On the polar Orlicz-Minkowski problems and the $p$-capacitary Orlicz-Petty bodies

TL;DR

This paper investigates the existence and properties of solutions to the polar Orlicz-Minkowski problems and introduces the concept of $p$-capacitary Orlicz-Petty bodies, establishing their existence and continuity under certain conditions.

Contribution

It provides new existence results for the polar Orlicz-Minkowski problems and introduces the $p$-capacitary Orlicz-Petty bodies, analyzing their properties and continuity.

Findings

Existence of solutions under specific conditions on measure and function.

Introduction and analysis of $p$-capacitary Orlicz-Petty bodies.

Continuity of the $p$-capacitary Orlicz-Petty bodies proved.

Abstract

In this paper, we propose and study the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure $\mu$ and a continuous function $\varphi:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body $K\in\mathcal{K}_0$ such that $K$ is an optimizer of the following optimization problems: \begin{equation*} \inf/\sup \bigg\{\int_{S^{n-1}}\varphi\big( h_L \big) \,d \mu: L \in \mathcal{K}_{0} \ \text{and}\ |L^\circ|=\omega_{n}\bigg\}. \end{equation*} The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on $\varphi,$ the existence of a solution is proved for a nonzero finite measure $\mu$ on $S^{n-1}$ which is not concentrated on any hemisphere of $S^{n-1}.$ Another part of this paper deals with the $p$-capacitary Orlicz-Petty bodies. In particular, the existence of the $p$-capacitary Orlicz-Petty bodies is established and the continuity of the $p$-capacitary Orlicz-Petty bodies is proved.

Equations

$$\inf/\sup\bigg{\{}\int_{S^{n-1}}\varphi\big{(}h_{L}\big{)}\,d\mu:L\in\mathscr{K}_{0}\ \text{and}\ |L^{\circ}|=\omega_{n}\bigg{\}}.$$

$$V_{q}(K,L)=\displaystyle{\frac{1}{n}\int_{S^{n-1}}\Big{(}\frac{h_{L}(u)}{h_{K}(u)}\Big{)}^{q}h_{K}(u)\,dS(K,u)},$$

$$V_q(K,L)\ge |K{|}^{\frac{n-q}{n}}|L{|}^{\frac{q}{n}};$$

$$\inf\bigg{\{}\int_{S^{n-1}}\Big{(}\frac{h_{L}(u)}{h_{K}(u)}\Big{)}^{q}h_{K}(u)\,dS(K,u):L\in\mathscr{K}_{0}\ \ \text{and}\ \ |L|=\omega_{n}\bigg{\}}$$

$$\inf\bigg{\{}\frac{1}{n}\int_{S^{n-1}}h_{L}\,d\mu:\ L\in\mathscr{K}_{0}\ \ \text{and}\ \ |L|=\omega_{n}\bigg{\}},$$

$$G_{q}(K)=\inf\bigg{\{}\int_{S^{n-1}}\Big{(}\frac{h_{L}(u)}{h_{K}(u)}\Big{)}^{q}h_{K}(u)\,dS(K,u):L\in\mathscr{K}_{0}\ \ \text{and}\ \ |L^{\circ}|=\omega_{n}\bigg{\}},$$

$$L^{\circ }=\{y\in {\mathbb{R}}^n:⟨x,y⟩\le 1\text{for any}x\in L\}$$

$$\Omega_{q}(K)=\inf\bigg{\{}\int_{S^{n-1}}\Big{(}\frac{1}{\rho_{L}(u)h_{K}(u)}\Big{)}^{q}h_{K}(u)\,dS(K,u):L\in\mathscr{S}_{0}\ \ \text{and}\ \ |L|=\omega_{n}\bigg{\}}$$

$$V_{\phi }(K,L)$$

$$V_{\phi ,\varphi }(K,L)$$

$$\widehat{V}_{\varphi}(K,L)=\inf\bigg{\{}\lambda>0:\int_{S^{n-1}}\varphi\Big{(}\frac{n|K|\cdot h_{L}(u)}{\lambda\cdot h_{K}(u)}\Big{)}h_{K}(u)\,dS(K,u)\leq n|K|\bigg{\}},$$

$$V_{q}(K,L)=V_{\varphi}(K,L)=V_{\varphi,\phi}(K,L)=n^{-q}\cdot|K|^{1-q}\big{(}\widehat{V}_{\varphi}(K,L)\big{)}^{q}.$$

$$V_{\varphi}(K,L)\geq|K|\cdot\varphi\bigg{(}\bigg{(}\frac{|L|}{|K|}\bigg{)}^{\frac{1}{n}}\bigg{)},$$

$${\hat{V}}_{\phi }(K,L)\ge n|K{|}^{\frac{n-1}{n}}|L{|}^{\frac{1}{n}}.$$

$$\displaystyle\inf\Big{\{}V_{\varphi}(K,L):L\in\mathscr{K}_{0}\ \ \text{and}\ \ |L|=\omega_{n}\Big{\}}\ \ \mathrm{and}\ \ \inf\Big{\{}\widehat{V}_{\varphi}(K,L):L\in\mathscr{K}_{0}\ \ \text{and}\ \ |L|=\omega_{n}\Big{\}}$$

$$G_{\phi }^{orlicz}(K)$$

$${\hat{G}}_{\phi }^{orlicz}(K)$$

$$G_{\phi }^{orlicz}(K)=n{V}_{\phi }(K,T_{\phi }K)\text{and}{\hat{G}}_{\phi }^{orlicz}(K)={\hat{V}}_{\phi }(K,{\hat{T}}_{\phi }K).$$

$$\inf/\sup\bigg{\{}\int_{S^{n-1}}\varphi\big{(}h_{L}\big{)}\,d\mu:L\in\mathscr{K}_{0}\ \text{and}\ |L^{\circ}|=\omega_{n}\bigg{\}}.$$

$$\int_{S^{n-1}}\varphi(h_{M}(u))d\mu(u)=\inf\bigg{\{}\int_{S^{n-1}}\varphi(h_{L}(u))\,d\mu(u):L\in\mathscr{K}_{0}\ \ \text{and}\ \ |L^{\circ}|=\omega_{n}\bigg{\}}.$$

$$C_{p,\phi }(K,L)$$

$$1$$

$$\displaystyle C_{p,\varphi}(K,L)\geq C_{p}(K)\cdot\varphi\bigg{(}\bigg{(}\frac{C_{p}(L)}{C_{p}(K)}\bigg{)}^{\frac{1}{n-p}}\bigg{)}\ \ \ \text{and}\ \ \ \widehat{C}_{p,\varphi}(K,L)\geq C_{p}(K)\bigg{(}\frac{C_{p}(L)}{C_{p}(K)}\bigg{)}^{\frac{1}{n-p}},$$

$$\displaystyle\sup/\inf\Big{\{}C_{p,\varphi}(K,L):L\in\mathscr{K}_{0}\ \ \text{and}\ \ |L^{\circ}|=\omega_{n}\Big{\}},$$

$$\displaystyle\sup/\inf\Big{\{}\widehat{C}_{p,\varphi}(K,L):L\in\mathscr{K}_{0}\ \ \text{and}\ \ |L^{\circ}|=\omega_{n}\Big{\}}.$$

$$C_{p,\varphi}(K,M)=\inf\Big{\{}C_{p,\varphi}(K,L):L\in\mathscr{K}_{0}\ \ \mbox{and}\ \ |L^{\circ}|=\omega_{n}\Big{\}}.$$

$$\widehat{C}_{p,\varphi}(K,\widehat{M})=\inf\Big{\{}\widehat{C}_{p,\varphi}(K,L):L\in\mathscr{K}_{0}\ \ \mbox{and}\ \ |L^{\circ}|=\omega_{n}\Big{\}}.$$

$$K+L=\{x+y:x\in K,y\in L\}.$$

$$\text{vrad}(K)=\bigg{(}\frac{|K|}{\omega_{n}}\bigg{)}^{\frac{1}{n}}.$$

$$h_K(u)=\underset{x\in K}{\max }⟨x,u⟩\text{for any}u\in S^{n-1},$$

$$d_H(K,L)=\underset{u\in S^{n-1}}{\max }|h_K(u)-h_L(u)|.$$

$${\rho }_L(u)=\max \{r\ge 0:ru\in L\}\text{for any}u\in S^{n-1}.$$

Full text

[TABLE]

The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on $\varphi,$ the existence of a solution is proved for a nonzero finite measure $\mu$ on $S^{n-1}$ which is not concentrated on any hemisphere of $S^{n-1}.$ Another part of this paper deals with the $p$-capacitary Orlicz-Petty bodies. In particular, the existence of the $p$-capacitary Orlicz-Petty bodies is established and the continuity of the $p$-capacitary Orlicz-Petty bodies is proved. 2010 Mathematics Subject Classification: 52A20, 52A38, 52A39, 52A40, 53A15.

1 Introduction

Let $\mathscr{K}_{0}$ be the set of all convex bodies in $\mathbb{R}^{n}$ with the origin $o$ in their interiors, i.e., $K\in\mathscr{K}_{0}$ is a convex compact subset of $\mathbb{R}^{n}$ such that $o\in\text{int}K,$ the interior of $K.$ For $0\neq q\in\mathbb{R},$ the $L_{q}$ mixed volume of $K,L\in\mathscr{K}_{0}$ (see e.g. [32, 52]) can be defined as

[TABLE]

where $S(K,\cdot)$ is the surface area measure of $K$ (see e.g. [1, 11]) and $h_{L}$ is the support function of $L$ defined on $S^{n-1},$ the unit sphere of $\mathbb{R}^{n}$ (see Section 2 for more details on the notations). When $q=1$, one gets the classical mixed volume (see e.g. [15, 43]). The classical and $L_{q}$ Minkowski inequalities [12, 15, 32, 43] lie at the heart of the rapidly developing $L_{q}$ Brunn-Minkowski theory of convex bodies. These inequalities read: for all $q\geq 1$ and all $K,L\in\mathscr{K}_{0},$

[TABLE]

equality holds if and only if $K$ is dilate of $L$ for $q>1$ and $K$ is homothetic to $L$ for $q=1$. Hereafter $|K|$ refers to the volume of $K\in\mathscr{K}_{0}$ and $\omega_{n}=|B^{n}_{2}|$ denotes the volume of the unit ball $B^{n}_{2}$ in $\mathbb{R}^{n}.$ The $L_{q}$ Minkowski inequality (1.2) implies that, for any fixed $K\in\mathscr{K}_{0}$ and $q\geq 1,$ the following optimization problem

[TABLE]

has a unique solution for $q>1$ and a unique solution up to translation for $q=1$.

Related to (1.3) is the celebrating $L_{q}$ Minkowski problem: for $0\neq q\in\mathbb{R}$, under what conditions on a given nonzero finite measure $\mu$ defined on $S^{n-1},$ there is a convex body $K$ (ideally with the origin $o$ in its interior) such that $h^{q-1}_{K}d\mu=dS(K,\cdot)$? The $L_{q}$ Minkowski problem is a popular problem in geometry and has attracted considerable attention (see e.g. [3, 4, 20, 23, 32, 35, 46, 58, 59, 60]). Solutions to the $L_{q}$ Minkowski problems have fundamental applications in, for instance, establishing the $L_{q}$ Sobolev type inequalities (see e.g. [5, 17, 34, 55]). When $q=1$, the $L_{q}$ Minkowski problem becomes the classical Minkowski problem. It has been proved in, e.g. [38, 39], that there exists a unique convex body $K$ (up to a translation) such that $dS(K,\cdot)=d\mu,$ if $\mu$ is not concentrated on any hemisphere of $S^{n-1}$ and has the centroid at the origin. In fact, to solve the classical Minkowski problem is equivalent to find an optimizer of the following optimization problem:

[TABLE]

which can be obtained from (1.3) with $S(K,\cdot)$ replaced by $\mu$ and $q=1$.

The classical and $L_{q}$ geominimal surface areas [33, 41, 42, 52] are important concepts in convex geometry, which are closely related to (1.3). For instance, for $q>0$ and $K\in\mathscr{K}_{0}$, the $L_{q}$ geominimal surface area of $K$, denoted by $G_{q}(K)$, can be formulated by

[TABLE]

where $L^{\circ}$ denotes the polar body of $L,$ i.e.,

[TABLE]

with $\langle\cdot,\cdot\rangle$ the standard inner product on $\mathbb{R}^{n}$. Note that $|L|=\omega_{n}$ in (1.3) is replaced by $|L^{\circ}|=\omega_{n}$ in (1.5). However, (1.3) and (1.5) are completely different, and both of them play fundamental roles in the $L_{q}$ Brunn-Minkowski theory of convex bodies. It has been proved [33, 41, 57] that there exist convex bodies $T_{q}K\in\mathscr{K}_{0}$ for $q>0$, the $L_{q}$ Petty bodies of $K,$ such that $|(T_{q}K)^{\circ}|=\omega_{n}$ and $G_{q}(K)=nV_{q}(K,T_{q}K).$ Indeed, a variation problem of (1.5), with $\mathscr{K}_{0}$ replaced by $\mathscr{S}_{0}$ (the set of all star bodies about the origin $o$), defines the $L_{q}$ affine surface area of $K$ [31, 33, 52]:

[TABLE]

with $\rho_{L}:S^{n-1}\rightarrow(0,\infty)$ the radial function of $L\in\mathscr{S}_{0}$. The $L_{q}$ affine surface area has equivalent convenient integral formulas (see e.g. [2, 22, 37, 44, 45]) and important applications in, such as, the valuation theory, the approximation of convex bodies by polytopes, the $f$-divergence of convex bodies and the $L_{q}$ affine isoperimetric inequalities (see e.g. [14, 24, 28, 29, 30, 40, 45, 47, 48, 49, 56]).

Extension from the $L_{q}$ Brunn-Minkowski theory to the Orlicz theory is rather dedicated. In view of (1.1)-(1.5), a major task is to get the “right” formula of the Orlicz mixed volume. In the Orlicz theory, there are at least 3 different ways to define the Orlicz mixed volume and each of them has its own advantages. These Orlicz mixed volumes are given as follows: for $K,L\in\mathscr{K}_{0},$ and $\phi,\ \varphi:(0,\infty)\rightarrow(0,\infty)$ continuous functions,

[TABLE]

and if in addition $\varphi\in\mathscr{I},$

[TABLE]

where $\mathscr{I}$ refers to the set of continuous functions $\varphi:(0,\infty)\rightarrow(0,\infty)$ such that $\varphi$ is strictly increasing, $\lim_{t\to 0^{+}}\varphi(t)=0,$ $\varphi(1)=1$ and $\lim_{t\to\infty}\varphi(t)=\infty.$ The $L_{q}$ mixed volume ($q>0$) defined by (1.1) is a special case of (1.6)-(1.8), namely, for $\varphi(t)=t^{q}$ and $\phi(t)=t^{q-1}$,

[TABLE]

Note that $V_{\varphi}(\cdot,\cdot)$ and $V_{\varphi,\phi}(\cdot,\cdot)$ have geometric interpretations (see e.g. [13, 50, 57]); while there are no geometric interpretations for $\widehat{V}_{\varphi}(\cdot,\cdot)$ in literature. Also note that $\widehat{V}_{\varphi}(\cdot,\cdot)$ is homogeneous [57] but $V_{\varphi}(\cdot,\cdot)$ and $V_{\varphi,\phi}(\cdot,\cdot)$ are not homogeneous. For $V_{\varphi}(\cdot,\cdot)$ and $\widehat{V}_{\varphi}(\cdot,\cdot)$, one has the Orlicz-Minkowski inequalities [13, 50]: for $K,L\in\mathscr{K}_{0}$ and $\varphi$ being a convex function, then

[TABLE]

and if in addition $\varphi\in\mathscr{I}$,

[TABLE]

Equality holds if and only if $K$ and $L$ are dilates if $\varphi$ is also strictly convex. The Orlicz-Minkowski inequalities (1.9) and (1.10) imply that, if $\varphi\in\mathscr{I}$ is strictly convex,

[TABLE]

have unique solutions, respectively. It seems intractable to pose the Minkowski type problems related to $V_{\varphi}(\cdot,\cdot)$ and $\widehat{V}_{\varphi}(\cdot,\cdot)$. However, the Orlicz-Minkowski problem can be asked (related to $V_{\varphi,\phi}(\cdot,\cdot)$): under what conditions on a nonzero finite measure $\mu$ on $S^{n-1}$ and a continuous function $\varphi:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body $K$, such that, $\tau\,dS(K,\cdot)=\varphi(h_{K})\,d\mu$ for some positive constant $\tau$? Solutions of this Orlicz-Minkowski problem can be found in [16, 21, 27].

Ye [53] and Zhu, Hong and Ye [57] investigated the following optimization problems and gave detailed studies of the (nonhomogeneous and homogeneous) Orlicz geominimal surface areas $G^{orlicz}_{\varphi}(\cdot)$ and $\widehat{G}^{orlicz}_{\varphi}(\cdot)$: under certain conditions on $\varphi,$ let

[TABLE]

In particular, Zhu, Hong and Ye [57] proved that, under certain conditions on $\varphi,$ there exist convex bodies $T_{\varphi}K$ and $\widehat{T}_{\varphi}K,$ called the Orlicz-Petty bodies of $K,$ such that $|(T_{\varphi}K)^{\circ}|=|(\widehat{T}_{\varphi}K)^{\circ}|=\omega_{n},$

[TABLE]

One can also see [54] for a special case.

Motivated by (1.11)-(1.13) and the relations between (1.3)-(1.5), we propose and study the following problems in Section 3. The polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure $\mu$ and a function $\varphi:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body $K\in\mathscr{K}_{0}$ such that $K$ is an optimizer of the following optimization problems:

[TABLE]

The polar Orlicz-Minkowski problems are not the same as (1.12)-(1.13) because the measure $\mu$ is not assumed to be related to any convex bodies, and are also completely different from the Orlicz-Minkowski problems.

Our main result in Section 3 is summarized in the following theorem. Let $\Omega$ be the set of all nonzero finite Borel measures on $S^{n-1}$ that are not concentrated on any hemisphere of $S^{n-1}$.

Theorem 1.1**.**

Let $\mu\in\Omega$ and $\varphi\in\mathscr{I}.$ Then there exists a convex body $M\in\mathscr{K}_{0}$ such that $|M^{\circ}|=\omega_{n}$ and

[TABLE]

Moreover, if $\varphi\in\mathscr{I}$ is convex, then $M$ is the unique solution to the polar Orlicz-Minkowski problem.

In Section 4, we replace $V_{\varphi}(\cdot,\cdot)$ and $\widehat{V}_{\varphi}(\cdot,\cdot)$ by their $p$-capacitary counterparts, and study the existence and continuity of the $p$-capacitary Orlicz-Petty bodies. Hereafter, for $K,L\in\mathscr{K}_{0},$ $p\in(1,n)$ and $\varphi:(0,\infty)\rightarrow(0,\infty),$ the Orlicz mixed $p$-capacities of $K$ and $L$ are given by:

[TABLE]

where $\mu_{p}(K,\cdot)$ and $\mu^{*}_{p}(K,\cdot)$ are measures given by (2.4) and (2.6), respectively. Related to $C_{p,\varphi}(\cdot,\cdot)$ and $\widehat{C}_{p,\varphi}(\cdot,\cdot)$, there are the $p$-capacitary Orlicz-Minkowski inequalities: for $p\in(1,n),$ $K,L\in\mathscr{K}_{0}$ and $\varphi\in\mathscr{I}$ being convex [18], one has,

[TABLE]

with equality if and only if $K$ and $L$ are dilates if $\varphi$ is strictly convex. One can also ask the $p$-capacitary $L_{q}$ and Orlicz Minkowski problems (i.e., with $S(K,\cdot)$ replaced by $\mu_{p}(K,\cdot)$); these problems have received extensive attention (see e.g. [8, 18, 25, 26, 61]).

Our major interest in Section 4 is the following problems: for $p\in(1,n),$ find optimizers of the following optimization problems:

[TABLE]

The main result in Section 4 is stated as follows.

Theorem 1.2**.**

Let $K\in\mathscr{K}_{0}$ be a convex body and $\varphi\in\mathscr{I}.$

(i) There exists a convex body $M\in\mathscr{K}_{0}$ such that $|M^{\circ}|=\omega_{n}$ and

[TABLE]

(ii) There exists a convex body $\widehat{M}\in\mathscr{K}_{0}$ such that $|\widehat{M}^{\circ}|=\omega_{n}$ and

[TABLE]

In addition, if $\varphi\in\mathscr{I}$ is convex, then both $M$ and $\widehat{M}$ are unique.

The convex bodies $M$ and $\widehat{M}$ in Theorem 1.2 are called the $p$-capacitary Orlicz-Petty bodies of $K$. The continuity of the $p$-capacitary Orlicz-Petty bodies is provided in Theorem 4.2.

2 Preliminaries and Notations

A subset $K\subseteq\mathbb{R}^{n}$ is said to be convex if $\lambda x+(1-\lambda)y\in K$ for any $\lambda\in[0,1]$ and $x,y\in K$. A convex body is a convex compact subset of $\mathbb{R}^{n}$ with nonempty interior. A convex body $K$ is said to be origin-symmetric if $-x\in K$ for any $x\in K$. We use $\mathscr{K}$ and $\mathscr{K}_{0}\subseteq\mathscr{K}$ to denote the set of all convex bodies, respectively, and the set of all convex bodies with the origin in their interiors. The Minkowski sum of $K,L\in\mathscr{K}$, denoted by $K+L$, is defined by

[TABLE]

For $\lambda\in\mathbb{R}$, the scalar product of $\lambda$ and $K,$ denoted by $\lambda K,$ is defined by $\lambda K=\{\lambda x:x\in K\}.$ For $K\in\mathscr{K},$ $|K|$ refers to the volume of $K$. In particular, $\omega_{n}$ represents the volume of the unit ball $B_{2}^{n}\subseteq\mathbb{R}^{n}.$ For $K\in\mathscr{K},$ one can define the volume radius of $K$ by

[TABLE]

The support function of a nonempty convex compact set $K\subseteq\mathbb{R}^{n}$, $h_{K}:S^{n-1}\rightarrow\mathbb{R}$, is defined by

[TABLE]

where $S^{n-1}$ is the unit sphere. It can be easily checked that, for any $\lambda\geq 0$ and $K,L\in\mathscr{K},$ $h_{\lambda K}(u)=\lambda h_{K}(u)$ and $h_{K+L}(u)=h_{K}(u)+h_{L}(u)\ \text{for any}\ u\in S^{n-1}.$ The Hausdorff distance between two convex compact sets $K,L\subseteq\mathbb{R}^{n}$, denoted by $d_{H}(K,L)$, is given by

[TABLE]

For a sequence ${\left\{K_{i}\right\}}_{i=1}^{\infty}\subseteq\mathscr{K}_{0}$ and a convex compact set $K$, $K_{i}\rightarrow K$ as $i\rightarrow\infty$ with respect to the Hausdorff metric means that $d_{H}(K_{i},K)\rightarrow 0$ as $i\rightarrow\infty$.

A subset $L\subseteq\mathbb{R}^{n}$ is called a star-shaped set about the origin if for $\text{any}\ x\in L$, the line segment from the origin $o$ to $x$ is contained in $L$. The radial function of a star-shaped set $L$ about the origin $o$, $\rho_{L}:S^{n-1}\rightarrow[0,\infty)$, is defined by

[TABLE]

A star-shaped set $L\subseteq\mathbb{R}^{n}$ about the origin $o$ is called a star body about the origin $o$ if the radial function $\rho_{L}$ is positive and continuous on $S^{n-1}.$ Denote by $\mathscr{S}_{0}$ the set of all star bodies about the origin $o$. Obviously, $\mathscr{K}_{0}\subseteq\mathscr{S}_{0}$.

The polar body $K^{\circ}$ of $K\in\mathscr{K}_{0}$ is defined by

[TABLE]

where $\langle\cdot,\cdot\rangle$ is the standard inner product in $\mathbb{R}^{n}$. By $K^{\circ\circ}$, we mean the polar body of $K^{\circ}$, and hence $K^{\circ\circ}=K$ if $K\in\mathscr{K}_{0}$ [43, Theorem 1.6.1]. It can be proved [43] that for any $K\in\mathscr{K}_{0},$

[TABLE]

For $K\in\mathscr{S}_{0}$, the following volume formula for $|K|$ holds:

[TABLE]

where $\sigma(\cdot)$ is the spherical measure on $S^{n-1}.$ Associated to each $K\in\mathscr{K}_{0}$, the surface area measure of $K$ on $S^{n-1}$, denoted by $S(K,\cdot),$ is defined by: for any measurable subset $A\subseteq S^{n-1}$,

[TABLE]

where $\nu_{K}^{-1}:S^{n-1}\rightarrow\partial K$ is the inverse Gauss map of $K$ and $\mathscr{H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure on $\partial K.$

Denote by $C(S^{n-1})$ the set of all continuous functions on $S^{n-1}.$ Let ${\left\{\mu_{i}\right\}}_{i=1}^{\infty}$ and $\mu$ be measures on $S^{n-1}$. We say $\mu_{i}\rightarrow\mu$ weakly if for any $f\in C(S^{n-1})$,

[TABLE]

The following lemma regarding the weak convergence $\mu_{i}\rightarrow\mu$ is often used.

Lemma 2.1**.**

If a sequence of measures ${\left\{\mu_{i}\right\}}_{i=1}^{\infty}$ on $S^{n-1}$ converges weakly to a finite measure $\mu$ on $S^{n-1}$ and a sequence of functions ${\left\{f_{i}\right\}}_{i=1}^{\infty}\subseteq C(S^{n-1})$ converges uniformly to a function $f\in C(S^{n-1})$, then

[TABLE]

We shall also need the following lemma.

Lemma 2.2**.**

Let ${\left\{K_{i}\right\}}_{i=1}^{\infty}\subseteq\mathscr{K}_{0}$ be a uniformly bounded sequence such that the sequence ${\left\{|K_{i}^{\circ}|\right\}}_{i=1}^{\infty}$ is bounded. Then, there exists a subsequence $\{K_{i_{j}}\}_{j=1}^{\infty}$ of ${\left\{K_{i}\right\}}_{i=1}^{\infty}$ and a convex body $K\in\mathscr{K}_{0}$ such that $K_{i_{j}}\rightarrow K$. Moreover, if $|K_{i}^{\circ}|=\omega_{n}$ for all $i=1,2,\cdots$, then $|K^{\circ}|=\omega_{n}$.

Proof.

Lutwak [33] proved that, if ${\left\{K_{i}\right\}}_{i=1}^{\infty}\subseteq\mathscr{K}_{0}$ and $K$ a convex compact set satisfy that $K_{i}\rightarrow K$ as $i\rightarrow\infty$ with respect to the Hausdorff metric and the sequence ${\left\{|K_{i}^{\circ}|\right\}}_{i=1}^{\infty}$ is bounded, then $K\in\mathscr{K}_{0}$. On the other hand, the Blaschke selection theorem [43] asserts that, if ${\left\{K_{i}\right\}}_{i=1}^{\infty}$ is a bounded sequence of convex compact sets in $\mathbb{R}^{n}$, then there exist a subsequence $\{K_{i_{j}}\}_{j=1}^{\infty}$ of $\{K_{i}\}_{i=1}^{\infty}$ and a convex compact set $K$, such that, $K_{i_{j}}\rightarrow K$ as $j\rightarrow\infty$ with respect to the Hausdorff metric. Combining the above two statements, one can immediately get $K_{i_{j}}\rightarrow K\in\mathscr{K}_{0}.$ Moreover, if $|K_{i}^{\circ}|=\omega_{n}$ for all $i=1,2,\cdots$, then $|K^{\circ}|=\lim_{j\rightarrow\infty}|K_{i_{j}}^{\circ}|=\omega_{n}$, due to the continuity of volume under the Hasdorff metric. ∎

Now we provide some basic background for the $p$-capacity, and all results can be found in e.g. [8, 9, 10]. Let $C^{\infty}_{c}(\mathbb{R}^{n})$ be the set of all infinitely differentiable functions on $\mathbb{R}^{n}$ with compact supports. For $x\in\mathbb{R}^{n}$, we use $|x|$ to mean the Euclidean norm of $x.$ For a compact subset $E\subseteq\mathbb{R}^{n}$ and $1<p<n$, define $C_{p}(E)$, the $p$-capacity of $E$, by

[TABLE]

where $\nabla f$ denotes the gradient of $f$. Clearly, for two compact sets $E\subseteq F\subseteq\mathbb{R}^{n},$ one has $C_{p}(E)\leq C_{p}(F).$ Moreover, for any compact set $E\subseteq\mathbb{R}^{n}$, $C_{p}(\lambda E)=\lambda^{n-p}C_{p}(E)$ for any $\lambda>0,$ and $C_{p}(E+x_{0})=C_{p}(E)$ for any $x_{0}\in\mathbb{R}^{n}$. Besides, the functional $C_{p}(\cdot)$ is continuous on $\mathscr{K}_{0}$ with respect to the Hausdorff metric.

By the $p$-Laplace equation for $p\in(1,n)$, we mean the equation $\text{div}(|\nabla U|^{p-2}\nabla U)=0,$ with $\text{div}(\cdot)$ the divergence of vector fields in $\mathbb{R}^{n}$. The $p$-Laplace equation has a fundamental solution $U_{0}(x)=|x|^{\frac{p-n}{p-1}}(x\neq o).$ For $K\in\mathscr{K}_{0}$, the $p$-capacitary function of $K$ is a weak solution of the following $p$-Laplace equation with the boundary conditions:

[TABLE]

It has been proved that there exists a unique solution $U_{K}$ to (2.3) such that

[TABLE]

and $U_{K}\in C(\mathbb{R}^{n}\setminus\text{int}K)\cap C^{\infty}(\mathbb{R}^{n}\setminus K)$ (see more details in [9]).

Let $K\in\mathscr{K}_{0}.$ Define $\mu_{p}(K,\cdot)$, the $p$-capacitary measure of $K$ on $S^{n-1}$, by (see e.g. [8, (1.11)])

[TABLE]

where $U_{K}$ is the $p$-capacitary function of $K$. For any $\lambda>0$, one can easily get

[TABLE]

The translation invariance of $C_{p}(\cdot)$ yields that for any $K\in\mathscr{K}_{0}$, the centroid of $\mu_{p}(K,\cdot)$ is at the origin, i.e.,

[TABLE]

Moreover, $\mu_{p}(K,\cdot)$ is not concentrated on any hemisphere of $S^{n-1}$ [51, Theorem 1], i.e.,

[TABLE]

where $a_{+}=\max\{a,0\}$ for all $a\in\mathbb{R}$. The $p$-capacity of $K\in\mathscr{K}_{0}$ can be calculated by the famous Poincaré formula (see e.g. [8]):

[TABLE]

In particular, for any $p\in(1,n)$,

[TABLE]

From (2.5), for any $K\in\mathscr{K}_{0}$, one can define a probability measure $\mu^{*}_{p}(K,\cdot)$ on $S^{n-1}$ by

[TABLE]

According to [36, (8.9)], one has the following isocapacitary inequality:

[TABLE]

holds for any $K\in\mathscr{K}_{0}$.

The $p$-capacity and the volume belong to a large family of functionals defined on $\mathscr{K}_{0}$. Such a family of functionals will be called the variational functionals compatible with the mixed volume [19] (see also [6, 7]). We summarize its definition below.

Definition 2.1**.**

A variational functional $\mathscr{V}:\mathscr{K}_{0}\rightarrow(0,\infty)$ is said to be compatible with the mixed volume if $\mathscr{V}$ satisfies the following properties:

(i) homogeneous, i.e., there exists a constant $\alpha\neq 0$ such that $\mathscr{V}(\lambda K)=\lambda^{\alpha}\mathscr{V}(K)$ for any $\lambda>0$ and any $K\in\mathscr{K}_{0}$;

(ii) translation invariant, i.e., $\mathscr{V}(K+x)=\mathscr{V}(K)$ for any $x\in\mathbb{R}^{n}$ and any $K\in\mathscr{K}_{0}$;

(iii) monotone increasing, i.e., $\mathscr{V}^{\frac{1}{\alpha}}(K)\leq\mathscr{V}^{\frac{1}{\alpha}}(L)$ if $K\subseteq L$;

(iv) the Brunn-Minkowski inequality, i.e., for any $\lambda\in[0,1]$ and $K,L\in\mathscr{K}_{0}$,

[TABLE]

with equality if and only if $K$ and $L$ are homothetic to each other;

(v) there exists a nonzero finite measure $S_{\mathscr{V}}(K,\cdot)$ on $S^{n-1}$, such that, for any $L\in\mathscr{K}_{0}$,

[TABLE]

and $S_{\mathscr{V}}(K,\cdot)$ is not concentrated on any hemisphere of $S^{n-1}$. Moreover, the convergence of $K_{i}\rightarrow K$ with respect to the Hausdorff metric implies that $S_{\mathscr{V}}(K_{i},\cdot)$ converges weakly to $S_{\mathscr{V}}(K,\cdot)$.

Besides the $p$-capacity and the volume, there are many other functionals on $\mathscr{K}_{0}$ satisfying the conditions in Definition 2.1, such as $\tau(K),$ the torsional rigidity of $K$, whose definition is given by (see e.g. [6]):

[TABLE]

where $W^{1,2}(\text{int}K)$ refers to the Sobolev space of the functions in $L^{2}(\text{int}K)$ whose first order weak derivatives belong to $L^{2}(\text{int}K),$ and $W^{1,2}_{0}(\text{int}K)$ denotes the closure of $C^{\infty}_{c}(\text{int}K)$ in the Sobolev space $W^{1,2}(\text{int}K).$ By the definition of torsional rigidity, one can easily get that $\tau(K)\leq\tau(L)$ if $K\subseteq L.$ Moreover, for any $K\in\mathscr{K}_{0},$

[TABLE]

The torsional rigidity satisfies the Brunn-Minkowski inequality, i.e., for any $\lambda\in[0,1]$ and $K,L\in\mathscr{K}_{0}$,

[TABLE]

with equality if and only if $K$ and $L$ are homothetic to each other; please refer to [6] for more details. For any $K\in\mathscr{K}_{0},$ there exists a unique solution $U_{\tau,K}\in C^{\infty}(\text{int}K)\cap C(K)$ to the following boundary value equation:

[TABLE]

One can define $\mu_{\tau}(K,\cdot),$ a nonnegative Borel measure on $S^{n-1},$ as follows (see e.g. [7]): for any measurable subset $A\subseteq S^{n-1}$,

[TABLE]

It follows from [7, Corollary 1] and [7, Theorem 6] that the measure $\mu_{\tau}(K,\cdot)$ satisfies condition (v) in Definition 2.1.

3 The polar Orlicz-Minkowski problems

In this section, we prove that the polar Orlicz-Minkowski problem (1.14) is solvable under the assumptions $\varphi\in\mathscr{I}$ and $\mu\in\Omega.$ Recall that $\Omega$ is the set of all nonzero finite Borel measures on $S^{n-1}$ that are not concentrated on any hemisphere of $S^{n-1}$, namely, for any $\mu\in\Omega$ and for any $v\in S^{n-1}$, one has

[TABLE]

The continuity of the solutions to the polar Orlicz-Minkowski problems is also provided. For convenience, if $\varphi\in\mathscr{I}$, let

[TABLE]

Clearly, as $\varphi(1)=1$,

[TABLE]

Moreover, due to |(\text{vrad}(L^{\circ})L)^{\circ}|=\Big{|}\displaystyle{\frac{L^{\circ}}{\text{vrad}(L^{\circ})}}\Big{|}=\omega_{n} and $h_{\text{vrad}(L^{\circ})L}=\text{vrad}(L^{\circ})h_{L}$ for any $L\in\mathscr{K}_{0},$ one has

[TABLE]

Theorem 3.1**.**

Let $\mu\in\Omega$ and $\varphi\in\mathscr{I}$. Then there exists a convex body $M\in\mathscr{K}_{0}$ such that $|M^{\circ}|=\omega_{n}$ and

[TABLE]

Moreover, if $\varphi\in\mathscr{I}$ is convex, then $M$ is the unique solution to the polar Orlicz-Minkowski problem (3.1).

Proof.

The fact that $\mu$ is not concentrated on any hemisphere of $S^{n-1},$ together with the monotone convergence theorem, implies that, for any given $v\in S^{n-1}$,

[TABLE]

Thus, there exists an integer $j_{0}\geq 1$ such that

[TABLE]

Let $\{M_{i}\}_{i=1}^{\infty}\subseteq\mathscr{K}_{0}$ be a sequence of convex bodies such that $|M^{\circ}_{i}|=\omega_{n}$ for any $i\geq 1$ and

[TABLE]

Let $R_{i}=\rho_{M_{i}}(u_{i})=\max_{u\in S^{n-1}}\{\rho_{M_{i}}(u)\}$. Obviously, $h_{M_{i}}(u)\geq R_{i}\cdot\langle u,u_{i}\rangle_{+}$ for any $u\in S^{n-1}$ and any $i\geq 1.$ Since $S^{n-1}$ is compact, we can assume $u_{i}\rightarrow v\in S^{n-1}$ as $i\rightarrow\infty$. We now prove $\sup_{i\geq 1}R_{i}<\infty$. This will follow if we can get a contradiction by assuming $R_{i}\rightarrow\infty$ as $i\rightarrow\infty$ (or, more precisely, some subsequence $R_{i_{j}}\rightarrow\infty$ as $j\rightarrow\infty$). To this end, by the monotonicity of $\varphi,$ Fatou’s Lemma, and $h_{M_{i}}(u)\geq R_{i}\cdot\langle u,u_{i}\rangle_{+}$, one has, for any positive constant $C>0$,

[TABLE]

Letting $C\rightarrow\infty$, it follows from (3.2) and $\varphi\in\mathscr{I}$ that $\widehat{\mathscr{G}}_{\varphi}(\mu)\geq\infty$, which is impossible. Hence $\sup_{i\geq 1}R_{i}<\infty$ and $\{M_{i}\}_{i=1}^{\infty}$ is bounded.

By Lemma 2.2, there exists a subsequence of $\{M_{i}\}_{i=1}^{\infty}$ which converges to some convex body $M\in\mathscr{K}_{0}$ with $|M^{\circ}|=\omega_{n}.$ Without loss of generality, we assume $M_{i}\rightarrow M$ as $i\rightarrow\infty$. Thus there exist two positive constants $r_{0}$ and $R_{0}$ such that for any $i\geq 1$ and any $u\in S^{n-1}$,

[TABLE]

From the fact that $\varphi$ is continuous on $[r_{0},R_{0}]$ and the dominated convergence theorem, one has

[TABLE]

Together with (3.3), one has

[TABLE]

Hence $M$ is a solution to the polar Orilicz-Minkowski problem (3.1).

For the uniqueness, let $M_{1}$ and $M_{2}$ be two convex bodies such that $|{M^{\circ}_{1}}|=|{M^{\circ}_{2}}|=\omega_{n}$ and

[TABLE]

Let $M_{0}=\displaystyle{\frac{M_{1}+M_{2}}{2}}.$ Clearly, due to the fact that $t^{-n}$ is strictly convex, (2.1) and (2.2), $\text{vrad}(M^{\circ}_{0})\leq 1$ with $\text{vrad}(M^{\circ}_{0})=1$ if and only if $M_{1}=M_{2}.$ By the strict monotonicity of $\varphi$ and the fact that $\varphi$ is convex, one has

[TABLE]

This implies $\text{vrad}(M^{\circ}_{0})=1$ and hence $M_{1}=M_{2}$. ∎

The following proposition states that the solutions to the polar Orlicz-Minkowski problem (3.1) for discrete measures must be polytopes.

Proposition 3.1**.**

Let $\varphi\in\mathscr{I}$ and $\mu\in\Omega$ be a discrete measure on $S^{n-1}$ whose support $\{u_{1},u_{2},\cdots,u_{m}\}\subseteq S^{n-1}$ is not concentrated on any hemisphere of $S^{n-1}.$ If $M\in\mathscr{K}_{0}$