overline{\#}_p\varepsilon \circ L)=\Gamma _{-p}K\widetilde{+}_p\varepsilon \cdot \Gamma _{-p}L. \end{aligned}$$

(3.14)

Proof

From (2.2), (3.8), (3.1), (2.5), and (2.2) again, we obtain

$$\begin{aligned} \rho ^{-p}_{\Gamma _{-p}(K\overline{\#}_p\varepsilon \circ L)}(u)&=\frac{1}{|K\overline{\#}_p\varepsilon \circ L|}\int _{S^{n-1}}|u\cdot v|^pdS_p(K\overline{\#}_p\varepsilon \circ L,v)\\&=\frac{1}{|K|}\int _{S^{n-1}}|u\cdot v|^pdS_p(K,v)+\frac{\varepsilon }{|K|}\int _{S^{n-1}}|u\cdot v|^pdS_p(L,v)\\&=\rho ^{-p}_{\Gamma _{-p}K}(u)+\varepsilon \rho ^{-p}_{\Gamma _{-p}L}(u). \end{aligned}$$

$\square$

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

$$\begin{aligned} A_q(K,L)=\frac{-q}{n}\lim _{\varepsilon \rightarrow 0^+} \frac{|\Gamma _{-q}(K\overline{\#}_q\varepsilon \circ L)| -|\Gamma _{-q}K|}{\varepsilon }. \end{aligned}$$

(3.15)

From definition (3.15), Lemma 3.1, (3.8), and (3.12), it follows that, for $\phi \in \mathrm {SL}(n)$,

$$\begin{aligned} A_q(\phi K, \phi L)=A_q(K, L). \end{aligned}$$

(3.16)

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

$$\begin{aligned} A_q(K,L)=\widetilde{V}_{-q}(\Gamma _{-q}K,\Gamma _{-q}L). \end{aligned}$$

(3.17)

Proof

Since the convergence as $\varepsilon \rightarrow 0^+$ in

$$\begin{aligned} \frac{\rho ^n_{\Gamma _{-q}(K\overline{\#}_q\varepsilon \circ L)}-\rho ^n_{\Gamma _{-q}K}}{\varepsilon }\quad \mathrm { and}\quad \frac{\rho ^{-q}_{\Gamma _{-q}(K\overline{\#}_q\varepsilon \circ L)}-\rho ^{-q}_{\Gamma _{-q}K}}{\varepsilon } \end{aligned}$$

is uniform on $S^{n-1}$, it follows that

$$\begin{aligned} A_q(K,L)&=\frac{-q}{n}\left. \frac{d}{d\varepsilon }\right| _{\varepsilon =0^+}| \Gamma _{-q}(K\overline{\#}_q\varepsilon \circ L)|\\&=\frac{-q}{n}\lim _{\varepsilon \rightarrow 0^+}\frac{1}{n}\int _{S^{n-1}} \frac{\rho ^n_{\Gamma _{-q}(K\overline{\#}_q\varepsilon \circ L)}(u)-\rho ^n_{\Gamma _{-q}K}(u)}{\varepsilon }du\\&=\frac{1}{n}\int _{S^{n-1}}\lim _{\varepsilon \rightarrow 0^+} \frac{\rho ^{-q}_{\Gamma _{-q}(K\overline{\#}_q\varepsilon \circ L)}(u)-\rho ^{-q}_{\Gamma _{-q}K}(u)}{\varepsilon }\rho ^{n+q}_{\Gamma _{-q}K}(u)du\\&=\frac{1}{n}\int _{S^{n-1}}\rho ^{n+q}_{\Gamma _{-q}K}(u)\rho ^{-q}_{\Gamma _{-q}L}(u)du. \end{aligned}$$

$\square$

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

$$\begin{aligned} A_q(E,L)\le nc_{n,q}^{n+q}\widetilde{V}_{n+q}(L,E). \end{aligned}$$

(3.18)

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

$$\begin{aligned} A_q(E,L)&=\widetilde{V}_{-q}(\Gamma _{-q}E,\Gamma _{-q}L)\\&=c_{n,q}^{n+q}\widetilde{V}_{-q}(E,\Gamma _{-q}L)\\&\le nc_{n,q}^{n+q}\widetilde{V}_{-q}(E,L)\\&=nc_{n,q}^{n+q}\widetilde{V}_{n+q}(L,E). \end{aligned}$$

$\square$

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

$$\begin{aligned} n^{\frac{1}{q}}c_{n,q}\overline{A}_q(K,L)&=\left( \frac{A_q(K,L)}{|\Gamma _{-q}K|}\right) ^{1/q}\nonumber \\&=\left( \frac{1}{n|\Gamma _{-q}K|}\int _{S^{n-1}}\left( \frac{\rho _{\Gamma _{-q}K}(v)}{\rho _{\Gamma _{-q}L}(v)}\right) ^q\rho _{\Gamma _{-q}K}^n(v)dv\right) ^{1/q}\nonumber \\&=\left( \int _{S^{n-1}}\left( \frac{\rho _{\Gamma _{-q}K}(v)}{\rho _{\Gamma _{-q}L}(v)}\right) ^qd\widetilde{V}_{\Gamma _{-q}K} (v)\right) ^{1/q}. \end{aligned}$$

(3.19)

For the case where $q\rightarrow \infty$, from (3.10) we see that

$$\begin{aligned} \overline{A}_\infty (K,L)&=\lim _{q\rightarrow \infty }\overline{A}_q(K,L)\nonumber \\&=\max \left\{ \frac{\rho _{\Gamma _{-\infty }K}(u)}{\rho _{\Gamma _{-\infty }L}(u)}: u\in S^{n-1}\right\} \nonumber \\&=\widehat{V}_{-\infty }(K,L). \end{aligned}$$

(3.20)

From (3.11) and definitions (3.19)–(3.20) we see immediately that for $\lambda >0$ and $q\in [1,\infty ]$,

$$\begin{aligned} \overline{A}_q(\lambda K,L)=\lambda \overline{A}_q(K,L)\quad \mathrm {and}\quad \overline{A}_q(K,\lambda L)=\lambda ^{-1}\overline{A}_q(K,L). \end{aligned}$$

(3.21)

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

$$\begin{aligned} \widetilde{V}_{\phi \Gamma _{-q}K}(\omega )=\widetilde{V}_{\Gamma _{-q}K} (\langle \phi ^{-1}\omega \rangle ). \end{aligned}$$

This together with Lemma 3.4 and (3.20) gives

$$\begin{aligned} \overline{A}_q(\phi K, \phi L)=\overline{A}_q(K,L), \end{aligned}$$

(3.22)

for $\phi \in \mathrm {SL}(n)$.

Therefore, from (3.21) and (3.22) we conclude that if $q\in [1,\infty ]$ and then for $\phi \in \mathrm {GL}(n)$,

$$\begin{aligned} \overline{A}_q(\phi K, \phi L)=\overline{A}_q(K,L). \end{aligned}$$

(3.23)

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

$$\begin{aligned} \min _{\phi \in \mathrm {SL}(n)}\overline{A}_q(B_2^n,\phi L). \end{aligned}$$

(4.1)

This minimization problem stems from (3.12) and the following dual $L_q$ Minkowski mixed volume inequality: If $1\le q\le \infty$, then

$$\begin{aligned} n^{\frac{1}{q}}c_{n,q}\overline{A}_{q}(B_2^n,L) \ge \left( \frac{|\Gamma _{-q}B_2^n|}{|\Gamma _{-q}L|}\right) ^{1/n}, \end{aligned}$$

(4.2)

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

$$\begin{aligned} \min _{\phi \in \mathrm {SL}(n)}\overline{A}_q(B_2^n, \phi L)&=\min _{\phi \in \mathrm {SL}(n)}\overline{A}_q(\phi ^{-1}B_2^n,L)\\&=\min _{|E|=\omega _n}\overline{A}_q(E,L). \end{aligned}$$

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

$$\begin{aligned} \overline{A}_q(E,L)E \end{aligned}$$

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

$$\begin{aligned} (\omega _n/|E'|)^{1/n}E' \end{aligned}$$

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

$$\begin{aligned} \min _{\phi \in \mathrm {SL}(n)}\overline{A}_1(B_2^n,\phi L)&=\min _{\phi \in \mathrm {SL}(n)} \frac{n^{-1}c^{-1}_{n,1}}{n|\Gamma _{-1}B_2^n|}\int _{S^{n-1}}\rho ^{-1}_{\Gamma _{-1} \phi L}(u)\rho _{\Gamma _{-1}B_2^n}^{n+1}(u)du\\&=\frac{1}{n^2}\min _{\phi \in \mathrm {SL}(n)}\int _{S^{n-1}}h_{\Pi \phi L}(u)du\\&=\frac{1}{n^2}\min _{\phi \in \mathrm {SL}(n)}\int _{S^{n-1}}|\phi L|u^\perp |du\\&=\frac{\omega _{n-1}}{n^2}\min _{\phi \in \mathrm {SL}(n)}S(\phi L), \end{aligned}$$

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

$$\begin{aligned} \mathop {\mathop {\lim }\limits _{\phi \in \mathrm {SL}(n)}} \limits _{\Vert \phi \Vert \rightarrow \infty } \widetilde{V}_i(\phi K,L)=\infty ,\quad i\in (n,\infty ). \end{aligned}$$

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

$$\begin{aligned} \phi \mu (\omega )=\mu (\langle \phi ^{-1}\omega \rangle ), \end{aligned}$$

(4.3)

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.,

$$\begin{aligned} |\phi \mu |=|\mu |, \end{aligned}$$

(4.4)

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

$$\begin{aligned} \int _{S^{n-1}}f(u)d\phi \mu (u)=\int _{S^{n-1}}f(\langle \phi u\rangle )d\mu (u). \end{aligned}$$

(4.5)

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

$$\begin{aligned} A_q(E,L)\rho _{E^*}^{-2}(x)=\int _{S^{n-1}}|x\cdot v|^2\rho _E^2(v)\left( \frac{\rho _{\Gamma _{-q}E}(v)}{\rho _{\Gamma _{-q}L}(v)}\right) ^q\rho _{\Gamma _{-q}E}^n(v)dv, \end{aligned}$$

(4.6)

for all $x\in \mathbb {R}^n$. An ellipsoid E solves $LJ_q$ if and only if it satisfies

$$\begin{aligned} |\Gamma _{-q}E|\rho _{E^*}^{-2}(x)=\int _{S^{n-1}}|x\cdot v|^2\rho _E^2(v)\left( \frac{\rho _{\Gamma _{-q}E}(v)}{\rho _{\Gamma _{-q}L}(v)}\right) ^q\rho _{\Gamma _{-q}E}^n(v)dv, \end{aligned}$$

(4.7)

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

$$\begin{aligned} E_\varepsilon =\phi ^{-1}_\varepsilon B_2^n=\left( \frac{I_n+\varepsilon \psi }{|I_n+\varepsilon \psi |^{1/n}} \phi _o\right) ^{-1}B_2^n. \end{aligned}$$

Thus,

$$\begin{aligned} \overline{A}_q(E_o,L)\le \overline{A}_q(E_\varepsilon ,L) \end{aligned}$$

(4.8)

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

$$\begin{aligned} 0&=\left. \frac{d}{d\varepsilon }\right| _{\varepsilon =0}\int _{S^{n-1}} \rho _{\Gamma _{-q}E_\varepsilon }^{n+q}(u)\rho _{\Gamma _{-q}L}^{-q}(u)du\\&=\left. \frac{d}{d\varepsilon }\right| _{\varepsilon =0}c_{n,q}^{n+q} \int _{S^{n-1}}|\phi _\varepsilon u|^{-n-q}\rho _{\Gamma _{-q}L}^{-q}(u)du\\&=(-n-q)c_{n,q}^{n+q}\int _{S^{n-1}}|\phi _ou|^{-n-q-1}\rho _{\Gamma _{-q}L}^{-q}(u) \left( \left. \frac{d}{d\varepsilon }\right| _{\varepsilon =0}| \phi _\varepsilon u|\right) du\\&=(-n-q)c_{n,q}^{n+q}\int _{S^{n-1}}\left( \langle \phi _ou\rangle \cdot \psi \langle \phi _ou\rangle -\frac{\mathrm {tr}\psi }{n}\right) |\phi _ou|^{-n-q} \rho _{\Gamma _{-q}L}^{-q}(u)du\\&=(-n-q)\int _{S^{n-1}}\left( \langle \phi _ou\rangle \cdot \psi \langle \phi _ou\rangle -\frac{\mathrm {tr}\psi }{n}\right) \rho _{\Gamma _{-q}E_o}^{n+q}(u)\rho _{\Gamma _{-q}L}^{-q}(u)du. \end{aligned}$$

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

$$\begin{aligned} \frac{1}{|\nu |}\int _{S^{n-1}}u\cdot \psi ud\phi _o\nu (u)=\frac{\mathrm {tr}\psi }{n}. \end{aligned}$$

Let $\psi =\psi _{ij}$ in the above equation, where $\psi _{ij}e_k=\delta _{jk}e_i$, then we obtain

$$\begin{aligned} \int _{S^{n-1}}(e_i\cdot u)(e_j\cdot u)d\phi _o\nu (u)=\frac{|\nu |}{n}\delta _{ij}. \end{aligned}$$

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

$$\begin{aligned} A_q(E_o,L)|x|^2=\int _{S^{n-1}}|\phi _o^tx\cdot u|^2\rho _{E_o}^2(u)\rho _{\Gamma _{-q}E_o}^{n+q}(u)\rho _{\Gamma _{-q}L}^{-q}(u)du. \end{aligned}$$

Equivalently,

$$\begin{aligned} A_q(E_o,L)|\phi _o^{-t}x|^2=\int _{S^{n-1}}|x\cdot u|^2 \rho _{E_o}^{2}(u)\left( \frac{\rho _{\Gamma _{-q}E_o}(u)}{\rho _{\Gamma _{-q}L}(u)}\right) ^q\rho _{\Gamma _{-q}E_o}^n(u)du. \end{aligned}$$

This proves (4.6).

Conversely, we suppose that (4.6) holds and shall prove that if $|E|=\omega _n$, then

$$\begin{aligned} \overline{A}_q(E, L)\ge \overline{A}_q(E_o, L), \end{aligned}$$

(4.9)

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

$$\begin{aligned} d\nu (u)&=\rho _{\Gamma _{-q}E_o}^{n+q}(u)\rho _{\Gamma _{-q}L}^{-q}(u)du\\&=c_{n,q}^{n+q}|\phi _ou|^{-n-q}\rho _{\Gamma _{-q}L}^{-q}(u)(u)du. \end{aligned}$$

On one hand,

$$\begin{aligned} A_q(E,L)&=\frac{1}{n}\int _{S^{n-1}}\rho _{\Gamma _{-q}E}^{n+q} \rho _{\Gamma _{-q}L}^{-q}(u)\nonumber \\&=\frac{c_{n,q}^{n+q}}{n}\int _{S^{n-1}}|\phi u|^{-n-q}\rho _{\Gamma _{-q}L}^{-q}(u)du\nonumber \\&=\frac{1}{n}\int _{S^{n-1}}|\phi \phi _o^{-1} \langle \phi _ou\rangle |^{-n-q}d\nu (u)\nonumber \\&=\frac{1}{n}\int _{S^{n-1}}|\phi \phi _o^{-1} u|^{-n-q}d\phi _o\nu (u). \end{aligned}$$

(4.10)

On another hand,

$$\begin{aligned} A_q(E_o,L)=|\nu |/n=|\phi _o\nu |/n. \end{aligned}$$

(4.11)

Thus, to prove (4.9), it suffices to show that if (4.6) holds then

$$\begin{aligned} \frac{1}{|\phi _o\nu |}\int _{S^{n-1}}|\phi \phi _o^{-1}u|^{-n-q}d\phi _o\nu (u)\ge 1, \end{aligned}$$

(4.12)

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

$$\begin{aligned} \frac{1}{|\phi _o\nu |}\int _{S^{n-1}}|Pu|^{-n-q}d\phi _o\nu (u)\ge 1, \end{aligned}$$

(4.13)

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

$$\begin{aligned} \int _{S^{n-1}}|Pu|^{-n-q}d\phi _o\nu (u)=\int _{S^{n-1}} |DOu|^{-n-q}d\phi _o\nu (u)=\int _{S^{n-1}}|Du|^{-n-q}dO\phi _o\nu (u). \end{aligned}$$

Observing that (4.6) is equivalent to

$$\begin{aligned} \int _{S^{n-1}}u_i^2d\phi _o\nu (u)=\frac{|\phi _o\nu |}{n}. \end{aligned}$$

(4.14)

This together with the fact that O is orthogonal gives

$$\begin{aligned} \int _{S^{n-1}}u_i^2dO\phi _o\nu (u)=\frac{|\phi _o\nu |}{n}. \end{aligned}$$

(4.15)

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

$$\begin{aligned} \frac{1}{|\phi _o\nu |}\int _{S^{n-1}}|Du|^{-n-q}dO\phi _o\nu (u)&=\frac{1}{|\phi _o\nu |}\int _{S^{n-1}}\left( \sum _{i=1}^n \lambda _i^2u_i^2\right) ^{-\frac{n+q}{2}}dO\phi _o\nu (u)\\&\ge \frac{1}{|\phi _o\nu |}\int _{S^{n-1}}\sum _{i=1}^nu_i^2 \lambda _i^{-n-q}dO\phi _o\nu (u)\\&=\frac{1}{n}\sum _{i=1}^n\lambda _i^{-n-q}\\&\ge 1. \end{aligned}$$

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

$$\begin{aligned} |x|^2=\frac{c_{n,q}^{n+q}}{A_q(B_2^n,L)}\int _{S^{n-1}}|x\cdot v|^2\rho _{\Gamma _{-q}L}^{-q}(v)dv,\quad \text {for}~~\text {all}~~x\in \mathbb {R}^n. \end{aligned}$$

Meanwhile, $B_2^n$ solves $LJ_q$ if and only if

$$\begin{aligned} |x|^2=\frac{c_{n,q}^{n+q}}{|\Gamma _{-q}B_2^n|}\int _{S^{n-1}}|x\cdot v|^2\rho _{\Gamma _{-q}L}^{-q}(v)dv,\quad \text {for}~~\text {all}~~x\in \mathbb {R}^n. \end{aligned}$$

(4.16)

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

$$\begin{aligned} \max _E|E|\quad \text {subject}~~\text {to}\quad \overline{A}_q(E,L)\le 1 \end{aligned}$$

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

$$\begin{aligned} \min _E\overline{A}_q(E,L)\quad \text {subject}~~\text {to}\quad |E|=\omega _n \end{aligned}$$

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 $\phi \in \mathrm {GL}(n)$,

$$\begin{aligned} \widetilde{E}_q\phi L=\phi \widetilde{E}_qL. \end{aligned}$$

It follows from (4.16) that $\widetilde{E}_qB_2^n=B_2^n$. This together with Lemma 4.6 gives

$$\begin{aligned} \widetilde{E}_qE=E, \end{aligned}$$

(4.17)

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 $1\le q<r\le \infty$,

$$\begin{aligned} \overline{A}_q(E,L)\le \overline{A}_r(E,L). \end{aligned}$$

(5.1)

Proof

If $q<r<\infty$, then from definition (3.19), Lemma 3.2, Jensen’s inequality, and definition (3.19) again, it follows that

$$\begin{aligned} \overline{A}_q(E,L)&=n^{-\frac{1}{q}}c_{n,q}^{-1} \left( \frac{1}{n|\Gamma _{-q}E|}\int _{S^{n-1}} \left( \frac{\rho _{\Gamma _{-q}E}(u)}{\rho _{\Gamma _{-q}L}(u)}\right) ^q \rho _{\Gamma _{-q}E}^n(u)du\right) ^{\frac{1}{q}}\\&=\left( \frac{1}{n|E|}\int _{S^{n-1}}\left( \frac{\rho _E(u)}{n^{{1}/{q}} \rho _{\Gamma _{-q}L}(u)}\right) ^q\rho _{E}^n(u)du\right) ^{\frac{1}{q}}\\&\le \left( \frac{1}{n|E|}\int _{S^{n-1}}\left( \frac{\rho _E(u)}{n^{{1}/{r}} \rho _{\Gamma _{-r}L}(u)}\right) ^q\rho _{E}^n(u)du\right) ^{\frac{1}{q}}\\&\le \left( \frac{1}{n|E|}\int _{S^{n-1}}\left( \frac{\rho _E(u)}{n^{{1}/{r}} \rho _{\Gamma _{-r}L}(u)}\right) ^r\rho _{E}^n(u)du\right) ^{\frac{1}{r}}\\&=n^{-\frac{1}{r}}c_{n,r}^{-1}\left( \frac{1}{n|\Gamma _{-r}E|}\int _{S^{n-1}} \left( \frac{\rho _{\Gamma _{-r}E}(u)}{\rho _{\Gamma _{-r}L}(u)}\right) ^r \rho _{\Gamma _{-r}E}^n(u)du\right) ^{\frac{1}{r}}\\&=\overline{A}_r(E,L). \end{aligned}$$

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

$$\begin{aligned} d_E=\max _{u\in S^{n-1}}h_{TB_2^n}(u)=\max _{u\in S^{n-1}}|T^tu|=\max _{u\in S^{n-1}}|Tu|=\Vert T\Vert . \end{aligned}$$

(5.2)

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

$$\begin{aligned} \mathcal {E}_R=\{E\in \mathcal {E}^n: |E|=\omega _n~~\mathrm {and}~~E\subseteq RB_2^n\}. \end{aligned}$$

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

$$\begin{aligned} \lim _{j\rightarrow \infty }\overline{A}_{q_j}(E,L)=\overline{A}_{q_o}(E,L) \end{aligned}$$

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

$$\begin{aligned} \overline{A}_{q_j}(E,L)\le \overline{A}_{q_{j+1}}(E,L). \end{aligned}$$

(5.3)

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

$$\begin{aligned} \overline{A}_{q_j}(\cdot ,L)\rightarrow \overline{A}_{q_o}(\cdot ,L),\quad \text {uniformly}~~\text {on}~~\mathcal {E}_R, \end{aligned}$$

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

$$\begin{aligned} \lim _{j\rightarrow \infty }\overline{A}_{q_j}(\overline{E}_{q_j}L,L) =\overline{A}_{q_o}(\overline{E}_{q_o}L,L). \end{aligned}$$

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

$$\begin{aligned} \lim _{j\rightarrow \infty }\overline{A}_{q_j}(\overline{E}_{q_j}L,L)&=\lim _{j\rightarrow \infty }\min _{|E|=\omega _n}\overline{A}_{q_j}(E,L)=\min _{|E|=\omega _n}\lim _{j\rightarrow \infty }\overline{A}_{q_j}(E,L)\\&=\min _{|E|=\omega _n}\overline{A}_{q_o}(E,L)=\overline{A}_{q_o}(\overline{E}_{q_o}L,L). \end{aligned}$$

$\square$

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

$$\begin{aligned} \lim _{j\rightarrow \infty }\overline{E}_{q_j}L=\overline{E}_{q_o}L. \end{aligned}$$

Proof

Assume that

$$\begin{aligned} \lim _{j\rightarrow \infty }\overline{E}_{q_j}L=E'\ne \overline{E}_{q_o}L. \end{aligned}$$

Since the $\overline{LJ}_q$ solution is unique, by the uniform convergence established in Lemma 5.2, we obtain

$$\begin{aligned} \overline{A}_{q_o}(\overline{E}_{q_o}L,L)<\overline{A}_{q_o}(E',L) =\lim _{j\rightarrow \infty }\overline{A}_{q_j}(\overline{E}_{q_j}L,L), \end{aligned}$$

which contradicts Lemma 5.3. $\square$

Noting that by Lemma 4.1,

$$\begin{aligned} \widetilde{E}_qL=\overline{A}_q(\overline{E}_qL,L)\overline{E}_qL. \end{aligned}$$

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

$$\begin{aligned} \lim _{j\rightarrow \infty }\widetilde{E}_{q_j}L=\widetilde{E}_{q_o}L. \end{aligned}$$

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

$$\begin{aligned} |K|/|\widetilde{E}_\infty K|\le 2^n/\omega _n, \end{aligned}$$

(6.1)

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

$$\begin{aligned} |\widetilde{E}_rL|\le |\widetilde{E}_qL|. \end{aligned}$$

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

$$\begin{aligned} |\widetilde{E}_\infty L|\le \cdots \le |\widetilde{E}_{q_j}L|\le \cdots \le |\widetilde{E}_qL|. \end{aligned}$$

$\square$

Theorem 6.3

If $L\in \mathcal {K}_e^n$ and $1\le q\le \infty$, then

$$\begin{aligned} |\Gamma _{-q}\widetilde{E}_qL|\le |\Gamma _{-q}L|, \end{aligned}$$

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

$$\begin{aligned} |\Gamma _{-q}\widetilde{E}_qL|=A_q(\widetilde{E}_qL,L) \ge |\Gamma _{-q}\widetilde{E}_qL|^{\frac{n+q}{n}}|\Gamma _{-q}L|^{-\frac{q}{n}}, \end{aligned}$$

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

$$\begin{aligned} |L|\le \frac{2^n}{\omega _n}|\widetilde{E}_qL|. \end{aligned}$$

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 $q=\infty$.