Orlicz–Legendre Ellipsoids

Abstract

The Orlicz–Legendre ellipsoids, which are in the framework of emerging dual Orlicz Brunn–Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz–John ellipsoids, and have largely generalized the classical Legendre ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz–Legendre ellipsoids and isotropy of measures is demonstrated.

1 Introduction

Corresponding to each body in Euclidean n-space $\mathbb R^n$, there is a unique ellipsoid with the following property: The moment of inertia of the ellipsoid and the moment of inertia of the body are the same about every 1-dimensional subspace of $\mathbb R^n$. This ellipsoid is called the Legendre ellipsoid of the body. The Legendre ellipsoid is a well-known concept from classical mechanics, and is closely related to the long-standing unsolved maximal slicing problem. See, e.g., Lindenstrauss and Milman [27], and Milman and Pajor [46].

The Legendre ellipsoid is an object in the dual Brunn–Minkowski theory, which was originated by Lutwak [31] and achieved great development since 1980s. See, e.g., [9–11, 32, 33, 55, 56]. It is remarkable that for each convex body (compact convex subset with non-empty interior) K in $\mathbb R^n$, Lutwak, Yang and Zhang [36] introduced a new ellipsoid by using the notion of $L_{2}$-curvature, which is now called the LYZ ellipsoid and is precisely the dual analogue of the Legendre ellipsoid.

Following LYZ [36], we write $\Gamma _{2} K$ and $\Gamma _{-2} K$ for the Legendre ellipsoid and LYZ ellipsoid, respectively. In [39], LYZ extended the domain of $\Gamma _{-2}$ to star-shaped sets and showed the relationship between the two ellipsoids: If K is a star-shaped set, then $\Gamma _{-2} K\subset \Gamma _2 K$, with equality if and only if K is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities in information theory: the Cramer–Rao inequality. When viewed as suitably normalized matrix-valued operators on the space of convex bodies, it was proved by Ludwig [28] that the Legendre ellipsoid and the LYZ ellipsoid are the only linearly invariant operators that satisfy the inclusion-exclusion principle. The Legendre ellipsoid also has applications in Finsler geometry [45].

In the geometry of convex bodies, many extremal problems of an affine nature often have ellipsoids as extremal bodies. Besides the above-mentioned Legendre ellipsoid and LYZ ellipsoid, the John ellipsoid $\mathrm{J}K$ [24] and the Löwner ellipsoid $\mathrm{L} K$ are of fundamental importance. Since the object considered in this paper is dual to the John ellipsoid, in what follows, we recall the John ellipsoid in detail.

Associated with each convex body K in $\mathbb R^n$, its John ellipsoid $\mathrm{J}K$ is the unique ellipsoid of maximal volume contained in K. The John ellipsoid has many applications in convex geometry, functional analysis, PDEs, etc. Particularly, by combining the isotropic characterization of the John ellipsoid and the celebrated Brascamp–Lieb inequality, it has a powerful effect on attacking reverse isoperimetric problems. See, e.g., [1–3, 40–42].

Since 2005, the family of John ellipsoids has expanded rapidly, and experienced the $L_p$ stage [41] and the very recent Orlicz stage [59]. It is interesting that with the expansion of the family, several ellipsoids, including the LYZ ellipsoid, are found to be close relatives of the John ellipsoid. We do a bit of review on this point.

Motivated by the study of geometry of $L_p$ Brunn–Minkowski theory (see, e.g., [34, 35, 37]), LYZ [41] introduced a family of ellipsoids, called the $L_p$ John ellipsoids $\mathrm{E}_p K$, $p>0$. It is striking that the bodies $\mathrm{E}_p K$ form a spectrum linking several fundamental objects in convex geometry: If the John point of K, i.e., the center of $\mathrm{J}K$, is at the origin, then $\mathrm{E}_\infty K$ is precisely the classical John ellipsoid $\mathrm{J}K$. The $L_2$ John ellipsoid $\mathrm{E}_2 K$ is just the LYZ ellipsoid. The $L_1$ John ellipsoid $\mathrm{E}_1 K$ is the so-called Petty ellipsoid. The volume-normalized Petty ellipsoid is obtained by minimizing the surface area of K under $\mathrm{SL}(n)$ transformations of K [14, 47].

Throughout this paper, we consider convex $\varphi :[0,\infty )\rightarrow [0,\infty )$ that is strictly increasing and satisfies $\varphi (0)=0$. Along the line of extension, the authors of this paper originally introduced the Orlicz–John ellipsoids [59] $\mathrm{E}_\varphi K$ for each convex body K with the origin in its interior, in the framework of booming Orlicz Brunn–Minkowski theory (see, e.g., [12, 13, 21, 29, 43, 44]). The new Orlicz–John ellipsoids $\mathrm{E}_\varphi K$ generalize LYZ’s $L_p$ John ellipsoids $\mathrm{E}_p K$ to the Orlicz setting, analogous to the way that Orlicz norms [50] generalize $L_p$ norms. Indeed, if $\varphi (t)=t^p$, $1\le p<\infty$, then $\mathrm{E}_\varphi K$ precisely turns to the $L_p$ John ellipsoid $\mathrm{E}_p K$. If $p\rightarrow \infty$, then $\mathrm{E}_{\varphi ^p} K$ approaches $\mathrm{E}_\infty K$.

The Löwner ellipsoid $\mathrm{L} K$ is the unique ellipsoid of minimal volume containing K, which is investigated widely in the field of convex geometry and local theory of Banach spaces. We refer to, e.g., [1, 2, 14, 17–20, 24–27, 30, 49, 58].

As LYZ [39] pointed out, there is in fact a “dictionary” correspondence between the Brunn–Minkowski theory [51] and its dual [31]. In retrospect, the John ellipsoid, LYZ ellipsoid and Petty ellipsoid are objects within the Brunn–Minkowski theory; while the Legendre ellipsoid and Löwner ellipsoid are objects within the dual Brunn–Minkowski theory. Along the idea of dictionary relation, we are tempted to consider the naturally posed problem: What is the dual analogue of the newly found Orlicz–John ellipsoid?

One of the main tasks in this paper is to demonstrate this existence of such a dual analogue of the Orlicz–John ellipsoid. Incidentally, it precisely acts as the spectrum linking the Legendre ellipsoid and Löwner ellipsoid. So, this paper is a sequel of [59].

For star bodies K, L in $\mathbb R^n$, define the normalized dual Orlicz mixed volume ${\bar{\tilde{V}}}_\varphi (K,L)$ of K and L with respect to $\varphi$ by

$$\begin{aligned} {\bar{\tilde{V}}}_\varphi (K,L) = {\varphi ^{ - 1}}\left( {\int _{{S^{n - 1}}} {\varphi \left( {\frac{{{\rho _K}}}{{{\rho _L}}}} \right) dV_K^*} } \right) . \end{aligned}$$

Here, $S^{n-1}$ is the unit sphere in $\mathbb R^n$; $\rho _K$ and $\rho _L$ are the radial functions of K and L, respectively; $V_K^*$ is the normalized dual conical measure of K, defined by

$$\begin{aligned} dV_K^*=\frac{\rho _K^n}{nV(K)}dS, \end{aligned}$$

where S is the spherical Lebesgue measure on $S^{n-1}$.

Enlightened by our work on Orlicz–John ellipsoids [59], we focus on

Problem ${\tilde{S}}_\varphi$ Suppose K is a star body in $\mathbb R^n$. Find an ellipsoid E, among all origin-symmetric ellipsoids, which solves the following constrained minimization problem:

$$\begin{aligned} \mathop {{\mathrm{min}}}\limits _E V(E)\quad {\mathrm{subject}}\;{\mathrm{to}}\quad {\bar{\tilde{V}}}_\varphi (K,E) \le 1. \end{aligned}$$

In Sect. 4, we prove that there exists a unique ellipsoid which solves the above minimization problem. It is called the Orlicz–Legendre ellipsoid of K with respect to $\varphi$, and denoted by $\mathrm{L}_\varphi K$. If $\varphi (t)=t^2$, then $\mathrm{L}_\varphi K$ is precisely the Legendre ellipsoid $\Gamma _2 K$.

It is interesting that the Orlicz–Legendre ellipsoid mirrors the Orlicz–John ellipsoid.

Similar to the important property of the Orlicz–John ellipsoid $\mathrm{E}_\varphi K$, in Sect. 5 we show that the Orlicz–Legendre ellipsoid $\mathrm{L}_\varphi K$ is jointly continuous in $\varphi$ and K.

In Sect. 6, we establish a characterization of Orlicz–Legendre ellipsoids, which is closely related to the isotropy of measures.

In general, Orlicz–Legendre ellipsoids $\mathrm{L}_\varphi K$ do not contain K. In Sect. 7, we prove that: If K is a star body (about the origin) in $\mathbb R^n$, then

$$\begin{aligned} V(\mathrm{L}_\varphi K)\ge V(K), \end{aligned}$$

with equality if and only if K is an ellipsoid centered at the origin.

If $\varphi (t)=t^2$, it reduces to the celebrated inequality: $V(\Gamma _2 K)\ge V(K)$, which goes back to Blaschke [6], John [23], Milman and Pajor [46], Petty [48], and also LYZ [36].

2 Preliminaries

2.1 Notation

The setting will be the Euclidean n-space $\mathbb R^n$. As usual, $x\cdot y$ denotes the standard inner product of x and y in $\mathbb R^n$, and V denotes the n-dimensional volume.

In addition to its denoting absolute value, without confusion we often use $|\cdot |$ to denote the standard Euclidean norm, on occasion the total mass of a measure, and the absolute value of the determinant of an $n\times n$ matrix.

For a continuous real function f defined on $S^{n-1}$, write $\Vert f\Vert _\infty$ for the $L_\infty$ norm of f. Let $\mathscr {L}^n$ denote the space of linear operators from $\mathbb R^n$ to $\mathbb R^n$. For $T\in \mathscr {L}^n$, $T^t$ and $\Vert T\Vert$ denote the transpose and norm of T, respectively.

A finite positive Borel measure $\mu$ on $S^{n-1}$ is said to be isotropic if

$$\begin{aligned} \frac{n}{|\mu |}\int _{{S^{n - 1}}} {{{(u \cdot v)}^2}d\mu (u)}= 1,\quad \mathrm{for\; all}\; v\in S^{n-1}. \end{aligned}$$

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

$$\begin{aligned} {\mathop {{\mathrm{tr}}}\nolimits } (x \otimes x) = |x{|^2}. \end{aligned}$$

Equivalently, $\mu$ is isotropic if

$$\begin{aligned} \frac{n}{|\mu |}\int _{{S^{n - 1}}} {u \otimes ud\mu (u)} = {I_n}, \end{aligned}$$

where $I_n$ denotes the identity operator on $\mathbb R^n$. For more information on the isotropy of measures, we refer to [5, 14, 15, 46].

2.2 Orlicz Norms

Throughout this paper, $\Phi$ denotes the class of convex functions $\varphi : [0,\infty )\rightarrow [0,\infty )$ that are strictly increasing and satisfy $\varphi (0)=0$.

We say a sequence $\{\varphi _i\}_{i\in \mathbb N}\subset \Phi$ is such that ${\varphi _i} \rightarrow {\varphi _0}\in \Phi$, provided

$$\begin{aligned} |{\varphi _i} - {\varphi _0}{|_I} = \mathop {\max }\limits _{t \in I} |{\varphi _i}(t) - {\varphi _0}(t)| \rightarrow 0, \end{aligned}$$

for each compact interval $I\subset [0,\infty )$.

Let $\mu$ be a finite positive Borel measure on $S^{n-1}$. For a continuous function $f: S^{n-1}\rightarrow [0,\infty )$, the Orlicz norm ${\left\| f: \mu \right\| _\varphi }$ of f is defined by

$$\begin{aligned} {\left\| f:\mu \right\| _\varphi } = \inf \left\{ {\lambda > 0:\frac{1}{{|\mu |}}\int _{{S^{n - 1}}} {\varphi \left( {\frac{f}{\lambda }} \right) d\mu } \le \varphi (1)} \right\} . \end{aligned}$$

If $\varphi (t)=t^p$, $1\le p<\infty$, then $\Vert f:\mu \Vert _\varphi$ is just the classical $L_p$ norm. According to the context, without confusion we write $\Vert f\Vert _\varphi$ for $\Vert f:\mu \Vert _\varphi$.

Lemma 2.1 was previously proved in [21], which will be used frequently.

Lemma 2.1

Suppose $\mu$ is a finite positive Borel measure on $S^{n-1}$ and the function $f: S^{n-1}\rightarrow [0,\infty )$ is continuous and such that $\mu (\{{f\ne 0}\})>0$. Then the function

$$\begin{aligned} \psi (\lambda ) = \int _{{S^{n - 1}}} {\varphi \left( {\frac{f}{\lambda }} \right) d\mu }\;\;\lambda \in (0,\infty ), \end{aligned}$$

has the following properties:

• (1) $\psi$ is continuous and strictly decreasing in $(0,\infty )$;

• (2) $\mathop {\lim }\limits _{\lambda \rightarrow {0^ + }} \psi (\lambda ) = \infty$;

• (3) $\mathop {\lim \limits _{\lambda \rightarrow \infty }}\psi (\lambda ) = 0$;

• (4) $0 < {\psi ^{ - 1}}( a ) < \infty$ for each $a\in (0,\infty )$.

Consequently, the Orlicz norm ${\left\| f\right\| }_\varphi$ is strictly positive. Moreover,

$$\begin{aligned} {\left\| f \right\| _\varphi } = {\lambda _0}\quad \Longleftrightarrow \quad \frac{1}{{|\mu |}}\int _{{S^{n - 1}}} {\varphi \left( {\frac{f}{{{\lambda _0}}}} \right) d\mu } = \varphi (1). \end{aligned}$$

2.3 Convex Bodies and Star Bodies

The support function $h_K$ of a compact convex set K in $\mathbb R^n$ is defined by

$$\begin{aligned} h_K(x)=\max \{x\cdot y: y\in K\},\quad \mathrm{for}\; x\in \mathbb R^n. \end{aligned}$$

For $T\in \mathrm{GL}(n)$, the support function of the image $TK=\{Tx:x\in K\}$ is given by

$$\begin{aligned} {h_{TK}}(x) = {h_K}({T^t}x). \end{aligned}$$

As usual, a body is a compact set with non-empty interior. Write $\mathcal K^n_o$ for the class of convex bodies in $\mathbb R^n$ that contain the origin in their interiors. $\mathcal K^n_o$ is often equipped with the Hausdorff metric $\delta _H$, which is defined by

$$\begin{aligned} \delta _H(K_1,K_2)=\Vert h_{K_1}-h_{K_2}\Vert _\infty , \quad \mathrm{for}\; K_1, K_2\in \mathcal K^n_o. \end{aligned}$$

Next, we turn to some basics on star bodies.

A set $K\subseteq \mathbb R^n$ is star-shaped, if $\lambda x\in K$ for any $(\lambda , x)\in [0,1]\times K$. For a non-empty, compact and star-shaped set K in $\mathbb R^n$, its radial function $\rho _K$ is defined by

$$\begin{aligned} \rho _K(x)=\sup \{\lambda \ge 0: \lambda x\in K\},\quad \mathrm{for}\; x\in \mathbb R^n{\setminus } \{o\}. \end{aligned}$$

It is easily seen that $\rho _K$ is homogeneous of degree $-1$. For $T\in \mathrm{GL}(n)$, we obviously have

$$\begin{aligned} \rho _{TK}(x)=\rho _K(T^{-1}x). \end{aligned}$$

(2.1)

A star-shaped set K is called a star body about the origin o, if $o\in \mathrm{int}K$ and $\rho _K$ is continuous on $S^{n-1}$. Write $\mathcal {S}^n_o$ for the class of star bodies about the origin o in $\mathbb R^n$. $\mathcal {S}^n_o$ is often equipped with the dual metric $\tilde{\delta }_H$, which is defined by

$$\begin{aligned} \tilde{\delta }_H(K_1,K_2)=\Vert \rho _{K_1}-\rho _{K_2}\Vert _\infty , \quad \mathrm{for}\; K_1, K_2\in \mathcal {S}^n_o. \end{aligned}$$

The dual conical measure $\tilde{V}_K$ of a star body $K\in \mathcal {S}^n_o$ is a Borel measure on $S^{n-1}$ defined by

$$\begin{aligned} d\tilde{V}_K=\frac{\rho _K^n}{n}dS. \end{aligned}$$

It is convenient to use its normalization $V^*_K$, given by $V^*_K=\frac{\tilde{V}_K}{V(K)}$. Observe that $V^*_K$ was first introduced by LYZ [44] to define Orlicz centroid bodies. Note that the dual conical measure differs from the cone-volume measure (see, e.g., [7, 8, 21, 22, 38, 43, 52, 53, 60]), but both are outgrowth from the cone measure (see, e.g., [4, 16]).

Note that for each Borel subset $\omega \subseteq S^{n-1}$, we also have

$$\begin{aligned} \tilde{V}_K(\omega )=V\left( K\cap \{su : s\ge 0\; \mathrm{and }\; u\in \omega \}\right) . \end{aligned}$$

Thus, it follows that

$$\begin{aligned} {\tilde{V}}_{TK}(\omega )={\tilde{V}}_K(\langle T^{-1}\omega \rangle ), \quad \mathrm{for}\; T\in \mathrm{SL}(n), \end{aligned}$$

(2.2)

where $\langle T^{-1}\omega \rangle =\{\frac{T^{-1}u}{|T^{-1}u|}: u\in \omega \}$.

For $K\in \mathcal {K}^n_o$, its polar body $K^*$ of K is defined by

$$\begin{aligned} K^*=\{x\in \mathbb R^n: x\cdot y\le 1,\; \mathrm{for\; }\; y\in K\}. \end{aligned}$$

For $K\in \mathcal {K}^n_o$, we have

$$\begin{aligned} \rho _{K^*}(u)=\frac{1}{h_K(u)}\quad \mathrm{and}\quad h_{K^*}(u)=\frac{1}{\rho _K(u)}, \quad \mathrm{for}\; u\in S^{n-1}, \end{aligned}$$

(2.3)

and

$$\begin{aligned} (TK)^*=T^{-t}K^*, \quad \mathrm{for}\; T\in \mathrm{GL}(n). \end{aligned}$$

(2.4)

2.4 Ellipsoids and Linear Operators

Throughout, $\mathcal E^n$ is used exclusively to denote the class of n-dimensional origin-symmetric ellipsoids in $\mathbb R^n$.

For $E\in \mathcal E^n$, let d(E) denote its maximal principal radius. Two facts are in order. First, $T\in \mathscr {L}^n$ is non-degenerated, if and only if the ellipsoid TB is non-degenerated. Second, for $T\in \mathscr {L}^n$, since

$$\begin{aligned} \left\| T \right\| = \mathop {\max }\limits _{u \in {S^{n - 1}}} |Tu| = \mathop {\max }\limits _{u \in {S^{n - 1}}} |{T^t}u| =\left\| {{T^t}} \right\| , \end{aligned}$$

it follows that

$$\begin{aligned} d(TB)= \mathop {\max }\limits _{u \in {S^{n - 1}}} {h_{TB}(u)}=\mathop {\max }\limits _{u \in {S^{n - 1}}} |{T^t}u| =\mathop {\max }\limits _{u \in {S^{n - 1}}} |Tu| = \mathop {\max }\limits _{u \in {S^{n - 1}}} {h_{{T^t}B}(u)} = d({T^t}B). \end{aligned}$$

Let

$$\begin{aligned} d_n(T_1,T_2)=\Vert T_1-T_2\Vert ,\quad \mathrm{for}\; T_1, T_2\in \mathscr {L}^n. \end{aligned}$$

Then the metric space $(\mathscr {L}^n, d_n)$ is complete. Since $\mathscr {L}^n$ is of finite dimension, a set in $(\mathscr {L}^n, d_n)$ is compact if and only if it is bounded and closed.

We conclude this section with the following basic known facts, which will be used in Sects. 4 and 5.

Lemma 2.2

Suppose $\{T_j\}_{j\in \mathbb N}\subset \mathrm{SL}(n)$. Then

$$\begin{aligned} \Vert T_j\Vert \rightarrow \infty \quad \Longleftrightarrow \quad \Vert T_j^{-1}\Vert \rightarrow \infty . \end{aligned}$$

Thus, $\{T_j\}_{j\in \mathbb N}$ is bounded from above, if and only if $\{T_j^{-1}\}_{j\in \mathbb N}$ is bounded from above.

Lemma 2.3

Suppose $E_0\in \mathcal {E}^n$, $\{E_j\}_{j\in \mathbb N}\subset \mathcal {E}^n$ and $V(E_j)=a$, $\forall j \in \mathbb N$, $a>0$. Then $E_j\rightarrow E_0$ with respect to $\delta _H$, if and only if $E_j\rightarrow E_0$ with respect to ${\tilde{\delta }}_H$.

3 Dual Orlicz Mixed Volumes

In order to define Orlicz–Legendre ellipsoids, we make some necessary preparations.

Definition 3.1

Suppose $K,L\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. The geometric quantity

$$\begin{aligned} {\tilde{V}}_{\varphi }(K,L) = \int _{{S^{n - 1}}} {\varphi \left( {\frac{{{\rho _K}}}{{{\rho _L}}}} \right) d{\tilde{V}}_K} \end{aligned}$$

is called the dual Orlicz mixed volume of K and L with respect to $\varphi$. The quantity

$$\begin{aligned} {\bar{\tilde{V}}}_{\varphi }(K,L) = {\varphi ^{ - 1}}\left( {\frac{{{{\tilde{V}}_\varphi }(K,L)}}{{V(K)}}} \right) = {\varphi ^{ - 1}}\left( {\int _{{S^{n - 1}}} {\varphi \left( {\frac{{{\rho _K}}}{{{\rho _L}}}} \right) d V_K^*} } \right) \end{aligned}$$

is called the normalized dual Orlicz mixed volume of K and L with respect to $\varphi$.

It is noted that dual Orlicz mixed volumes were previously introduced in [57].

Obviously, $\tilde{V}_{\varphi }(K,K)=\varphi (1)V(K)$, and ${\bar{\tilde{V}}}(K,K)=1$. If $\varphi (t)=t^p$, $1\le p<\infty$, then ${\tilde{V}}_{\varphi }(K,L)$ reduces to the classical dual mixed volume

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

and ${\bar{\tilde{V}}}_{\varphi }(K,L)$ reduces to normalized dual mixed volume [54]

$$\begin{aligned} { \bar{\tilde{V}}}_{-p}(K,L) = {\left[ {\frac{{{{\tilde{V}}_p}(K,L)}}{{V(K)}}} \right] ^{\frac{1}{p}}} = {\left( {\int _{{S^{n - 1}}} {{{\left( {\frac{{{\rho _K}}}{{{\rho _L}}}} \right) }^p}dV_K^*} } \right) ^{\frac{1}{p}}}. \end{aligned}$$

Combining Definition 3.1 with (2.1) and (2.2), we have the following.

Lemma 3.2

Suppose $K, L\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Then $\tilde{V}_\varphi (TK,L)= |T| \tilde{V}_\varphi (K,T^{-1}L)$, for $T \in \mathrm{GL}(n)$.

Along with the functional $\tilde{V}_\varphi (K,L)$, we introduce

Definition 3.3

Suppose $K, L\in \mathcal {S}^n_o$ and $\varphi \in \Phi$; define

$$\begin{aligned} {O_\varphi }(K,L) = {\left\| {\frac{{{\rho _K}}}{{{\rho _L}}}:{{\tilde{V}}_K}} \right\| _\varphi }= \inf \left\{ {\lambda > 0:{\varphi ^{ - 1}}\left( {\int _{{S^{n - 1}}} {\varphi \left( {\frac{{{\rho _K}}}{{\lambda {\rho _L}}}} \right) dV_K^*} } \right) \le 1} \right\} . \end{aligned}$$

Obviously, $O_\varphi (K,K)=1$. If $\varphi (t)=t^p$, $1\le p<\infty$, then $O_\varphi (K,L)={\bar{\tilde{V}}}_{-p}(K,L)$.

From Definition 3.3 and Definition 3.1, we have

$$\begin{aligned} {O_\varphi }(K,L)&= \inf \left\{ {\lambda > 0:\frac{{{{\tilde{V}}_\varphi }( K, \lambda L)}}{{V(K)}} \le \varphi (1)} \right\} \\&= \inf \left\{ {\lambda > 0:{{\bar{\tilde{V}}}_\varphi }(K, \lambda L) \le 1} \right\} . \end{aligned}$$

Combining this with Lemma 3.2, we immediately obtain

Lemma 3.4

Suppose $K,L\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Then

• (1) $O_\varphi (TK,L)=O_\varphi (K,T^{-1}L)$, for all $T\in \mathrm{GL}(n)$.

• (2) $O_\varphi (\lambda K, L)= O_\varphi (K,\lambda ^{-1}L)=\lambda O_\varphi (K,L)$, for all $\lambda >0$.

The next lemma provides a simple but powerful identity.

Lemma 3.5

Suppose $K,L\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Then

$$\begin{aligned} {\bar{\tilde{V}}}_\varphi (K,{O_\varphi }(K,L)L) = 1. \end{aligned}$$

Consequently, there is the following equivalence

$$\begin{aligned} {\bar{\tilde{V}}}_\varphi (K, L) = 1\quad \Longleftrightarrow \quad {O_\varphi }(K,L)=1. \end{aligned}$$

Proof

From Definition 3.1, Definition 3.3, together with Lemma 2.1, it follows that

$$\begin{aligned} \varphi \left( {{{\bar{\tilde{V}}}_\varphi }(K,{O_\varphi }(K,L)L)} \right) = \int _{{S^{n - 1}}} {\varphi \left( {\frac{{{\rho _K}}}{{{O_\varphi }(K,L){\rho _K}}}} \right) dV_K^*} = \varphi (1). \end{aligned}$$

Thus, ${{\bar{\tilde{V}}}_\varphi }(K,{O_\varphi }(K,L)L) = 1$. By Lemma 2.1 again, the desired equivalence follows. $\square$

What follows establishes the dual Orlicz Minkowski inequalities.

Lemma 3.6

Suppose $K, L\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Then

$$\begin{aligned} {{\bar{\tilde{V}}}_\varphi }(K,L) \ge {\left( {\frac{{V(K)}}{{V(L)}}} \right) ^{\frac{1}{n}}}, \end{aligned}$$

(3.1)

and

$$\begin{aligned} {O_\varphi }(K,L) \ge {\left( {\frac{{V(K)}}{{V(L)}}} \right) ^{\frac{1}{n}}}. \end{aligned}$$

(3.2)

Each equality holds in the above inequalities if and only if K and L are dilates.

Proof

From Definition 3.1, the fact that $\varphi ^{-1}$ is strictly increasing in $(0,\infty )$ together with the convexity of $\varphi$ and Jensen’s inequality, the definition of $V_K^*$, and Hölder’s inequality, we have

$$\begin{aligned} {{\bar{\tilde{V}}}_\varphi }(K,L)&= {\varphi ^{ - 1}}\left( {\int _{{S^{n - 1}}} {\varphi \left( {\frac{{{\rho _K}}}{{{\rho _L}}}} \right) dV_K^*} } \right) \\&\ge \frac{1}{{nV(K)}}\int _{{S^{n - 1}}} {\frac{{\rho _K^{n + 1}}}{{{\rho _L}}}dS} \\&\ge \frac{1}{{V(K)}}{\left( {\frac{1}{n}\int _{{S^{n - 1}}} {\rho _K^{(n + 1) \cdot \frac{n}{{n + 1}}}dS} } \right) ^{\frac{{n + 1}}{n}}}{\left( {\frac{1}{n}\int _{{S^{n - 1}}} {\rho _L^{ - 1 \cdot ( - n)}dS} } \right) ^{ - \frac{1}{n}}} \\&= {\left( {\frac{{V(K)}}{{V(L)}}} \right) ^{\frac{1}{n}}}. \end{aligned}$$

By the equality condition of Hölder’s inequality, the equality in the fourth line occurs only if $\rho _K/\rho _L$ is a positive constant on $S^{n-1}$. Thus, the equality holds in (3.1) only if K and L are dilates. Conversely, if $K=sL$ for some $s>0$, then ${{\bar{\tilde{V}}}_\varphi }(K,L)=s=\left( V(K)/V(L))\right) ^{1/n}$.

By Lemma 3.5 with inequality (3.1), we can derive inequality (3.2) directly. $\square$

The next lemma is crucial to proving the continuity of the functionals ${\tilde{V}}_\varphi (K,L)$, ${\bar{\tilde{V}}}_\varphi (K,L)$ and $O_\varphi (K,L)$ in $(K,L,\varphi )$.

Lemma 3.7

Suppose $f_i, f$ are strictly positive and continuous functions on $S^{n-1}$; $\varphi _k, \varphi \in \Phi$; $\mu _l,\mu$ are Borel probability measures on $S^{n-1}$; $i,k,l\in \mathbb N$. If $f_i\rightarrow f$ uniformly, $\varphi _k\rightarrow \varphi$ uniformly on each compact set, and $\mu _l\rightarrow \mu$ weakly, then

$$\begin{aligned}&\int _{{S^{n - 1}}} {{\varphi _k}\left( {{f_i}} \right) d{\mu _l}} \rightarrow \int _{{S^{n - 1}}} {\varphi \left( f \right) d\mu }, \end{aligned}$$

(3.3)

$$\begin{aligned}&\varphi _k^{ - 1}\left( {\int _{{S^{n - 1}}} {{\varphi _k}\left( {{f_i}} \right) d{\mu _l}} } \right) \rightarrow {\varphi ^{ - 1}}\left( {\int _{{S^{n - 1}}} {\varphi \left( f \right) d\mu }} \right) , \end{aligned}$$

(3.4)

and

$$\begin{aligned} {\left\| {{f_i}:{\mu _l}} \right\| _{{\varphi _k}}} \rightarrow {\left\| {f:\mu } \right\| _\varphi }. \end{aligned}$$

(3.5)

Proof

Since $f_i\rightarrow f$ uniformly, there exists an $N_0\in \mathbb N$, such that

$$\begin{aligned} \frac{1}{2}\mathop {\min }_{u \in {S^{n - 1}}} f(u) \le {f_i} \le 2\mathop {\max }_{u \in {S^{n - 1}}} f(u),\quad \mathrm{for}\; i>N_0. \end{aligned}$$

Let

$$\begin{aligned} {c_m} = \min \left\{ {\frac{1}{2}\mathop {\min }_{u \in {S^{n - 1}}} f(u),\mathop {\min }_{u \in {S^{n - 1}}} {f_i}(u),\;{\mathrm{with}}\;i \le N_0} \right\} , \end{aligned}$$

and

$$\begin{aligned} {c_M} = \max \left\{ {2\mathop {\max }_{u \in {S^{n - 1}}} f(u),\mathop {\max }_{u \in {S^{n - 1}}} {f_i}(u),\;{\mathrm{with}}\;i \le N_0} \right\} . \end{aligned}$$

So, by the strict positivity and continuity of $f_i$ and f, we have

$$\begin{aligned} f(u), f_i(u)\in [c_m, c_M]\subset (0,\infty ),\quad \mathrm{for}\; u\in S^{n-1}\; \mathrm{and}\; i\in \mathbb {N}. \end{aligned}$$

(3.6)

From the continuity and uniform convergence of $f_i$ and $\varphi _k$ on $[c_m, c_M]$, it follows that as $i,k\rightarrow \infty$,

$$\begin{aligned} \varphi _k(f_i)\rightarrow \varphi (f),\quad \mathrm{uniformly\; on}\; S^{n-1}. \end{aligned}$$

Added that $\mu _l\rightarrow \mu$ weakly as $l\rightarrow \infty$, one immediately concludes (3.3) and (3.4).

Finally, we conclude to show (3.5).

From (3.6) together with the strict monotonicity of $\varphi$ and $\varphi ^{-1}$, Lemma 2.1, and (3.6) together with the strict monotonicity of $\varphi$ and $\varphi ^{-1}$ again, it follows that

$$\begin{aligned} \frac{{{c_m}}}{{{{\left\| {{f_i}:{\mu _l}} \right\| }_{{\varphi _k}}}}} \le \varphi _k^{ - 1}\left( {\int _{{S^{n - 1}}} {{\varphi _k}\left( {\frac{{{f_i}}}{{{{\left\| {{f_i}:{\mu _l}} \right\| }_{{\varphi _k}}}}}} \right) d{\mu _l}} } \right) = 1 \le \frac{{{c_M}}}{{{{\left\| {{f_i}:{\mu _l}} \right\| }_{{\varphi _k}}}}}, \end{aligned}$$

which immediately gives

$$\begin{aligned} {c_m} \le {\left\| {{f_i}:{\mu _l}} \right\| _{{\varphi _k}}} \le {c_M},\quad {\mathrm{for}}\;i,k,l \in \mathbb N. \end{aligned}$$

Since $\left\{ \Vert f_i:\mu _l\Vert _{\varphi _k}: i,k,l\in \mathbb N\right\}$ is bounded, to prove (3.5), it suffices to prove that each convergent subsequence $\{\Vert f_{i_p}:\mu _{l_r}\Vert _{\varphi _{k_q}}\}_{p,q,r\in \mathbb N}$ of $\left\{ \Vert f_i:\mu _l\Vert _{\varphi _k}: i,k,l\in \mathbb N\right\}$ necessarily converges to $\Vert f:\mu \Vert _\varphi$, as $i_p,k_q,l_r\rightarrow \infty$.

Assume $\mathop {\lim }\limits _{p,q,r \rightarrow \infty } {\left\| {{f_{{i_p}}}:{\mu _{{l_r}}}} \right\| _{{\varphi _{{k_q}}}}} = {\lambda _0}$. Then, $\frac{{{f_{{i_p}}}}}{{{{\left\| {{f_{{i_p}}}:{\mu _{{l_r}}}} \right\| }_{{\varphi _{{k_q}}}}}}} \rightarrow \frac{f}{{{\lambda _0}}}$ uniformly on $S^{n-1}$. Hence, by (3.4), we have

$$\begin{aligned} \mathop {\lim }_{p,q,r \rightarrow \infty } \varphi _{{k_q}}^{ - 1}\left( {\int _{{S^{n - 1}}} {{\varphi _{{k_q}}}\left( {\frac{{{f_{{i_p}}}}}{{{{\left\| {{f_{{i_p}}}:{\mu _{{l_r}}}} \right\| }_{{\varphi _{{k_q}}}}}}}} \right) d{\mu _{{l_r}}}} } \right) = {\varphi ^{ - 1}}\left( {\int _{{S^{n - 1}}} {\varphi \left( {\frac{f}{{{\lambda _0}}}} \right) d\mu } } \right) . \end{aligned}$$

Meanwhile, since

$$\begin{aligned} \varphi _{{k_q}}^{ - 1}\left( {\int _{{S^{n - 1}}} {{\varphi _{{k_q}}}\left( {\frac{{{f_{{i_p}}}}}{{{{\left\| {{f_{{i_p}}}:{\mu _{{l_r}}}} \right\| }_{{\varphi _{{k_q}}}}}}}} \right) d{\mu _{{l_r}}}} } \right) = 1, \quad \mathrm{for\; each}\; (p,q,r), \end{aligned}$$

it yields that ${\varphi ^{ - 1}}\left( {\int _{{S^{n - 1}}} {\varphi \left( {\frac{f}{{{\lambda _0}}}} \right) d\mu } } \right) = 1$. From Lemma 2.1, it follows that $\lambda _0=\Vert f:\mu \Vert _\varphi$, which concludes (3.5). $\square$

Using Lemma 3.7, we immediately obtain

Lemma 3.8

Suppose $K,K_i,L,L_j\in \mathcal {S}^n_o$ and $\varphi ,\varphi _k\in \Phi$, $i,j,k\in \mathbb N$. If $K_i\rightarrow K$, $L_j\rightarrow L$ and $\varphi _k\rightarrow \varphi$; then

$$\begin{aligned} \mathop {\lim }\limits _{i,j,k \rightarrow \infty } {{\tilde{V}}_{{\varphi _k}}}({K_i},{L_j})= & {} {{\tilde{V}}_\varphi }(K,L),\\ \mathop {\lim }\limits _{i,j,k \rightarrow \infty } {{\bar{\tilde{V}}}_{{\varphi _k}}}({K_i},{L_j})= & {} {{\bar{\tilde{V}}}_\varphi }(K,L), \end{aligned}$$

and

$$\begin{aligned} \mathop {\lim }\limits _{i,j,k \rightarrow \infty } {O_{{\varphi _k}}}({K_i},{L_j}) = {O_\varphi }(K,L). \end{aligned}$$

Proof

That $K_i\rightarrow K$ and $L_j\rightarrow L$ yields $\rho _{K_i}/\rho _{L_j}$ and $\rho _K/\rho _L$ are strictly positive continuous on $S^{n-1}$; $\rho _{K_i}/\rho _{L_j}\rightarrow \rho _{K}/\rho _L$ uniformly on $S^{n-1}$; $\tilde{V}_{K_i}\rightarrow \tilde{V}_{K}$ weakly, and $V_{K_i}^*\rightarrow V_K^*$ weakly. Combining these facts and applying Lemma 3.7, it yields the desired limits. $\square$

Recall that

$$\begin{aligned} {\bar{\tilde{V}}}_{-1}(K,L)=\int _{S^{n-1}}{\frac{\rho _K}{\rho _L} dV_K^*},\quad \mathrm{for}\; K,L\in \mathcal {S}^n_o. \end{aligned}$$

The next lemma will be used in Sect. 5.

Lemma 3.9

Suppose $K,L\in \mathcal {S}^n_o$, $\varphi \in \Phi$ and $p\in [1,\infty )$. Then for $1<p<q<\infty$,

$$\begin{aligned} {\bar{\tilde{V}}}_{-1}(K,L)\le O_{\varphi }(K,L)\le O_{\varphi ^p}(K,L)\le O_{\varphi ^q}(K,L)\le \left\| \frac{\rho _K}{\rho _L} \right\| _\infty = \lim _{p\rightarrow \infty } O_{\varphi ^p}(K,L). \end{aligned}$$

Proof

For $\lambda \in (0,\infty )$, let $g_p(\lambda )={{{\left( {\int _{{S^{n - 1}}} {\varphi {{\left( {\frac{{{\rho _K}}}{{\lambda {\rho _L}}}} \right) }^p}dV_K^*} } \right) }^{1/p}}}$,    $1\le p<\infty$.

From the definition of $\bar{\tilde{V}}_{-1}(K,\lambda L)$ together with the convexity of $\varphi$ and Jensen’s inequality, Hölder’s inequality, and finally the fact that $\lim _{p \rightarrow \infty } {g_p}(\lambda ) = \varphi \left( \frac{1}{\lambda } \Vert \frac{\rho _K}{\rho _L} \Vert _\infty \right)$, it follows that

$$\begin{aligned} \varphi \left( {{\bar{\tilde{V}}_{ - 1}}(K,\lambda L)} \right) \le {g_1}(\lambda ) \le {g_p}(\lambda ) \le {g_q}(\lambda ) \le \varphi \left( \frac{1}{\lambda } {{{\left\| {\frac{{{\rho _K}}}{{{\rho _L}}}} \right\| }_\infty }} \right) ,\quad \mathrm{for}\; 1<p<q<\infty . \end{aligned}$$

Thus, from the definitions of $O_{\varphi ^p}(K,L)$ and $g_p(\lambda )$, and the monotonicity of $\varphi ^{-1}$, it yields the desired inequalities except the equality on the right.

Finally, we proceed to show $\left\| \frac{\rho _K}{\rho _L} \right\| _\infty = \lim _{p\rightarrow \infty } O_{\varphi ^p}(K,L)$.

Take $\{p_j\}_{j\in \mathbb {N}}\subset \mathbb {N}$ with $\lim _{j\rightarrow \infty }p_j=\infty$. For brevity, let $\lambda _\infty =\Vert \frac{\rho _K}{\rho _L}\Vert _\infty$, ${\lambda _j} = O_{\varphi ^{p_j}}(K,L)$, and ${g_\infty }(\lambda ) = \varphi \left( \frac{1}{\lambda } \Vert \frac{\rho _K}{\rho _L} \Vert _\infty \right)$. Recall that $g_{p_j}$ and $g_\infty$ are positive and continuous on $[\lambda _1,\lambda _\infty ]$, and $g_{p_j}$ increases to $g_\infty$ pointwise on $[\lambda _1,\lambda _\infty ]$. By Dini’s theorem, $g_{p_j}\rightarrow g_\infty$, uniformly on $[\lambda _1,\lambda _\infty ]$. Consequently,

$$\begin{aligned} \lim _{j \rightarrow \infty } {g_j}({\lambda _j}) = \left( { \lim _{j \rightarrow \infty } {g_j}} \right) \left( \lim _{j \rightarrow \infty } {\lambda _j}\right) = {g_\infty }\left( \lim _{j \rightarrow \infty } {\lambda _j}\right) . \end{aligned}$$

Added that $g_\infty$ is strictly decreasing on $[\lambda _1,\lambda _\infty ]$, and $g_{p_j}(\lambda _j)=\varphi (1)$ for each j, we obtain

$$\begin{aligned} \lim _{j\rightarrow \infty }\lambda _j=\lambda _\infty , \end{aligned}$$

which completes the proof. $\square$

4 Orlicz–Legendre Ellipsoids

Let $K\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. For any $T\in \mathrm{SL}(n)$, by Lemma 3.6 it gives

$$\begin{aligned} {{\bar{\tilde{V}}}_\varphi }(K,TB) \ge {\left( {\frac{{V(K)}}{{{\omega _n}}}} \right) ^{\frac{1}{n}}}\quad \mathrm{and}\quad O_\varphi (K,TB)\ge {\left( {\frac{{V(K)}}{{{\omega _n}}}} \right) ^{\frac{1}{n}}}. \end{aligned}$$

In view of the intimate connection between $\bar{\tilde{V}}_\varphi$ and $O_\varphi$, to find the so-called Orlicz–Legendre ellipsoids, we also consider the following three problems, which are closely related to our originally posed Problem ${\tilde{S}}_\varphi$.

Problem $\mathrm{P}_1$ Find an ellipsoid E, among all origin-symmetric ellipsoids, which solves the constrained minimization problem

$$\begin{aligned} \min \limits _E {\bar{\tilde{V}}}_\varphi (K,E)\quad {\mathrm{subject}}\;{\mathrm{to}}\quad V(E) \le {\omega _n}. \end{aligned}$$

Problem $\mathrm{P}_2$ Find an ellipsoid E, among all origin-symmetric ellipsoids, which solves the constrained minimization problem

$$\begin{aligned} \min \limits _E O_\varphi (K,E)\quad {\mathrm{subject}}\;{\mathrm{to}}\quad V(E) \le {\omega _n}. \end{aligned}$$

The homogeneity of the volume functional and Orlicz norm prompts us to consider the following Problem $\mathrm{P}_3$, which is in some sense dual to Problem $\mathrm{P}_2$.

Problem $\mathrm{P}_3$ Find an ellipsoid E, among all origin-symmetric ellipsoids, which solves the constrained maximization problem

$$\begin{aligned} \min \limits _E V(E) \quad {\mathrm{subject}}\;{\mathrm{to}}\quad O_\varphi (K,E)\le 1. \end{aligned}$$

For convenient comparison, we restate Problem ${\tilde{S}}_\varphi$ as the following.

Problem ${\tilde{S}}_\varphi$ Find an ellipsoid E, among all origin-symmetric ellipsoids, which solves the constrained maximization problem

$$\begin{aligned} \min \limits _E V(E) \quad {\mathrm{subject}}\;{\mathrm{to}}\quad {\bar{\tilde{V}}}_\varphi (K,E)\le 1. \end{aligned}$$

This section is organized as follows. Lemma 4.2 and Theorem 4.3 demonstrate the existence and uniqueness of the solution to $\mathrm{P}_1$, respectively. The connection between $\mathrm{P}_1$ and $\mathrm{P}_2$ is established by Lemma 4.4. Theorem 4.5 demonstrates the existence and uniqueness of a solution to $\mathrm{P}_2$. Theorem 4.6 shows that the solutions to $\mathrm{P}_2$ and $\mathrm{P}_3$ only differ by a scalar factor. Lemma 4.7 reveals that $\mathrm{P}_3$ and $\tilde{S}_\varphi$ are essentially identical. Therefore, the notion of Orlicz–Legendre ellipsoids is ready to come out.

From Definition 3.1 together with the fact that $\varphi ^{-1}$ is strictly increasing in $(0,\infty )$, the objective functional in $\mathrm{P}_1$ can be replaced by $\tilde{V}_\varphi (K,E)$.

Lemma 4.1

Suppose $T\in \mathrm{SL}(n)$. Then

$$\begin{aligned} {\mathop {\mathop {\lim }\limits _{\scriptstyle T \in {\mathrm{SL}}(n)}}\limits _{\scriptstyle \left\| T \right\| \rightarrow \infty }} {{\tilde{V}}_\varphi }(K,TB) = \infty \quad \mathrm{and}\quad {\mathop {\mathop {\lim }\limits _{\scriptstyle T \in {\mathrm{SL}}(n)}}\limits _{\scriptstyle \left\| T \right\| \rightarrow \infty }} O_\varphi (K,TB) = \infty . \end{aligned}$$

Proof

Let $T\in \mathrm{SL}(n)$ and $r_K=\min \limits _{S^{n-1}}\rho _K$. Then $r_KB\subseteq K$. For $|\alpha |>n$, it is known that

$$\begin{aligned} {\mathop {\mathop {\lim }\limits _{\scriptstyle T \in {\mathrm{SL}}(n)}}\limits _{\scriptstyle \left\| T \right\| \rightarrow \infty }} \int _{{S^{n - 1}}} {|Tu{|^\alpha }dS(u)} = \infty . \end{aligned}$$

(4.1)

From the definition of $\tilde{V}_\varphi (K,TB)$ together with the convexity of $\varphi$ and Jensen’s inequality, the facts

$$\begin{aligned} \int _{S^{n-1}}\frac{\rho _{K}^{n+1}}{\rho _{TB}} dS=\int _{S^{n-1}}\rho _{T^{-1}K}^{n+1}dS\quad \mathrm{and}\quad T^{-1}K\supseteq T^{-1}(r_K B) \end{aligned}$$

together with the monotonicity of $\varphi$, and finally (2.1), it follows that

$$\begin{aligned} \frac{{{{\tilde{V}}_\varphi }(K,TB)}}{{V(K)}}&\ge \varphi \left( {\frac{1}{{nV(K)}}\int _{{S^{n - 1}}} {\frac{{\rho _K^{n + 1}}}{{{\rho _{TB}}}}dS} } \right) \\&\ge \varphi \left( {\frac{1}{{nV(K)}}\int _{{S^{n - 1}}} {\rho _{{T^{ - 1}}({r_K}B)}^{n + 1}dS} } \right) \\&= \varphi \left( {\frac{{r_K^{n + 1}}}{{nV(K)}}\int _{{S^{n - 1}}} {|Tu{|^{n + 1}}dS(u)} } \right) . \end{aligned}$$

Thus, by the monotonicity of $\varphi$ and (4.1), it concludes the first limit.

To prove the second limit, we argue by contradiction and assume it to be false. Then, there is a constant $c\in (0,\infty )$ such that $\sup \left\{ O_\varphi (K,T_jB): j\in \mathbb {N} \right\} <c$, for any sequence $\{T_j\}_j\subset \mathbb {N}$ with $\lim _{j\rightarrow \infty }\Vert T_j\Vert =\infty$. By using arguments similar to those above, we can show that for each $j\in \mathbb {N}$,

$$\begin{aligned} \frac{{{{\tilde{V}}_\varphi }(K,{O_\varphi }(K,{T_j}B){T_j}B)}}{{V(K)}} \ge \varphi \left( {\frac{{r_K^{n + 1}}}{{ncV(K)}}\int _{{S^{n - 1}}} {|{T_j}u{|^{n + 1}}dS(u)} } \right) . \end{aligned}$$

Combining this with (4.1), we have

$$\begin{aligned} \mathop {\lim }\limits _{j \rightarrow \infty } \frac{{{{\tilde{V}}_\varphi }(K,{O_\varphi }(K,{T_j}B){T_j}B)}}{{V(K)}} = \infty . \end{aligned}$$

However, this obviously contradicts Lemma 3.5. $\square$

Now, using Lemma 4.1, we can prove the existence of a solution to problem $\mathrm{P}_1$.

Lemma 4.2

Suppose $K\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Then there exists a solution to $\mathrm{P}_1$.

Proof

Note that any $E\in \mathcal {E}^n$ with $V(E)<\omega _n$ cannot be a solution to $\mathrm{P}_1$. Hence, Problem $\mathrm{P}_1$ can be equivalently restated as

$$\begin{aligned}\inf \left\{ {{{\tilde{V}}_\varphi }(K,TB):T \in {\mathrm{SL}}(n)} \right\} .\end{aligned}$$

Observe that the infimum exists, since

$$\begin{aligned} V(K)\varphi \left( {{{\left( {\frac{{V(K)}}{{{\omega _n}}}} \right) }^{\frac{1}{n}}}} \right) \le \inf \left\{ {{{\tilde{V}}_\varphi }(K,TB):T \in {\mathrm{SL}}(n)} \right\} \le {\tilde{V}_\varphi }(K,B) < \infty , \end{aligned}$$

where the left inequality follows from Lemma 3.6 and Definition 3.1.

Let

$$\begin{aligned} {\mathcal T} = \left\{ {T \in {\mathrm{SL}}(n):{{\tilde{V}}_\varphi }(K,TB) \le {{\tilde{V}}_\varphi }(K,B)} \right\} . \end{aligned}$$

From the definition of $d_n$ (see Sect. 2.4) and Lemma 3.8, ${\tilde{V}}_\varphi (K,TB)$ is continuous in $T\in (\mathrm{SL}(n),d_n)$. Thus, the set ${\mathcal T}$ is closed in $(\mathrm{SL}(n),d_n)$. Meanwhile, the definition of ${\mathcal T}$ and Lemma 4.1 guarantee that ${\mathcal T}$ is bounded in $(\mathrm{SL}(n),d_n)$. Hence, $\mathcal T$ is compact.

The continuity of ${\tilde{V}}_\varphi (K,TB)$ on $({\mathcal T},d_n)$ implies that there exists a $T_0\in {\mathcal T}$ such that

$$\begin{aligned} {\tilde{V}}_\varphi (K,T_0B)=\min \{{\tilde{V}}_\varphi (K,TB): T\in {\mathcal T}\}= \inf \{{\tilde{V}}_\varphi (K,TB): T\in \mathrm{SL}(n)\}, \end{aligned}$$

which completes the proof. $\square$

Theorem 4.3

Suppose $K\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Then, modulo orthogonal transformations, there exists a unique $\mathrm{SL}(n)$ transformation solving the extremal problem

$$\begin{aligned} \min \left\{ {{{\tilde{V}}_\varphi }(K,TB):T \in {\mathrm{SL}}(n)} \right\} . \end{aligned}$$

Equivalently, there exists a unique solution to Problem $\mathrm{P}_1$.

Proof

Lemma 4.2 shows the existence. We prove the uniqueness by contradiction.

Assume that $T_1, T_2\in \mathrm{SL}(n)$ both solve the considered minimization problem. Let $E_1=T_1B$, $E_2=T_2B$. It is known that each $T\in \mathrm{SL}(n)$ can be represented in the form $T=PQ$, where P is symmetric, positive definite and Q is orthogonal. So, without loss of generality, we may assume that $T_1, T_2$ are symmetric and positive definite.

By the Minkowski inequality for symmetric and positive definite matrices, we have

$$\begin{aligned} \det \left( \frac{T_1^{-1}+T_2^{-1}}{2} \right) ^{\frac{1}{n}} > \frac{1}{2}\det \left( T_1^{-1}\right) ^{\frac{1}{n}} + \frac{1}{2}\det \left( T_2^{-1}\right) ^{\frac{1}{n}}=1. \end{aligned}$$

Let

$$\begin{aligned} T_3^{-1}= \det \left( \frac{T_1^{-1}+T_2^{-1}}{2} \right) ^{-\frac{1}{n}} \frac{T_1^{-1}+T_2^{-1}}{2}. \end{aligned}$$

Then $T_3\in \mathrm{SL}(n)$ is symmetric. Moreover, for all $u\in S^{n-1}$, we have

$$\begin{aligned} h_{T_3^{-1}B}(u) < h_{\frac{T_1^{-1}+T_2^{-1}}{2} B}(u)\le \frac{1}{2} h_{T_1^{-1}B}(u) + \frac{1}{2} h_{T_1^{-1}B}(u). \end{aligned}$$

Thus, from the fact that $\varphi$ is strictly increasing and convex in $[0,\infty )$, it implies that

$$\begin{aligned} \varphi \left( \rho _K h_{T_3^{-1}B} \right) < \frac{1}{2} \varphi \left( \rho _K h_{T_1^{-1}B} \right) + \frac{1}{2} \varphi \left( \rho _K h_{T_2^{-1}B} \right) . \end{aligned}$$

Hence, letting $E_3=T_3B$ and using the fact

$$\begin{aligned} {\tilde{V}}_\varphi (K,E_i)=\int _{S^{n-1}}{ \varphi \left( \rho _K h_{T_i^{-1}B}\right) d{\tilde{V}}_K },\quad i=1,2,3, \end{aligned}$$

it gives

$$\begin{aligned} {\tilde{V}}_\varphi (K,E_3)< {\tilde{V}}_\varphi (K,E_1)={\tilde{V}}_\varphi (K,E_2). \end{aligned}$$

However, from the fact that $T_3\in \mathrm{SL}(n)$ and the assumption on $E_1$ and $E_2$, we also have

$$\begin{aligned} {\tilde{V}}_\varphi (K,E_3) \ge {\tilde{V}}_\varphi (K,E_1)={\tilde{V}}_\varphi (K,E_2), \end{aligned}$$

which contradicts the above. This completes the proof. $\square$

Lemma 4.4

Suppose $E_0\in \mathcal {E}^n$ and $V(E_0)=\omega _n$. Then, for any $T\in \mathrm{SL}(n)$,

$$\begin{aligned} {{\tilde{V}}_\varphi }\left( {K,{O_\varphi }(K,{E_0}){E_0}} \right) \le {{\tilde{V}}_\varphi }\left( {K,{O_\varphi }(K,{E_0})T{E_0}} \right) \end{aligned}$$

if and only if

$$\begin{aligned} {O_\varphi }(K,{E_0}) \le {O_\varphi }(K,T{E_0}). \end{aligned}$$

Proof

From Definition 3.1 together with the strict monotonicity of $\varphi ^{-1}$, Lemma 3.5, and Lemma 2.1 together with Definition 3.3, it follows that

$$\begin{aligned}&{{\tilde{V}}_\varphi }\left( {K,{O_\varphi }(K,{E_0}){E_0}} \right) \le {{\tilde{V}}_\varphi }\left( {K,{O_\varphi }(K,{E_0})T{E_0}} \right) \\&\quad \quad \Longleftrightarrow \quad {{\bar{\tilde{V}}}_\varphi }\left( {K,{O_\varphi }(K,{E_0}){E_0}} \right) \le {{\bar{\tilde{V}}}_\varphi }\left( {K,{O_\varphi }(K,{E_0})T{E_0}} \right) \\&\quad \quad \Longleftrightarrow \quad 1 \le {{\bar{\tilde{V}}}_\varphi }\left( {K,{O_\varphi }(K,{E_0})T{E_0}} \right) \\&\quad \quad \Longleftrightarrow \quad {{\bar{\tilde{V}}}_\varphi }\left( {K,{O_\varphi }(K,{TE_0}){TE_0}} \right) \le {{\bar{\tilde{V}}}_\varphi }\left( {K,{O_\varphi }(K,{E_0})T{E_0}} \right) \\&\quad \quad \Longleftrightarrow \quad { {\tilde{V}}_\varphi }\left( {K,{O_\varphi }(K,{TE_0}){TE_0}} \right) \le { {\tilde{V}}_\varphi }\left( {K,{O_\varphi }(K,{E_0})T{E_0}} \right) \\&\quad \quad \Longleftrightarrow \quad {O_\varphi }(K,{E_0}) \le {O_\varphi }(K,T{E_0}), \end{aligned}$$

as desired. $\square$

From Theorem 4.3 and Lemma 4.4, we can prove the following

Theorem 4.5

Suppose $K\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Then there exists a unique solution to Problem $\mathrm{P}_2$.

Proof

First, we prove the existence of a solution to problem $\mathrm{P}_2$. Note that the constraint condition in $\mathrm{P}_2$ can be turned into $V(E)=\omega _n$.

Let ${\lambda _0} = \inf \left\{ {{O_\varphi }(K,TB):T \in {\mathrm{SL}}(n)} \right\}$. From Lemma 3.6, we have

$$\begin{aligned} 0 < {\left( {\frac{{V(K)}}{{{\omega _n}}}} \right) ^{\frac{1}{n}}} \le {\lambda _0} \le O_\varphi (K,B) < \infty . \end{aligned}$$

Similar to the proof of Lemma 4.2, we can show the set

$$\begin{aligned} \{T\in \mathrm{SL}(n): O_\varphi (K,TB)\le O_\varphi (K,B)\} \end{aligned}$$

is compact. Combining this with the continuity of $O_\varphi (K,TB)$, the existence of solution to $\mathrm{P}_2$ is demonstrated.

Now, we proceed to prove the uniqueness.

Assume ellipsoid $E_0$ is a solution to $\mathrm{P}_2$. Then, $O_\varphi (K,E_0)\le O_\varphi (K,T E_0)$, for $T\in \mathrm{SL}(n)$. Thus, by Lemma 4.4, it follows that

$$\begin{aligned} {{\tilde{V}}_\varphi }\left( {K,O_\varphi (K,E_0){E_0}} \right) \le {{\tilde{V}}_\varphi }\left( {K,O_\varphi (K,E_0)T{E_0}} \right) ,\quad \mathrm{for}\; T\in \mathrm{SL}(n). \end{aligned}$$

Thus, $E_0$ is a solution to Problem $\mathrm{P}_1$ for star body $\lambda _0^{-1}K$. Hence, by Theorem 4.3, the solution to $\mathrm{P}_2$ is unique. $\square$

Theorem 4.6

Suppose $K\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Then

• (1) If $E_0$ is the unique solution to Problem $\mathrm{P}_2$, then $O_\varphi (K,E_0)E_0$ is a solution to Problem $\mathrm{P}_3$.

• (2) If $E_1$ is a solution to Problem $\mathrm{P}_3$, then $\left( \frac{\omega _n}{V(E_1)}\right) ^{\frac{1}{n}}E_1$ is a solution to Problem $\mathrm{P}_2$.

Consequently, there exists a unique solution to Problem $\mathrm{P}_3$.

Proof

(1) Let $E\in \mathcal {E}^n$ with $O_\varphi (K,E)\le 1$. Since $\left( \frac{\omega _n}{V(E)}\right) ^{\frac{1}{n}} E$ satisfies the constraint condition of $\mathrm{P}_2$, by Lemma 3.4 (2), the fact $V(E_0)=\omega _n$, and the assumption $O_\varphi (K,E)\le 1$, we have

$$\begin{aligned} V\left( {{O_\varphi }(K,{E_0}){E_0}} \right)&= {O_\varphi }{(K,{E_0})^n}V({E_0}) \\&\le {O_\varphi }{\left( {K,{{\left( {\frac{{{\omega _n}}}{{V(E)}}} \right) }^{\frac{1}{n}}}E} \right) ^n}V({E_0}) \\&= \frac{{V(E)}}{{{\omega _n}}}{O_\varphi }{\left( {K,E} \right) ^n}V({E_0}) \\&= V(E){O_\varphi }{\left( {K,E} \right) ^n} \\&\le V(E), \end{aligned}$$

which shows that $O_\varphi (K,E_0)E_0$ solves Problem $\mathrm{P}_3$.

(2) Note that a solution $E_1$ to $\mathrm{P}_3$ must satisfy $O_\varphi (K,E_1)=1$.

Let $E'\in {\mathcal E}^n$ with $V(E')\le \omega _n$. By Lemma 3.4 (2), ${O_\varphi }\left( {K,{O_\varphi }(K,E')E'}\right) = 1$. Thus $O_\varphi (K,E')E'$ satisfies the constraint condition of Problem $\mathrm{P}_3$. Since $E_1$ is a solution to Problem $\mathrm{P}_3$, it follows that

$$\begin{aligned} V\left( O_\varphi (K,E')E' \right) \ge V({E_1}). \end{aligned}$$

So, by the assumption $V(E')\le \omega _n$, the fact $O_\varphi (K,E_1)= 1$ and Lemma 3.4 (2), we have

$$\begin{aligned} {O_\varphi }(K,E') \ge {\left( {\frac{{V({E_1})}}{{V(E')}}} \right) ^{\frac{1}{n}}} \ge {\left( {\frac{{V({E_1})}}{{{\omega _n}}}} \right) ^{\frac{1}{n}}} = {O_\varphi }\left( {K,{{\left( {\frac{{{\omega _n}}}{{V({E_1})}}} \right) }^{\frac{1}{n}}}{E_1}} \right) , \end{aligned}$$

which shows that $\left( \frac{\omega _n}{V(E_1)}\right) ^{\frac{1}{n}}E_1$ solves Problem $\mathrm{P}_2$. $\square$

Lemma 4.7

Suppose $K\in {\mathcal S}^n_o$ and $\varphi \in \Phi$. Then Problems $\mathrm{P}_3$ and ${\tilde{S}}_\varphi$ have the same solution.

Proof

From the facts

$$\begin{aligned} \min _{\left\{ {E \in {{\mathcal E}^n}: O_\varphi (K,E) \le 1} \right\} } V(E) = \min _{\left\{ {E \in {{\mathcal E}^n}:O_\varphi (K,E) = 1} \right\} } V(E) \end{aligned}$$

and

$$\begin{aligned} \min _{\left\{ {E \in {{\mathcal E}^n}:{\bar{\tilde{V}}}_\varphi (K,E) \le 1 } \right\} } V(E) = \min _{\left\{ {E \in {{\mathcal E}^n}:{\bar{\tilde{V}}}_\varphi (K,E) = 1 } \right\} } V(E) \end{aligned}$$

together with the implication that for $E \in {{\mathcal E}^n}$,

$$\begin{aligned} O_\varphi (K,E) = 1\quad \Longleftrightarrow \quad {\bar{\tilde{V}}}_\varphi (K,E) = 1, \end{aligned}$$

it follows that an ellipsoid $E \in {{\mathcal E}^n}$ solves Problem $\mathrm{P}_3$, if and only if it solves Problem $\tilde{S}_\varphi$. By Theorem 4.6, it concludes the proof. $\square$

For different dilations $\lambda _1 K$ and $\lambda _2 K$, $\lambda _1, \lambda _2>0$, Problems $\mathrm{P}_1$ do not generally have the identical solution. By contrast, the homogeneity of $O_\varphi (\lambda K,L)$ in $\lambda \in (0,\infty )$ guarantees that all Problems $\mathrm{P}_2$ for $\lambda K$ in $\lambda \in (0,\infty )$ have the identical unique solution. Problems $\mathrm{P}_3$ and ${\tilde{S}}_\varphi$ are identical, and Problem $\mathrm{P}_3$ is the dual problem of $\mathrm{P}_2$. Thus, Problem ${\tilde{S}}_\varphi$ is not the dual problem of $\mathrm{P}_1$ in general.

In view of Theorem 4.5, Theorem 4.6 and Lemma 4.7, we are in position to introduce a family of ellipsoids in the framework of dual Orlicz Brunn–Minkowski theory, which are extensions of Legendre ellipsoids.

Definition 4.8

Suppose $K\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Among all origin-symmetric ellipsoids E, the unique ellipsoid that solves the constrained minimization problem

$$\begin{aligned} \mathop {\min }\limits _E V(E)\quad \mathrm{subject\;to}\quad \bar{\tilde{V}}_\varphi (K,E)\le 1 \end{aligned}$$

is called the Orlicz–Legendre ellipsoid of K with respect to $\varphi$, and is denoted by $\mathrm{L}_\varphi K$.

Among all origin-symmetric ellipsoids E, the unique ellipsoid that solves the constrained minimization problem

$$\begin{aligned} \mathop {\min }\limits _E O_\varphi (K,E)\quad {\mathrm{subject}}\;{\mathrm{to}}\quad V(E) = {\omega _n} \end{aligned}$$

is called the normalized Orlicz–Legendre ellipsoid of K with respect to $\varphi$, and is denoted by $\overline{\mathrm{L}}_\varphi K$.

For the polar of $\mathrm{L}_\varphi K$ or ${\overline{\mathrm{L}}}_\varphi K$, we write $\mathrm{L}_\varphi ^* K$ or ${\overline{\mathrm{L}}}_\varphi ^* K$, rather than $(\mathrm{L}_\varphi K)^*$ or $({\overline{\mathrm{L}}}_\varphi K)^*$.

If $\varphi (t)=t^p$, $1\le p <\infty$, we write $\mathrm{L}_{\varphi } K$ and ${\overline{\mathrm{L}}}_\varphi K$ for $\mathrm{L}_p K$ and ${\overline{\mathrm{L}}}_pK$, respectively. Especially, $\mathrm{L}_2 K$ is precisely the Legendre ellipsoid $\Gamma _2 K$.

We observe that for the case $\varphi (t)=t^p$, Problems $\mathrm{P}_1$ and $\mathrm{P}_2$ are identical, and were previously solved by Bastero and Romance [5]. Yu [54] introduced the ellipsoids $\mathrm{L}_p K$ for convex bodies containing the origin in their interiors.

From Theorem 4.6, it is obvious that

$$\begin{aligned} \mathrm{L}_\varphi K=O_\varphi (K,{{\overline{\mathrm{L}}}_\varphi } K) {{\overline{\mathrm{L}}}_\varphi } K\quad \mathrm{and}\quad {{\overline{\mathrm{L}}}_\varphi } K= \left( \frac{\omega _n}{V(\mathrm{L}_\varphi K)}\right) ^{\frac{1}{n}}\mathrm{L}_\varphi K. \end{aligned}$$

(4.2)

Definition 4.8 combined with inequality (3.1) shows that for any $E\in \mathcal {E}^n$,

$$\begin{aligned} \mathrm{L}_\varphi E=E. \end{aligned}$$

From Definition 4.8 and Lemma 3.4, we easily know that the operator $\mathrm{L}_\varphi$ intertwines with elements of $\mathrm{GL}(n)$.

Lemma 4.9

Suppose $K\in \mathcal {S}^n$ and $\varphi \in \Phi$. Then for any $T\in \mathrm{GL}(n)$,

$$\begin{aligned} \mathrm{L}_\varphi (TK)=T(\mathrm{L}_\varphi K). \end{aligned}$$

To connect Orlicz–Legendre ellipsoids with the Löwner ellipsoid, it is necessary to introduce the following.

Definition 4.10

Suppose $K\in {\mathcal S}^n_o$. The unique origin-symmetric ellipsoid of minimal volume containing K is denoted by $\mathrm{L}_\infty K$.

Among all origin-symmetric ellipsoids E, the unique ellipsoid that uniquely solves the constrained minimization problem

$$\begin{aligned} \mathop {\min }\limits _E {\left\| {\frac{{{\rho _K}}}{{{\rho _E}}}} \right\| _\infty }\quad {\mathrm{subject}}\;{\mathrm{to}}\quad V(E)\le \omega _n \end{aligned}$$

is denoted by $\mathrm{\overline{L}}_\infty K$.

The notion of $\mathrm{L}_\infty K$ is well defined, since $(\mathrm{L}_\infty K)^*$ is the $L_\infty$ John ellipsoid $\mathrm{E}_\infty \left( \mathrm{conv } K\right) ^*$, i.e., the unique origin-symmetric ellipsoid of maximal volume contained in $\left( \mathrm{conv } K\right) ^*$. The extremal problems involved in the definition are dual to each other. Thus, we have

$$\begin{aligned} \mathrm{L}_\infty K=\left( \mathrm{E}_\infty \left( \mathrm{conv } K\right) ^*\right) ^*\quad \mathrm{and}\quad \mathrm{\overline{L}}_\infty K =\left( \frac{\omega _n}{V(\mathrm{L}_\infty K)}\right) ^{\frac{1}{n}}\mathrm{L}_\infty K. \end{aligned}$$

Here, $\mathrm{conv} K$ denotes the convex hull of K.

For a convex body $K\in \mathcal {K}^n_o$, if the John point of $K^*$ is at the origin, then $(\mathrm{L}_\infty K)^*$ is precisely the John ellipsoid $\mathrm{J}(K^*)$ of $K^*$. If K is an origin-symmetric star body in $\mathbb R^n$, then $\mathrm{L}_\infty K$ is precisely the Löwner ellipsoid of K.

5 The Continuity of Orlicz–Legendre Ellipsoids

In this section, we aim to show the continuity of Orlicz–Legendre ellipsoids $\mathrm{L}_\varphi K$ with respect to $\varphi$ and K.

Throughout this section, we suppose $\varphi \in \Phi$, $K,K_i\in {\mathcal S}^n_o$, $\varphi ,\varphi _j\in \Phi$, $i,j\in \mathbb N$, and $K_i\rightarrow K$ and $\varphi _j\rightarrow \varphi$. It is easily seen that there exist positive $r_m$ and $r_M$ such that

$$\begin{aligned} r_mB\subseteq K\subseteq r_MB\quad \mathrm{and}\quad r_mB\subseteq K_i\subseteq r_MB\quad \mathrm{for \; each}\; i\in \mathbb N. \end{aligned}$$

Lemma 5.1

$\sup \limits _{i,j \in {\mathbb N}} \left\{ {d\left( {{{\overline{\mathrm{L}} }_\varphi }K} \right) ,d\left( {{{\overline{\mathrm{L}} }_\varphi }{K_i}} \right) ,d\left( {{{\overline{\mathrm{L}} }_{{\varphi _j}}}K} \right) ,d\left( {{{\overline{\mathrm{L}} }_{{\varphi _j}}}{K_i}} \right) } \right\} < \infty$.

Proof

Let $E\in {\mathcal E}^n$. First, we prove the implication

$$\begin{aligned} O_\varphi (K,E)\le 1 \quad \Longrightarrow \quad d({E^*}) \le \frac{{n{\omega _n}}}{{2{r_m}{\omega _{n - 1}}}}{\varphi ^{ - 1}}\left( {{{\left( {\frac{{{r_M}}}{{{r_m}}}} \right) }^n}\varphi (1)} \right) . \end{aligned}$$

(5.1)

Assume $O_\varphi (K,E)\le 1$. From the definition of $O_\varphi (K,E)$ together with Lemma 2.1 and Lemma 3.5, the definition of ${\tilde{V}}_\varphi (K,E)$ together with the fact $r_mB\subseteq K\subseteq r_MB$ and the monotonicity of $\varphi$, the convexity of $\varphi$ together with Jensen’s inequality, the monotonicity of $\varphi$ again together with (2.3) and the fact $h_{E^*}(u)\ge d(E^*)|v_{E^*}\cdot u|$ for $u\in S^{n-1}$, and finally Cauchy’s projection formula, it follows that

$$\begin{aligned} \varphi (1)&\ge \frac{{{{\tilde{V}}_\varphi }(K,E)}}{{V(K)}} \\&\ge {\left( {\frac{{{r_m}}}{{{r_M}}}} \right) ^n}\frac{1}{{n{\omega _n}}}\int _{{S^{n - 1}}} {\varphi \left( {\frac{{{r_m}}}{{{\rho _E}}}} \right) dS} \\&\ge {\left( {\frac{{{r_m}}}{{{r_M}}}} \right) ^n}\varphi \left( {\frac{1}{{n{\omega _n}}}\int _{{S^{n - 1}}} {\frac{{{r_m}}}{{{\rho _E}}}dS} } \right) \\&\ge {\left( {\frac{{{r_m}}}{{{r_M}}}} \right) ^n}\varphi \left( {\frac{{{r_m}}}{{n{\omega _n}}}\int _{{S^{n - 1}}} {d({E^*})|{v_{{E^*}}} \cdot u|dS(u)} } \right) \\&= {\left( {\frac{{{r_m}}}{{{r_M}}}} \right) ^n}\varphi \left( {\frac{{2{r_m}{\omega _{n - 1}}}}{{n{\omega _n}}}d({E^*})}\right) , \end{aligned}$$

which, together with the monotonicity of $\varphi ^{-1}$, immediately yields (5.1).

Since that $\varphi _j\rightarrow \varphi$ implies $\varphi _j(1)\rightarrow \varphi (1)$ and $\varphi _j^{-1}\rightarrow \varphi ^{-1}$, it follows that

$$\begin{aligned} \varphi _j^{ - 1}\left( {{{\left( {\frac{{{r_M}}}{{{r_m}}}} \right) }^n}{\varphi _j}(1)} \right) \rightarrow {\varphi ^{ - 1}}\left( {{{\left( {\frac{{{r_M}}}{{{r_m}}}} \right) }^n}\varphi (1)} \right) , \end{aligned}$$

and therefore

$$\begin{aligned} \sup \limits _{j \in \mathbb {N}} \left\{ {{\varphi ^{ - 1}}\left( {{{\left( {\frac{{{r_M}}}{{{r_m}}}} \right) }^n}\varphi (1)} \right) ,\varphi _j^{ - 1}\left( {{{\left( {\frac{{{r_M}}}{{{r_m}}}} \right) }^n}{\varphi _j}(1)} \right) } \right\} < \infty . \end{aligned}$$

This, as well as (5.1), yields

$$\begin{aligned} \sup \limits _{i,j \in \mathbb {N}} \left\{ d\left( {\mathrm{L}_\varphi ^*K} \right) , d\left( {\mathrm{L}_\varphi ^*{K_i}} \right) ,d\left( {\mathrm{L}_{{\varphi _j}}^*K} \right) ,{d\left( {\mathrm{L}_{{\varphi _j}}^*{K_i}} \right) } \right\} < \infty . \end{aligned}$$

(5.2)

From (4.2), the inequality $V(\mathrm{L}_\varphi K)\le V(\mathrm{L}_\infty K)$ (which will be given by Theorem 7.2) together with Definition 4.10, we have

$$\begin{aligned} d\left( {\overline{\mathrm{L}}}^* _\varphi K \right) \le {r_M}d\left( {\mathrm{L}_\varphi ^*K} \right) . \end{aligned}$$

(5.3)

Observe that (5.3) also holds when $\varphi$ is replaced by $\varphi _j$ or K is replaced by $K_i$. Thus, from (5.2), it follows that

$$\begin{aligned} \sup \limits _{i,j \in {\mathbb N}} \left\{ {d\left( {\overline{\mathrm{L}} _\varphi ^*K} \right) , d\left( {\overline{\mathrm{L}} _\varphi ^*{K_i}} \right) , d\left( {\overline{\mathrm{L}} _{{\varphi _j}}^*K} \right) , d\left( {\overline{\mathrm{L}} _{{\varphi _j}}^*{K_i}} \right) } \right\} < \infty , \end{aligned}$$

which, together with Lemma 2.2, concludes the desired lemma. $\square$

Now, from Lemma 5.1, there exists a constant $R\in (0,\infty )$ such that all the ellipsoids ${\overline{\mathrm{L}}} _\varphi K$, ${\overline{\mathrm{L}}} _{\varphi _j} K$, ${\overline{\mathrm{L}}}_\varphi K_i$ and ${\overline{\mathrm{L}}} _{\varphi _j} K_i$ are in the set

$$\begin{aligned} \mathcal {E}_R =\left\{ E\in \mathcal {E}^n: V(E)=\omega _n\; \mathrm{and}\; E\subseteq RB \right\} . \end{aligned}$$

From the compactness of the sets $\mathcal {E}_R$ and $\{K\in \mathcal {S}^n_o: r_m B\subseteq K\subseteq r_M B\}$, together with Lemma 3.8, we immediately obtain:

Lemma 5.2

The limit $\lim _{i,j \rightarrow \infty } {O_{{\varphi _j}}}({K_i},E) = {O_\varphi }(K,E)$ is uniform in $E\in \mathcal {E}_R$.

Lemma 5.3

$\lim _{i,j \rightarrow \infty } {O_{{\varphi _j}}}({K_i},{\overline{\mathrm{L}} _{{\varphi _j}}}{K_i}) = {O_\varphi }(K,{\overline{\mathrm{L}} _\varphi }K)$.

Proof

From Definition 4.8 and Lemma 5.2, we have

$$\begin{aligned} \mathop {\lim }\limits _{i,j \rightarrow \infty } {O_{{\varphi _j}}}({K_i},{\overline{\mathrm{L}} _{{\varphi _j}}}{K_i})&= \mathop {\lim }\limits _{i,j \rightarrow \infty } \mathop {\min }\limits _{E \in {{\mathcal E}_R}} {O_{{\varphi _j}}}({K_i},E) \\&= \mathop {\min }\limits _{E \in {{\mathcal E}_R}} \mathop {\lim }\limits _{i,j \rightarrow \infty } {O_{{\varphi _j}}}({K_i},E) \\&= \mathop {\min }\limits _{E \in {{\mathcal E}_R}} {O_\varphi }(K,E) \\&= {O_\varphi }(K,{\overline{\mathrm{L}} _\varphi }K), \end{aligned}$$

as desired. $\square$

Lemma 5.4

$\lim _{i,j \rightarrow \infty } {\overline{\mathrm{L}} _{{\varphi _j}}}{K_i} = {\overline{\mathrm{L}} _\varphi }K$.

Proof

From the compactness of $\mathcal {E}_R$, to prove the lemma, it suffices to prove that any convergent subsequence $\{{\overline{\mathrm{L}}}_{\varphi _{j_q}} K_{i_p} \}_{p,q\in \mathbb N}$ of $\{\overline{\mathrm{L}}_{\varphi _j}K_i\}$ must converge to $\overline{\mathrm{L}}_\varphi K$.

From Lemma 3.8 and Lemma 5.3, it follows that

$$\begin{aligned} {O_\varphi }(K,\mathop {\lim }\limits _{p,q \rightarrow \infty } {\overline{\mathrm{L}} _{{\varphi _{{j_q}}}}}{K_{{i_p}}})&= \mathop {\lim }\limits _{p,q \rightarrow \infty } {O_\varphi }(K,{\overline{\mathrm{L}} _{{\varphi _{{j_q}}}}}{K_{{i_p}}}) \\&= \mathop {\lim }\limits _{p,q \rightarrow \infty } {O_{{\varphi _{{j_q}}}}}(K_{i_p},{\overline{\mathrm{L}} _{{\varphi _{{j_q}}}}}{K_{{i_p}}}) \\&= {O_\varphi }(K,{\overline{\mathrm{L}} _\varphi }K), \end{aligned}$$

which, together with the uniqueness of $\overline{\mathrm{L}}_\varphi K$, implies $\lim _{p,q \rightarrow \infty } {\overline{\mathrm{L}} _{{\varphi _{{j_q}}}}}{K_{{i_p}}} = {\overline{\mathrm{L}} _\varphi }K$. $\square$

Theorem 5.5

Suppose $K,K_i\in {\mathcal S}^n_o$ and $\varphi ,\varphi _j\in \Phi$, $i,j\in \mathbb N$. If $K_i\rightarrow K$ and $\varphi _j\rightarrow \varphi$, then

$$\begin{aligned} \mathop {\lim }\limits _{i,j \rightarrow \infty } {\mathrm{L}_{{\varphi _j}}}{K_i} = {\mathrm{L}_\varphi }K. \end{aligned}$$

Proof

From Lemma 5.3, Lemma 5.4, together with the identity

$$\begin{aligned} {\mathrm{L}_\varphi }K = {O_\varphi }(K,{\overline{\mathrm{L}} _\varphi }K){\overline{\mathrm{L}} _\varphi }K, \end{aligned}$$

the desired limit is immediately derived. $\square$

Note that $\mathrm{L}_{\varphi ^p}K$ and therefore $\mathrm{L}_pK$ are continuous in $(K,p)\in {\mathcal S}^n_o\times [1,\infty )$. We observe that although Yu et al. [54] first introduced the notion of $L_p$ Legendre ellipsoids, they did not consider the above continuity at all. We proceed to show the following.

Theorem 5.6

Suppose $K\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Then $\lim _{p \rightarrow \infty } \mathrm{L}_{{\varphi ^p}}K = \mathrm{L_\infty }K$.

Proof

By Lemma 5.1, there exists a constant $C\in (0,\infty )$ such that $\overline{\mathrm{L}}_\infty K$ and all the Orlicz–Legendre ellipsoids $\overline{\mathrm{L}}_{\varphi ^p}K$ are in the set

$$\begin{aligned} \mathcal {F}=\{E\in \mathcal {E}^n: V(E)=\omega _n\; \mathrm{and}\; E\subseteq CB\}. \end{aligned}$$

For $E \in \mathcal {F}$, let ${f_p}(E) =O_{{\varphi ^p}}(K,E)$ for $p\in [1,\infty )$, and let ${f_\infty }(E) = {\left\| {{{{\rho _K}}}/{{{\rho _E}}}} \right\| _\infty }$. By Lemma 3.9, the sequence $\{f_j\}$ of continuous functionals is increasing pointwise to $f_\infty$ on the compact set $\mathcal {F}$, and therefore by Dini’s theorem,

$$\begin{aligned} {f_j} \rightarrow {f_\infty },\quad \mathrm{uniformly\; on\;} \mathcal {F}. \end{aligned}$$

(5.4)

The theorem will be obtained after the following steps.

First, using (5.4) and an argument similar to the proofs of Lemmas 5.3 and 5.4, it will establish

$$\begin{aligned} \mathop {\lim }\limits _{p \rightarrow \infty } \mathrm{\overline{L}} _{{\varphi ^p}}K = \mathrm{\overline{L} _\infty }K. \end{aligned}$$

(5.5)

Second, using the definition of $f_p$, (5.4), (5.5), and the definition of $f_\infty$, it will establish

$$\begin{aligned} \mathop {\lim }\limits _{p \rightarrow \infty } O_{\varphi ^p}(K,\mathrm{\overline{L}}_{\varphi ^p}K)= {\left\| {\frac{{{\rho _K}}}{{{\rho _{{\mathrm{\overline{L} }_\infty }K}}}}} \right\| _\infty }. \end{aligned}$$

(5.6)

Finally, from the identities

$$\begin{aligned} O_{{\varphi ^p}}(K,\mathrm{\overline{L}}_{\varphi ^p}K)\mathrm{\overline{L}} _{{\varphi ^p}}K = \mathrm{L}_{{\varphi ^p}}K \quad \mathrm{and}\quad {\left\| {\frac{{{\rho _K}}}{{{\rho _{{\mathrm{\overline{L} }_\infty }K}}}}} \right\| _\infty }\mathrm{\overline{L} _\infty }K = \mathrm{L_\infty }K, \end{aligned}$$

together with (5.5) and (5.6), it concludes the desired limit. $\square$

6 A Characterization of Orlicz–Legendre Ellipsoids

In this section, we establish a connection linking the characterization of Orlicz–Legendre ellipsoids and the isotropy of measures.

Definition 6.1

Suppose $K\in \mathcal {S}^n_o$ and $\varphi \in \Phi \cap C^1[0,\infty )$, the Borel measure $\mu _\varphi (K,\cdot )$ on $S^{n-1}$ is defined by

$$\begin{aligned} d\mu _\varphi (K,\cdot )=\varphi '\left( \rho _K\right) \rho _K^{n+1}dS. \end{aligned}$$

The next theorem characterizes the Orlicz–Legendre ellipsoid $\mathrm{L}_\varphi K$.

Theorem 6.2

Suppose $K\in \mathcal {S}^n_o$, $\varphi \in \Phi \cap C^1[0,\infty )$, and $T\in \mathrm{GL}(n)$. Then, $\mathrm{L}_\varphi K=TB$, if and only if $\mu _\varphi \left( O_\varphi (T^{-1}K,B)^{-1}T^{-1}K,\cdot \right)$ is isotropic on $S^{n-1}$.

Proof

In terms of Lemma 4.9, without loss of generality, we may assume $T=I_n$. Then, $\overline{\mathrm{L}}_\varphi K=B$. Let $K'=O_\varphi (K,B)^{-1}K$. Note that by Lemma 4.4,

$$\begin{aligned} \overline{\mathrm{L}}_\varphi K=B\quad \Longleftrightarrow \quad \tilde{V}_\varphi (K',B)=\min \limits _{T\in \mathrm{SL}(n)}\tilde{V}_\varphi (K',TB). \end{aligned}$$

Hence, to prove the desired equivalence, it suffices to prove that

$$\begin{aligned} \tilde{V}_\varphi (K',B)=\min \limits _{T\in \mathrm{SL}(n)}\tilde{V}_\varphi (K',TB)\quad \Longleftrightarrow \quad \mu _\varphi \left( K',\cdot \right) \;\mathrm{is\; isotropic\; on\;} S^{n-1}. \end{aligned}$$

First, we assume $\tilde{V}_\varphi (K',B)=\min _{T\in \mathrm{SL}(n)}\tilde{V}_\varphi (K',TB)$, and prove the necessity by the variational method.

Let $L: \mathbb R^n\rightarrow \mathbb R^n$ be a linear transformation. Choose $\varepsilon _0>0$ sufficiently small so that for all $\varepsilon \in (-\varepsilon _0,\varepsilon _0)$ the matrix $I_n+\varepsilon L$ is invertible. For $\varepsilon \in (-\varepsilon _0,\varepsilon _0)$, define

$$\begin{aligned} L_\varepsilon =\frac{I_n+\varepsilon L}{|I_n+\varepsilon L|^{\frac{1}{n}}}. \end{aligned}$$

Then $L_\varepsilon \in \mathrm{SL}(n)$. Our assumption implies that for all $\varepsilon$,

$$\begin{aligned} \tilde{V}_\varphi (K',L_\varepsilon ^{-1}B)\ge {\tilde{V}}_\varphi (K',B). \end{aligned}$$

The fact $\frac{1}{{{\rho _{L_\varepsilon ^{ - 1}B}}(u)}} = {h_{L_\varepsilon ^tB}}(u)$ for $u\in S^{n-1}$, together with the definition of $\tilde{V}_\varphi (K',L_\varepsilon ^{-1}B)$, gives

$$\begin{aligned} {{\tilde{V}}_\varphi }(K',L_\varepsilon ^{ - 1}B) = \int \limits _{{S^{n - 1}}} {\varphi \left( {{\rho _{K'}}(u)\frac{{{{(1 + 2\varepsilon u \cdot Lu + {\varepsilon ^2}Lu \cdot Lu)}^{\frac{1}{2}}}}}{{|{I_n} + \varepsilon L{|^{\frac{1}{n}}}}}} \right) d{{\tilde{V}}_{K'}(u)}} . \end{aligned}$$

From the smoothness of $\varphi$ and $|L_\varepsilon u|$ in $\varepsilon$, the integrand depends smoothly on $\varepsilon$. Thus,

$$\begin{aligned} {\left. {\frac{d}{{d\varepsilon }}} \right| _{\varepsilon = 0}}{{\tilde{V}}_\varphi }(K',L_\varepsilon ^{ - 1}B) = 0. \end{aligned}$$

Calculating it directly, we have

$$\begin{aligned} 0 = \frac{1}{n}\int _{{S^{n - 1}}} {\left( { - \frac{{{\mathop {{\mathrm{tr}}}\nolimits } \;L}}{n} + u \cdot Lu} \right) d{\mu _\varphi }(K',u)}. \end{aligned}$$

Let $v\in S^{n-1}$ and $L=v\otimes v$. Using the facts $\mathrm{tr} (v\otimes v) =1$ and $u\cdot (v\otimes v) u= (u\cdot v)^2$, it gives

$$\begin{aligned} \int _{{S^{n - 1}}} {{{(u \cdot v)}^2}d{\mu _\varphi }(K',u)} = \frac{{|{\mu _\varphi }(K', \cdot )|}}{n}. \end{aligned}$$

Thus, $\mu _\varphi (K',\cdot )$ is isotropic on $S^{n-1}$.

Conversely, suppose that $\mu _\varphi (K',\cdot )$ is isotropic on $S^{n-1}$. We prove that if $E\in \mathcal {E}^n$ and $V(E)=\omega _n$, then

$$\begin{aligned} {\tilde{V}}_\varphi (K',E)\ge {\tilde{V}}_\varphi (K',B). \end{aligned}$$

The proof will be completed after two steps.

First, for $a=(a_1,\ldots ,a_n)\in [0,\infty )^n$, define

$$\begin{aligned} F(a) = \int _{{S^{n - 1}}} {\varphi \left( {{\rho _{K'}}(u)} \right) |{\mathop {{\mathrm{diag}}}\nolimits } ({a_1}, \ldots ,{a_n})u|d{{\tilde{V}}_{K'}}(u)} , \end{aligned}$$

where $\mathrm{diag}(a_1,\ldots ,a_n)$ denotes the $n\times n$ diagonal matrix with diagonal elements $a_1,\ldots ,a_n$. We aim to show that

$$\begin{aligned} F(a)\ge F(e),\quad \mathrm{whenever}\; \prod _{j=1}^na_j=1. \end{aligned}$$

(6.1)

Here, e denotes the point $(1,\ldots ,1)$.

From the smoothness of $\varphi$ and $|\mathrm{diag}(a_1,\ldots ,a_n) u|$ in $(a_1,\ldots ,a_n)$, we have

$$\begin{aligned} {\left. {\frac{\partial }{{\partial {a_j}}}} \right| _{a = e}}F(a) = \int _{{S^{n - 1}}} {u_j^2\varphi '\left( {{\rho _{K'}}(u)} \right) {\rho _{K'}}(u)d{{\tilde{V}}_{K'}}(u)}, \end{aligned}$$

where $(u_1,\ldots ,u_n)$ denotes the coordinates of $u\in S^{n-1}$. From the isotropy of $\mu _\varphi ({K'},\cdot )$, it follows that

$$\begin{aligned} {\left. {\frac{\partial }{{\partial {a_j}}}} \right| _{a = e}}F(a) = \frac{{|{\mu _\varphi }({K'}, \cdot )|}}{n}. \end{aligned}$$

Thus,

$$\begin{aligned} \nabla F(e) = \frac{{|{\mu _\varphi }({K'}, \cdot )|}}{n}e. \end{aligned}$$

(6.2)

It can be checked that the function $F: [0, \infty )^n\rightarrow [0,\infty )$ is continuous and convex, and $F(\lambda a)$ is strictly increasing in $\lambda \in [0,\infty )$, for $a\in {(0,\infty )^n}$. Thus, $F^{-1}([0, F(e)])$ is a convex body. Its boundary is given by the equation $F(a)=F(e)$ with $a\in [0,\infty )^n$, so (6.2) implies the vector e is an outer normal of the convex body $F^{-1}([0, F(e)])$ at the boundary point e.

Consequently, ${F^{ - 1}}([0,F(e)]) \subset \left\{ {a \in {\mathbb R^n}:a \cdot e \le n} \right\}$. That is to say, for all $a\in \mathbb [0,\infty )^n$, if $F(a)\le F(e)$, then $a\cdot e\le n$. In contrast, for all $b=(b_1,\ldots ,b_n)\in (0,\infty )^n$ with $b_1\cdots b_n =1$, the AM–GM inequality yields that $b\cdot e\ge n$, with equality if and only if $b=e$. Hence, (6.1) is derived.

Finally, represent each $T\in \mathrm{SL}(n)$ into the form $T=O_1^{-1}AO_2$, where $O_1$, $O_2$ are $n\times n$ orthogonal matrices, and $A=\mathrm{diag}(a_1,\ldots , a_n)$ is diagonal and positive definite with $a_1a_2\cdots a_n=1$. Note that ${{\tilde{V}}_\varphi }(K',TB) = {{\tilde{V}}_\varphi }(O_1K',AB)$, and $\mu _\varphi (O_1K',\cdot )$ is isotropic on $S^{n-1}$. So, applying (6.1) to the body $O_1K'$, it gives

$$\begin{aligned} {\tilde{V}_\varphi }({K'},TB) \ge {{\tilde{V}}_\varphi }({K'},B), \end{aligned}$$

which concludes the desired sufficiency. $\square$

Corollary 6.3

Suppose $K\in \mathcal {S}^n_o$ and $\varphi \in \Phi \cap C^1[0,\infty )$. Then, modulo orthogonal transformations, there exists an $\mathrm{SL}(n)$ transformation T such that the measure $\mu _\varphi (TK,\cdot )$ is isotropic on $S^{n-1}$.

7 Volume Ratio Inequalities

In general, the Orlicz–Legendre ellipsoid $\mathrm{L}_\varphi K$ does not contain K. However, we show that the volume functional over the class of Orlicz–Legendre ellipsoids of K is bounded by $V(\mathrm{L}_1K)$ from below and by $V(\mathrm{L}_\infty K)$ from above.

Theorem 7.1

Suppose $K\in \mathcal {S}^n_o$, $\varphi \in \Phi$ and $1\le p< q<\infty$. Then

$$\begin{aligned} V(\mathrm{L}_1K)\le V(\mathrm{L}_\varphi K)\le V(\mathrm{L}_{\varphi ^p} K)\le V(\mathrm{L}_{\varphi ^q} K)\le V(\mathrm{L}_\infty K). \end{aligned}$$

Proof

From Lemma 3.9, it follows that

$$\begin{aligned} \left\{ {E \in {\mathcal {E}^n}:{{\left\| {\frac{{{\rho _K}}}{{{\rho _E}}}:{V^*_K}} \right\| }_1} \le 1} \right\}&\supseteq \left\{ {E \in {\mathcal {E}^n}:{{\left\| {\frac{{{\rho _K}}}{{{\rho _E}}}:{V^*_K}} \right\| }_\varphi } \le 1} \right\} \\&\supseteq \left\{ {E \in {\mathcal {E}^n}:{{\left\| {\frac{{{\rho _K}}}{{{\rho _E}}}:{V^*_K}} \right\| }_{{\varphi ^p}}} \le 1} \right\} \\&\supseteq \left\{ {E \in {\mathcal {E}^n}:{{\left\| {\frac{{{\rho _K}}}{{{\rho _E}}}:{V^*_K}} \right\| }_{{\varphi ^q}}} \le 1} \right\} \\&\supseteq \left\{ {E \in {\mathcal {E}^n}:{{\left\| {\frac{{{\rho _K}}}{{{\rho _E}}}} \right\| }_\infty } \le 1} \right\} . \end{aligned}$$

From the above inclusions and the definition of Orlicz–Legendre ellipsoids, the desired inequalities are obtained. $\square$

Theorem 7.2

Suppose $K\in \mathcal {S}^n_o$ and $\varphi \in \Phi$. Then

$$\begin{aligned} V(\mathrm{L}_\varphi K)\ge V(K), \end{aligned}$$

with equality if and only if $K\in \mathcal {E}^n$.

Proof

From Lemma 3.6 and the fact that $\mathrm{L}_\varphi E=E$ for any $E\in \mathcal {E}^n$, it follows that

$$\begin{aligned} {O_\varphi }\left( {K,{\mathrm{L}_\varphi }K} \right) \ge {\left( {\frac{{V(K)}}{{V({\mathrm{L}_\varphi }K)}}} \right) ^{\frac{1}{n}}}, \end{aligned}$$

with equality if and only if $K\in \mathcal {E}^n$. Combining this with the fact $1 = {O_\varphi }\left( {K,{\mathrm{L}_\varphi }K} \right)$, it establishes the desired inequality. $\square$

If $\varphi (t)=t^p$, $1\le p<\infty$, then Theorem 7.2 implies that $V(\mathrm{L}_pK)\ge V(K)$, and in particular that $V(\Gamma _2K)\ge V(K)$.

A classical result on John’s ellipsoid is Ball’s volume ratio inequality [1, 2], which states: if K is an origin-symmetric convex body in $\mathbb R^n$, then

$$\begin{aligned} \frac{{V(K)}}{{V(\mathrm{J}K)}} \le \frac{{{2^n}}}{{{\omega _n}}}, \end{aligned}$$

with equality if and only if K is a parallelotope. The fact that equality holds in Ball’s inequality only for parallelotopes was established by Barthe [3]. He also established the outer volume-ratio inequality: if K is an origin-symmetric convex body in $\mathbb R^n$, then

$$\begin{aligned} \frac{{V(K)}}{{V(\mathrm{L}K)}} \ge \frac{{{2^n}}}{{n!{\omega _n}}}, \end{aligned}$$

with equality if and only if K is a cross-polytope.

Recall that when K is an origin-symmetric convex body, $\mathrm{L}_\infty K$ is just the Löwner ellipsoid $\mathrm{L}K$. Thus, combining Theorem 7.1 with Barthe’s outer volume ratio inequality, we immediately obtain

Theorem 7.3

Suppose $K\in \mathcal {K}^n_o$ is origin-symmetric and $\varphi \in \Phi$. Then

$$\begin{aligned} \frac{{V(K)}}{{V({\mathrm{L}_\varphi }K)}} \ge \frac{{{2^n}}}{{n!{\omega _n}}}. \end{aligned}$$