The Orlicz Electrostatic q-Capacity Minkowski Problem

Abstract

The existence of solutions to the discrete Orlicz electrostatic q-capacity Minkowski problem was given by Ji and Yang when $1<p<2$. Now, we have studied the problem by removing the assumption that the measure $\mu$ does not have a pair of antipodal point masses. By means of approximation, the sufficient condition is given for the existence of solutions to this problem for general measures when $1<q<2$, which is an extension of the $L_p$ electrostatic q-capacity Minkowski problem when $1<q<2$.

1. Introduction

A compact convex set with nonempty interior in ${\mathbb{R}}^n$ is called a convex body. A polytope in ${\mathbb{R}}^n$ is the convex hull of a finite set of points in ${\mathbb{R}}^n$. The collection of convex bodies will be denoted by ${\mathcal{K}}^n$. In convex geometric analysis, the Brunn–Minkowski theory, also known as the mixed volume theory, originates from the variation of volume. The variation of the volume of convex body in ${\mathbb{R}}^n$ gives rise to the classical surface area measure. The concept of mixed volume originated from the variation of volume. For $K,L\in {\mathcal{K}}^n$, their volume variation formula is

$${\displaystyle \frac{d}{dt}V(K+tL){|}_{t=0^{+}}={\int }_{{\mathbb{S}}^{n-1}}{h}_L\left(v\right)d{S}_K\left(v\right),}$$

(1)

where V is the n-dimensional volume (i.e., Lebesgue measure in ${\mathbb{R}}^n$), and $h_L$ is the support function of L. The surface area measure $S_K$ on the unit sphere ${\mathbb{S}}^{n-1}$ can be defined as follows, for the Borel set $\omega \subset {\mathbb{S}}^{n-1}$:

$$S_K\left(\omega \right)={\mathcal{H}}^{n-1}\left(g^{-1}\left(\omega \right)\right),$$

where $g^{-1}:{\mathbb{S}}^{n-1}\to \mathsf{\partial }K$ denotes the inverse Gauss map and ${\mathcal{H}}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure (see Section 2 for specific definitions). The first mixed volume Formula (1) represents a generalization of the volume formula,

$${\displaystyle V\left(K\right)=\frac{1}{n}{\int }_{{\mathbb{S}}^{n-1}}{h}_K\left(v\right)d{S}_K\left(v\right).}$$

(2)

A Minkowski problem pertains to the characterization of a geometric measure that is engendered by convex bodies. The classical Minkowski problem focuses on determining the necessary and/or sufficient conditions on a given finite Borel measure $\mu$ defined on ${\mathbb{S}}^{n-1}$ such that $\mu$ is the surface area measure of a convex body. Minkowski [1,2] first solved this problem in the case of discrete measures, and Aleksandrov [3,4] solved this problem for general measures. Proving the regularity of the solutions to this problem is extremely difficult and has given rise to a series of influential works, (see, e.g., Lewy [5], Nirenberg [6], Caffarelli [7,8]).

The Minkowski problem has a wide range of applications in convex geometry analysis. Now, let us give a brief introduction. Andrews [9] demonstrated Firey’s conjecture that convex surfaces moving by their Gauss curvature become spherical as they contract to points. Lutwak, Yang, and Zhang [10] established a sharp affine $L_p$ Sobolev inequality for functions on Euclidean n-space. Cianchi et al. [11] established an affine Moser–Trudinger inequality and an affine Morrey–Sobolev inequality. Colesanti [12] established the equality conditions for the first eigenvalue of the Laplacian and for the torsional rigidity. Additionally, they proved a Brunn–Minkowski inequality applicable to another class of variational functionals. Srivastava et al. [13] studied a general family of weighted fractional integral operators and utilized this general operator to establish numerous reverse Minkowski inequalities. Chen [14] gave a necessary and sufficient condition for the solvability of the $L_p$ Minkowski problem when $-2<p<0$. Chou and Wang [15] obtained a Kzadan–Warner-type obstruction for the Minkowski problem in centroaffine geometry. Lutwak [16,17] used the $L_p$ Minkowski sum to generalize the Brunn–Minkowski theory to the $L_p$ Brunn–Minkowski theory and to present the $L_p$ Minkowski problem. Lutwak, Yang, and Zhang [18] presented a volume-normalized formulation of the $L_p$ Minkowski problem. Zhu [19] proved the existence of solutions to the $L_p$ Minkowski problem for $p<0$. The logarithmic Minkowski problem prescribes the cone-volume measure (see [20,21]), and Minkowski-type problems prescribe curvature measures; see [22].

The electrostatic q-capacity Minkowski problem is a very significant variant among Minkowski problems. Jerison [23] solved the Minkowski problem for classical electrostatic capacity. Surprisingly, the conditions for the solution of this problem are quite similar to those of the classical Minkowski problem. For $K\in {\mathcal{K}}^n$, the classical electrostatic capacity $C_2\left(K\right)$ of K is defined by

$${\displaystyle C_2\left(K\right)=inf\{{\int }_{{\mathbb{R}}^n}{|\nabla v|}^{2}dx:v\in C_c^{\infty }\left({\mathbb{R}}^n\right)\mathrm{and}v\ge {\chi }_K\},}$$

(3)

where $C_c^{\infty }\left({\mathbb{R}}^n\right)$ represents the collection of smooth functions with compact supports, and ${\chi }_K$ stands for the characteristic function corresponding to K. The equilibrium potential $U=U_K$ of K is the unique solution to the boundary value problem

$$\left\{\begin{array}{cc}\Delta U\left(x\right)=0\hfill & \mathrm{in}{\mathbb{R}}^n\setminus K,\hfill \\ U\left(x\right)=1\hfill & \mathrm{on}\mathsf{\partial }K,\hfill \\ \underset{\left|x\right|\to \infty }{lim}U\left(x\right)=0.\hfill & \end{array}\right.$$

where Δ is the Laplace operator. The electrostatic capacitary measure ${\mu }_2(K,·)$ of K is a finite Borel measure on ${\mathbb{S}}^{n-1}$, defined for Borel $\omega \subset {\mathbb{S}}^{n-1}$ by

$${\displaystyle {\mu }_2(K,\omega )={\int }_{x\in g^{-1}\left(\omega \right)}{|\nabla U\left(x\right)|}^{2}d{\mathcal{H}}^{n-1}\left(x\right),}$$

(4)

Caffarelli et al. [24] proved the uniqueness of solutions to the electrostatic capacity Minkowski problem. The regularity of solutions to this problem depends on the regularity of the solution to the Monge–Ampère equation (see [25]). The Hadamard variational formula is as follows:

$${\displaystyle \frac{d{C}_2(K+tL)}{dt}{|}_{t=0^{+}}={\int }_{{\mathbb{S}}^{n-1}}{h}_L\left(v\right)d{\mu }_2(K,v),}$$

(5)

and the Poincaré capacity formula is as follows:

$${\displaystyle C_2\left(K\right)=\frac{1}{n-2}{\int }_{{\mathbb{S}}^{n-1}}{h}_K\left(v\right)d{\mu }_2(K,v).}$$

(6)

It played a crucial role in Jerison’s research and is highly similar to Formulas (1) and (2).

Jerison’s work has spurred a great deal of subsequent research regarding this topic. Colesanti et al. [26] extended Jerison’s research results to electrostatic q-capacity, where $1<q<n$. For $1<q<n$, the electrostatic q-capacity of a compact set K is defined by

$${\displaystyle C_q\left(K\right)=inf\{{\int }_{{\mathbb{R}}^n}{|\nabla v|}^{q}dx:v\in C_c^{\infty }\left({\mathbb{R}}^n\right)\mathrm{and}v\ge {\chi }_K\}.}$$

(7)

They established the Hadamard variational formula for electrostatic q-capacity,

$${\displaystyle \frac{d{C}_q(K+tL)}{dt}{|}_{t=0^{+}}=(q-1){\int }_{{\mathbb{S}}^{n-1}}{h}_L\left(v\right)d{\mu }_q(K,v),}$$

(8)

and the Poincaré q-capacity formula

$${\displaystyle C_q\left(K\right)=\frac{q-1}{n-q}{\int }_{{\mathbb{S}}^{n-1}}{h}_K\left(v\right)d{\mu }_q(K,v).}$$

(9)

Here, ${\mu }_q(K,·)$ is the electrostatic q-capacitary measure of K.

For $K\in {\mathcal{K}}^n$, the electrostatic q-capacitary measure ${\mu }_q(K,·)$ is a finite Borel measure on ${\mathbb{S}}^{n-1}$, defined for Borel $\omega \subset {\mathbb{S}}^{n-1}$ and $1<q<n$ by

$${\displaystyle {\mu }_q(K,\omega )={\int }_{x\in g^{-1}\left(\omega \right)}{|\nabla U\left(x\right)|}^{q}d{\mathcal{H}}^{n-1}\left(x\right),}$$

(10)

where $g^{-1}:{\mathbb{S}}^{n-1}\to \mathsf{\partial }K$ denotes the inverse Gauss map and U is the q-equilibrium potential of K, see [27] for details.

Colesanti et al. [26] demonstrated the existence and regularity of solutions to the electrostatic q-capacity Minkowski problem when $1<q<2$, as well as the uniqueness of the solution for $1<q<n$. Akman et al. [28] worked out the existence of the solution for $2<q<n$.

For $K\in {\mathcal{K}}^n$, the support function of K is defined by $h_K\left(v\right)=max\{x·v:x\in K\}$, where $x·v$ represents the inner product in ${\mathbb{R}}^n$. The collection of convex bodies which have the origin o in their interiors will be denoted by ${\mathcal{K}}_o^n$. Firey [29] first proposed the concept of the $L_p$ Minkowski sum of convex bodies in 1962. Zou [30] introduced the concept of the $L_p$ electrostatic q-capacitary measure and also started working on the $L_p$ electrostatic q-capacity Minkowski problem.

Suppose $p\in \mathbb{R}$ and $1<q<n$, the $L_p$ electrostatic q-capacitary measure of $K\in {\mathcal{K}}_o^n$ is a finite Borel measure on ${\mathbb{S}}^{n-1}$, defined for any Borel $\omega \subset {\mathbb{S}}^{n-1}$ by

$${\displaystyle {\mu }_{p,q}(K,\omega )={\int }_{\omega }{h}_K{\left(v\right)}^{1-p}d{\mu }_q(K,v).}$$

(11)

The variation in the electrostatic q-capacity functional of the $L_p$ Minkowski sum of convex bodies yields the $L_p$ electrostatic q-capacitary measure. For $p\ge 1$ and $K,L\in {\mathcal{K}}_o^n$ (see [30,31]),

$${\displaystyle \frac{d}{dt}{C}_q\left(K{+}_{p}t{·}_{p}L\right){|}_{t=0^{+}}=\frac{q-1}{p}{\int }_{{\mathbb{S}}^{n-1}}{h}_L^p\left(v\right)d{\mu }_{p,q}(K,v).}$$

(12)

The $L_p$ electrostatic q-capacity Minkowski problem. Given a finite Borel measure $\mu$ on ${\mathbb{S}}^{n-1}$, what are the necessary and sufficient conditions on $\mu$ so that $\mu$ is the $L_p$ electrostatic q-capacitary measure ${\mu }_{p,q}(K,·)$ of a convex body K in ${\mathbb{R}}^n$?

Wang [32] and Xiong [33] studied the discrete logarithmic Minkowski problem for electrostatic q-capacity for $p=0$ and $1<q<n$. Zou [30] solved the $L_p$ electrostatic q-capacity Minkowski problem for $p>1$ and $1<q<n$. Xiong et al. [31] solved the discrete $L_p$ electrostatic q-capacity Minkowski problem for $0<p<1$ and $1<q<2$. This conclusion was augmented for general measures by Feng et al. [34]. Chen [35] proved some geometric properties of $L_p$ q-capacitary measure and a $L_p$ Minkowski inequality for the q-capacity for any fixed $p\ge 1$ and $q>n$. Colesanti and Cuoghi [36] established a Brunn–Minkowski-type inequality, which is related to the n-dimensional logarithmic capacity. Colesanti and Salani [37] proved the Brunn–Minkowski-type inequality, which is related to the electrostatic q-capacity of convex bodies in ${\mathbb{R}}^n$, for $1<q<n$. Lu and Xiong [38] provided sufficient conditions for the existence of solutions to the discrete $L_p$ Minkowski problem for q-capacity when $0<p<1$ and $q\ge n$.

For $K\in {\mathcal{K}}^n$, Akman et al. [39] established the fact that a unique solution $U=U_K$ exists for the boundary value problem of q-Laplace equation:

$$\left\{\begin{array}{cc}\nabla ·\left(\right|\nabla v{|}^{q-2}\nabla v)=0\hfill & \mathrm{in}{\mathbb{R}}^n\setminus K,\hfill \\ v=1\hfill & \mathrm{on}\mathsf{\partial }K,\hfill \\ \underset{\left|x\right|\to \infty }{lim}v\left(x\right)=0.\hfill \end{array}\right.$$

Then,

$${\displaystyle C_q{\left(K\right)}^{\frac{1}{q-1}}=-\underset{\left|x\right|\to \infty }{lim}\frac{v\left(x\right)}{F\left(x\right)},}$$

where

$${\displaystyle F\left(x\right)={\left(n{\omega }_n\right)}^{\frac{1}{1-q}}\left(\frac{q-1}{q-n}\right){\left|x\right|}^{\frac{q-n}{q-1}},}$$

is the functional solution to q-Laplace equation, ${\omega }_n$ denotes the volume of unit ball $B^n$ in ${\mathbb{R}}^n$.

Studying the Orlicz electrostatic q-capacity Minkowski problem is crucial given the classical Minkowski problem and its various extensions. The concept of Orlicz addition was put forward by Gardner et al. [40] and independently by Xi et al. [41]. This was performed with the aim of laying the groundwork for the nascent Orlicz–Brunn–Minkowski theory regarding convex bodies, which stemmed from the research efforts [42,43] of Lutwak, Yang, and Zhang. Böröczky [44] proved a conjecture put forward by Lutwak, Yang, and Zhang regarding the equality case in the Orlicz–Petty projection inequality and provided an essentially optimal stability version. Haberl et al. [45] demonstrated the existence of solutions to the even Minkowski problem. Zhu et al. [46] established the dual Orlicz–Minkowski inequality, as well as the dual Orlicz–Brunn–Minkowski inequality. Xing and Ye [47] found solutions to the dual Orlicz–Minkowski problem for general measures. Ye [48] conducted research on the monotone properties of general affine surfaces in the context of the Steiner symmetrization. Zou and Xiong [49] revealed the relationship existing between the isotropy of measures and the characterization of Orlicz–John ellipsoids.

Hong et al. [50] combined the electrostatic q-capacity for $1<q<n$ with the Orlicz addition of convex domains to develop the electrostatic q-capacitary Orlicz–Brunn–Minkowski theory and proposed the concept of the Orlicz electrostatic q-capacitary measure of convex body. Let $\phi :(0,\infty )\to (0,\infty )$ be a given continuous function. For $K\in {\mathcal{K}}_o^n$ and $1<q<n$, the Orlicz electrostatic q-capacitary measure ${\mu }_{\phi ,q}(K,·)$, of K is defined by

$$d{\mu }_{\phi ,q}(K,·)=\phi \left(h_K\right)d{\mu }_q(K,·).$$

(13)

The Orlicz electrostatic q-capacity Minkowski problem. Given a continuous function $\phi :(0,\infty )\to (0,\infty )$ and a finite Borel measure $\mu$ on ${\mathbb{S}}^{n-1}$, what are the necessary and sufficient conditions on $\mu$ such that there exists a convex body K, satisfying

$${\displaystyle \mu =c{\mu }_{\phi ,q}(K,·),}$$

where $c>0$ is a constant?

Ji [51] demonstrated the existence part of the discrete Orlicz electrostatic q-capacity Minkowski problem for $1<q<2$. Xiong [52] solved the existence of solutions to the Orlicz electrostatic q-capacity Minkowski problem for $q>n$. Hu [53] proved the existence of the Orlicz–Minkowski problem for logarithmic capacity ($q=n$).

Our main goal is to solve the Orlicz electrostatic q-capacity Minkowski problem for $1<q<2$. Let $\mathcal{L}$ be a set of $\phi \in \mathcal{L}$ such that $\phi :(0,\infty )\to (0,\infty )$ is continuously decreasing, and $\phi \left(s\right)\to \infty ,\mathrm{as}s\to 0^{+}$.

Theorem 1.

Suppose $1<q<2$, and μ is a finite discrete Borel measure on ${\mathbb{S}}^{n-1}$, which is not concentrated on any closed hemisphere and $\phi \in \mathcal{L}$. Then, there exists a polytope P containing the origin and a constant $c>0$, such that

$$\mu =c{\mu }_{\phi ,q}(P,·).$$

Remark 1.

Ji [51] first studied the existence of the discrete Orlicz electrostatic q-capacity Minkowski problem for $1<q<2$, we removed the assumption that the measure μ does not have a pair of antipodal point masses, that is, if $\mu \left(\right\{v\left\}\right)>0$, then $\mu \left(\right\{-v\left\}\right)=0$ for $v\in {\mathbb{S}}^{n-1}$.

Theorem 2.

Suppose $1<q<2$, and μ is a finite Borel measure on ${\mathbb{S}}^{n-1}$, which is not concentrated on any closed hemisphere and $\phi \in \mathcal{L}$. Then, there exists a convex body K containing the origin and a constant $c>0$, such that

$$\mu =c{\mu }_{\phi ,q}(K,·).$$

Remark 2.

Let $\phi \left(s\right)=s^{1-p}$ with $p>1$ in Theorem 2, we obtain the existence of solutions to the $L_p$ electrostatic q-capacity Minkowski problem for $p>1$ and $1<q<2$.

This article is structured as follows: In the second section, we introduce some basic facts about convex body and the electrostatic q-capacity. In the third section, we investigate an extreme problem and elucidate the connection between its solution and the solution to the discrete Orlicz electrostatic q-capacity Minkowski problem. Next, we use approximation methods to solve the Orlicz electrostatic q-capacity Minkowski problem for general measures.

2. Preliminaries

For basic facts on convex bodies, we can consult [54,55,56].

We recall that the support function of $K\in {\mathcal{K}}^n$ is defined by $h_K\left(v\right)=max\{x·v:x\in K\}$. Write $S(K,·)$ for the classical surface area measure of K. For a set $K\subset {\mathbb{R}}^n$, $intK$ and $\mathsf{\partial }K$ denotes the interior and boundary of K, respectively. The Hausdorff metric of $K,L\in {\mathcal{K}}^n$ is defined by

$${\delta }_H(K,L)=max\{|h_K\left(v\right)-h_L\left(v\right)|:v\in {\mathbb{S}}^{n-1}\}.$$

The outer unit normal vector to $\mathsf{\partial }K$ at x, denoted by $g\left(x\right)$, is well defined for ${\mathcal{H}}^{n-1}$ almost all $x\in \mathsf{\partial }K$. The map $g:\mathsf{\partial }K\to {\mathbb{S}}^{n-1}$ is called the Gauss map of K. For $\omega \subset {\mathbb{S}}^{n-1}$, let

$$g^{-1}\left(\omega \right)=\{x\in \mathsf{\partial }K:g\left(x\right)\mathrm{is}\mathrm{defined}\mathrm{and}g\left(x\right)\in \omega \}.$$

If Borel set $\omega \subset {\mathbb{S}}^{n-1}$, then $g^{-1}\left(\omega \right)$ is ${\mathcal{H}}^{n-1}$-measurable (see [56], Chapter 2). For $K\in {\mathcal{K}}^n$, the support hyperplane $H(K,v)$ in the direction $v\in {\mathbb{S}}^{n-1}$ is defined by

$$H(K,v)=\{x\in {\mathbb{R}}^n:x·v=h(K,v)\}.$$

The support set $F(K,v)$ in the direction $v\in {\mathbb{S}}^{n-1}$ is defined by

$$F(K,v)=K\cap H(K,v).$$

Suppose the unit vectors $v_1,\dots ,v_N$ are not concentrated on any closed hemisphere. Let $P(v_1,\dots ,v_N)$ be the set with $P\in P(v_1,\dots ,v_N)$ such that for fixed $a_1,\dots ,a_N\ge 0$,

$${\displaystyle P=\bigcap _{k=1}^N\{x\in {\mathbb{R}}^n:x·v_k\le a_k\}.}$$

If $P\in P(v_1,\dots ,v_N)$, then P has at most N facets (i.e., ($n-1$)-dimensional faces) and the collection of the unit outer normals of P is a subset of $\{v_1,\dots ,v_N\}$. Let $P_N(v_1,\dots ,v_N)$ be the subset of $P(v_1,\dots ,v_N)$ such that a polytope $P\in P_N(v_1,\dots ,v_N)$, if $P\in P(v_1,\dots ,v_N)$ and P has exactly N facets.

For nonnegative continuous function f defined on ${\mathbb{S}}^{n-1}$, the Aleksandrov body $\left[f\right]$ is defined by

$${\displaystyle \left[f\right]=\bigcap _{u\in {\mathbb{S}}^{n-1}}\{x\in {\mathbb{R}}^n:x·u\le f\left(v\right)\}.}$$

This collection is also called the Wulff shape associated with f. Obviously, $\left[f\right]\in {\mathcal{K}}_o^n$ and $K=\left[h_K\right]$ for every $K\in {\mathcal{K}}_o^n$.

Some essential facts on the electrostatic q-capacity and the electrostatic q-capacitary measure are needed [23,26,30,31,38].

Let $K\in {\mathcal{K}}^n$ and $1<q<n$. The electrostatic q-capacity $C_q\left(K\right)$ is positively homogeneous of degree $n-q$, and the electrostatic q-capacitary measure ${\mu }_q(K,·)$ is positively homogeneous of degree $n-q-1$, that is,

$$C_q\left(tk\right)=t^{n-q}{C}_q\left(k\right),{\mu }_q(tK,·)=t^{n-q-1}{\mu }_q(K,·)\mathrm{for}t>0.$$

The electrostatic q-capacity $C_q\left(K\right)$ and the electrostatic q-capacitary measure ${\mu }_q(K,·)$ are translation invariant, that is,

$$C_q(K+x)=C_q\left(K\right),{\mu }_q(K+x,·)={\mu }_q(K,·)\mathrm{for}x\in {\mathbb{R}}^n.$$

The functional $C_q(·)$ is continuous on ${\mathcal{K}}^n$ with respect to the Hausdorff metric. The electrostatic q-capacitary measure ${\mu }_q(K,·)$ is absolutely continuous with respect to the surface area measure $S(K,·)$.

If $K,L\in {\mathcal{K}}^n$ and $K\subseteq L$, then $C_q\left(K\right)\le C_q\left(L\right)$.

If $K_j,K\in {\mathcal{K}}^n$ and $K_j\to K$, then ${\mu }_q(K_j,·)\to {\mu }_q(K,·)$ weakly as $j\to \infty$.

The following two lemmas are needed. For their proofs, we can refer to [26] and [57], respectively.

Lemma 1.

If $K\in {\mathcal{K}}^n$ and $1<q<n$, then

$${\displaystyle V{\left(K\right)}^{\frac{n-q}{n}}\le {\left(\frac{q-1}{n-q}\right)}^{q-1}{\left(n{\omega }_n\right)}^{-1}{C}_q\left(K\right),}$$

where ${\omega }_n$ denotes the n-dimensional volume of unit ball ${\mathcal{B}}^n$.

Lemma 2.

Let $I\subset \mathbb{R}$ be an interval containing 0 in its interior, and assume that $h_t\left(v\right)=h(t,v):I\times {\mathbb{S}}^{n-1}\to (0,\infty )$ is continuous, such that the convergence

$${\displaystyle h^{\prime }(0,v)=\underset{t\to \infty }{lim}\frac{h(t,v)-h(0,v)}{t}}$$

is uniform on ${\mathbb{S}}^{n-1}$. Then,

$${\displaystyle \frac{d{C}_q\left(\left[h_t\right]\right)}{dt}{|}_{t=0}=(q-1){\int }_{{\mathbb{S}}^{n-1}}{h}^{\prime }(0,v)d{\mu }_q(\left[h_0\right],v).}$$

3. An Extremal Problem

We recall that $\mathcal{L}$ be a set of $\phi \in \mathcal{L}$ such that $\phi :(0,\infty )\to (0,\infty )$ is continuous decreasing and $\phi \left(s\right)\to \infty ,\mathrm{as}s\to 0^{+}$. Let $\phi \in \mathcal{L}$. For $t>0$, define the function $\varphi$ by

$$\varphi \left(t\right)={\int }_0^t{\displaystyle \frac{1}{\phi \left(s\right)}}ds.$$

(14)

Suppose $\mu$ is the discrete Borel measure on ${\mathbb{S}}^{n-1}$ such that

$${\displaystyle \mu =\sum _{k=1}^N{\alpha }_k{\delta }_{v_k},}$$

where ${\alpha }_1,\dots ,{\alpha }_N\in (0,\infty )$, the unit vectors $v_1,\dots ,v_N$ are in general position and ${\delta }_{v_k}$ denotes the delta measure. For $P\in P(v_1,\dots ,v_N)$ containing the origin, define the function ${\mathsf{\Phi }}_{\mu }\left(P\right):P\to \mathbb{R}$ by

$${\displaystyle {\mathsf{\Phi }}_{\mu }\left(P\right)=\sum _{k=1}^N{\alpha }_k\varphi \left(h_P\left(v_k\right)\right)={\int }_{{\mathbb{S}}^{n-1}}\varphi \left(h_P\left(v_k\right)\right)d\mu \left(v\right).}$$

(15)

In the following, we consider an extreme problem for the functional ${\mathsf{\Phi }}_{\mu }$:

$$inf\{{\mathsf{\Phi }}_{\mu }\left(P\right):P\in P(v_1,\dots ,v_N),dimP=n,C_q\left(P\right)=1\},$$

(16)

and aim to show that the extremal problem (15) is closely connected with our concerned discrete Orlicz electrostatic q-capacity Minkowski problem.

Lemma 3.

If μ is a discrete Borel measure on ${\mathbb{S}}^{n-1}$ whose support set is in general position, then there exists a polytope P solving extremal problem (16).

Proof. 

Let

$$\gamma =inf\{{\mathsf{\Phi }}_{\mu }\left(P\right):P\in P(v_1,\dots ,v_N),dimP=n,C_q\left(P\right)=1\}<\infty .$$

Take a minimizing sequence ${\left\{P_i\right\}}_i$ such that $P_i\in P(v_1,\dots ,v_N),C_q\left(P\right)=1$ and ${\displaystyle \underset{i\to \infty }{lim}{\mathsf{\Phi }}_{\mu }\left(P\right)=\gamma .}$

We demonstrate that ${\left\{P_i\right\}}_i$ is bounded. We argue by contradiction. Assume that $P_i$ is unbounded. Given that the unit vectors $v_1,\dots ,v_N$ are in general position, based on the proof of Zhu [21], we can conclude that $V\left(P_i\right)$ is unbounded. However, from Lemma 1 and the fact that $C_q\left(P_i\right)=1$, we obtain

$$V(P_i)\le {\left({\displaystyle \frac{q-1}{n-q}}\right)}^{{\displaystyle \frac{n(q-1)}{n-q}}}{(n{\omega }_n^{{\displaystyle \frac{q}{n}}})}^{-{\displaystyle \frac{n}{n-q}}},$$

which contradicts the fact that $V\left(P_i\right)$ is unbounded. Therefore, ${\left\{P_i\right\}}_i$ is bounded.

According to the Blaschke selection theorem, ${\left\{P_{i_j}\right\}}_j$ is a convergent subsequence of ${\left\{P_i\right\}}_i$ converging to a polytope P.

In the following, we show that $dimP=n$ by contradiction.

If $dimP=n-1$, then there exists a $v_{i_0}\in \{v_1,\dots ,v_N\}$ such that $P\subseteq v_{i_0}^{\perp }$. Thus, $S(P,·)$ is concentrated at the points $\{\pm v_{i_0}\}$. Since the electrostatic q-capacitary measure ${\mu }_q(P,·)$ is absolutely continuous with respect to $S(P,·)$, we have the electrostatic q-capacitary measure ${\mu }_q(P,·)$ also concentrated at the points $\{\pm v_{i_0}\}$. From the fact that $h_p(\pm v_{i_0})=0$ and (9), we obtain $C_q\left(P\right)=0$, which contradicts the fact that $C_q\left(P\right)=1$.

If $dimP\le n-2<n-q$, it follows from ([58], p. 179) that $C_q\left(P\right)=0$, which contradicts the fact that $C_q\left(P\right)=1$.

Therefore, $dimP=n$. This completes the proof. □

Lemma 4.

If the polytope P is a solution to the extremal problem (16), then P contains the origin as an interior point.

Proof. 

From Colesanti et al. ([26], Lemma 2.18), there exists a constant $d=d(n,q,R)$ with $1\le d<\infty$, such that $|\nabla U\left(x\right)|\ge d^{-1}$ almost everywhere on $\mathsf{\partial }P$ with respect to $d{\mathcal{H}}^{n-1}$. Then, by (10) we obtain

$${\displaystyle {\mu }_q(P,\omega )={\int }_{g^{-1}\left(\omega \right)}{|\nabla U\left(x\right)|}^{q}d{\mathcal{H}}^{n-1}\left(x\right)\ge d^{-q}S(P,\omega ).}$$

Now, we prove that the origin is an interior point of P. We argue by contradiction. Suppose that $o\in \mathsf{\partial }P$. Without loss of generality, let the support set $F(P,v_{i_0})$ is a facet of P and $h_P\left(v_{i_0}\right)=0$, where $v_{i_0}\in \{v_1,\dots ,v_N\}$. For sufficiently small $\epsilon >0$, we write

$${\displaystyle P_{\epsilon }=\bigcap _{k=1}^N\{x\in {\mathbb{R}}^n:x·v_k\le h_P\left(v_k\right)+\epsilon {\delta }_{v_{i_0}}\},}$$

and

$${\displaystyle \tau \left(\epsilon \right)P_{\epsilon }=C_q{\left(P_{\epsilon }\right)}^{-\frac{1}{n-q}}{P}_{\epsilon }.}$$

Then, $C_q\left(\tau \left(\epsilon \right)P_{\epsilon }\right)=1\mathrm{and}\tau \left(\epsilon \right)P_{\epsilon }\to P$, as $\epsilon \to 0^{+}$. According to the construction of $P_{\epsilon }$, we obtain that

$$h_{P_{\epsilon }}\left(v_k\right)=h_P\left(v_k\right)+\epsilon {\delta }_{v_{i_0}},k=1,\dots ,N.$$

From (14), (15), Lemma 2, the fact that $h_P\left(v_{i_0}\right)=0$ and ${\mu }_q(P,\left\{v_{i_0}\right\})\ge d^{-q}S(P,\left\{v_{i_0}\right\})>0$, we obtain that

$$\begin{array}{cc}& {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}{|}_{\epsilon =0^{+}}{\mathsf{\Phi }}_{\mu }\left(\tau \left(\epsilon \right)P_{\epsilon }\right)}\hfill \\ \\ & {\displaystyle =\sum _{k=1}^N{\alpha }_k{\varphi }^{\prime }\left(h_P\left(v_k\right)\right)({\tau }^{\prime }\left(0\right)h_P\left(v_k\right)+{\delta }_{v_{i_0}})}\hfill \\ \\ & {\displaystyle =\sum _{k=1}^N\frac{{\alpha }_k}{\phi \left(h_P\left(v_k\right)\right)}\left(-\frac{q-1}{n-q}\sum _{j=1}^N{\delta }_{v_{i_0}}{\mu }_q(P,\left\{v_j\right\})h_P\left(v_k\right)+{\delta }_{v_{i_0}}\right)}\hfill \\ \\ & {\displaystyle =\frac{{\alpha }_{i_0}}{\phi \left(h_P\left(v_{i_0}\right)\right)}-\frac{q-1}{n-q}\left(\sum _{k=1}^N\frac{{\alpha }_k{h}_P\left(v_k\right)}{\phi \left(h_P\left(v_k\right)\right)}\right){\mu }_q(P,\left\{v_{i_0}\right\})<0.}\hfill \end{array}$$

Then, there exists a ${\epsilon }_0>0$ such that ${\mathsf{\Phi }}_{\mu }\left(\tau \left({\epsilon }_0\right)P_{{\epsilon }_0}\right)<{\mathsf{\Phi }}_{\mu }\left(P\right)$, which contradicts ${\mathsf{\Phi }}_{\mu }\left(P\right)$ is the minimum. Therefore, P contains the origin as an interior point. □

Lemma 5.

If the polytope P is a solution to the extremal problem (16), then P has exactly N facets whose outer normal vectors are $v_1,\dots ,v_N$.

Proof. 

Without loss of generality, suppose $v_{i_0}\in \{v_1,\dots ,v_N\}$ such that $F(P,v_{i_0})=P\cap H(P,v_{i_0})$ is not a facet of P. For sufficiently small $\epsilon >0$, let

$$P_{\epsilon }=P\cap \{x\in {\mathbb{R}}^n:x·v_{i_0}\le h_P\left(v_{i_0}\right)-\epsilon \}$$

and

$$\tau P_{\epsilon }=\tau \left(\epsilon \right)P_{\epsilon }=C_q{\left(P_{\epsilon }\right)}^{-{\displaystyle \frac{1}{n-q}}}{P}_{\epsilon }.$$

Thus, $C_q\left(\tau P_{\epsilon }\right)=1$ and $\tau P_{\epsilon }\to P$, as $\epsilon \to 0^{+}$. According to the construction of $P_{\epsilon }$, the unit outer normal vectors of $P_{\epsilon }$ and P differ only in the direction of $v_{i_0}$. Therefore, we obtain that

$$h_{P_{\epsilon }}\left(v_k\right)=h_P\left(v_k\right)-\epsilon {\delta }_{v_{i_0}},k=1,\dots ,N.$$

From (14), (15) and Lemma 2, we obtain

$$\begin{array}{cc}& {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}{|}_{\epsilon =0^{+}}{\mathsf{\Phi }}_{\mu }\left(\tau \left(\epsilon \right)P_{\epsilon }\right)}\hfill \\ \\ & {\displaystyle =\sum _{k=1}^N{\alpha }_k{\varphi }^{\prime }\left(h_P\left(v_k\right)\right)({\tau }^{\prime }\left(0\right)h_P\left(v_k\right)-{\delta }_{v_{i_0}})}\hfill \\ \\ & {\displaystyle =\sum _{k=1}^N\frac{{\alpha }_k}{\phi \left(h_P\left(v_k\right)\right)}\left(-\frac{q-1}{n-q}\sum _{j=1}^N{\delta }_{v_{i_0}}{\mu }_q(P,\left\{v_j\right\})h_P\left(v_k\right)-{\delta }_{v_{i_0}}\right)}\hfill \\ \\ & {\displaystyle =-\frac{{\alpha }_{i_0}}{\phi \left(h_P\left(v_{i_0}\right)\right)}-\frac{q-1}{n-q}\left(\sum _{k=1}^N\frac{{\alpha }_k{h}_P\left(v_k\right)}{\phi \left(h_P\left(v_k\right)\right)}\right){\mu }_q(P,\left\{v_{i_0}\right\}).}\hfill \end{array}$$

Since P degenerates in the direction of $v_{i_0}$, we calculate and obtain that ${\mu }_q(P,\left\{v_{i_0}\right\})=0$. Thus

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}{|}_{\epsilon =0^{+}}{\mathsf{\Phi }}_{\mu }\left(\tau \left(\epsilon \right)P_{\epsilon }\right)<0.}$$

Then, there exists a ${\epsilon }_0>0$ such that ${\mathsf{\Phi }}_{\mu }\left(\tau \left({\epsilon }_0\right)P_{{\epsilon }_0}\right)<{\mathsf{\Phi }}_{\mu }\left(P\right)$, which contradicts ${\mathsf{\Phi }}_{\mu }\left(P\right)$ is the minimum. Thus, P has N facets. □

4. Proof of the Main Results

In this section, we first prove the existence of the solutions to the discrete Orlicz electrostatic q-capacity Minkowski problem for $1<q<2$. Then, using the methods and techniques in [52], the existence of the solutions to this problem is obtained for the general measures.

Proof of Theorem 1.

By Lemma 3–5, there exists a polytope $P\in P_N(v_1,\dots ,v_N)$ that contains the origin as an interior point in the solution to the extremal problem (16).

For ${\lambda }_1,\dots ,{\lambda }_N\in \mathbb{R}$ and sufficiently small $\left|t\right|$, let polytope

$${\displaystyle P_t=\bigcap _{k=1}^N\{x\in {\mathbb{R}}^n:x·v_k\le h_P\left(v_k\right)+t{\lambda }_k\}.}$$

Since $v_1,\dots ,v_N$ are unit outer normal vectors of P, there exists a $\epsilon >0$, such that for any $\left|t\right|<\epsilon ,v_1,\dots ,v_N$ are normal vectors of $P_t$. Then,

$$h_{P_t}\left(v_k\right)=h_P\left(v_k\right)+t{\lambda }_k,k=1,\dots ,N.$$

Let

$$\tau \left(t\right)P_t=C_q{\left(P_t\right)}^{-{\displaystyle \frac{1}{n-q}}}{P}_t.$$

By Lemma 2 and $C_q\left(P\right)=1$ yields

$$\begin{array}{cc}\hfill {\tau }^{\prime }\left(0\right)& {\displaystyle =-\frac{1}{n-q}{C}_q{\left(P_t\right)}^{-\frac{1}{n-q}-1}{|}_{t=0}\underset{t\to 0}{lim}\frac{C_q\left(P_t\right)-C_q\left(P\right)}{t}}\hfill \\ \\ & {\displaystyle =-\frac{q-1}{n-q}{\int }_{{\mathbb{S}}^{n-1}}\underset{t\to 0}{lim}\frac{h_{P_t}\left(v\right)-h_P\left(v\right)}{t}d{\mu }_q(P,v)}\hfill \\ \\ & {\displaystyle =-\frac{q-1}{n-q}\sum _{j=1}^N{\lambda }_j{\mu }_q(P,\left\{v_j\right\}).}\hfill \end{array}$$

Since $C_q\left(\tau \left(t\right)P_t\right)=1,\tau \left(t\right)P_t\to P$ as $t\to 0$ and P is the solution to the extremal problem (16), this gives

$$\begin{array}{cc}\hfill 0& {\displaystyle =\frac{\mathsf{\partial }}{\mathsf{\partial }t}{|}_{t=0}{\mathsf{\Phi }}_{\mu }\left(\tau \left(t\right)P_t\right)}\hfill \\ \\ & {\displaystyle =\sum _{k=1}^N{\alpha }_k\frac{\mathsf{\partial }}{\mathsf{\partial }t}{|}_{t=0}\varphi \left(\tau \left(t\right)h_{P_t}\left(v_k\right)\right)}\hfill \\ \\ & {\displaystyle =\sum _{k=1}^N{\alpha }_k{\varphi }^{\prime }\left(\tau \left(t\right)h_{P_t}\left(v_k\right)\right){|}_{t=0}\left[{\tau }^{\prime }\left(t\right)h_{P_t}\left(v_k\right){|}_{t=0}+\tau \left(t\right){\lambda }_k{|}_{t=0}\right]}\hfill \\ \\ & {\displaystyle =\sum _{k=1}^N{\alpha }_k{\varphi }^{\prime }\left(h_P\left(v_k\right)\right)\left[{\tau }^{\prime }\left(0\right)h_P\left(v_k\right)+{\lambda }_k\right].}\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill 0& {\displaystyle =\sum _{k=1}^N\frac{{\alpha }_k}{\phi \left(h_P\left(v_k\right)\right)}\left(-\frac{q-1}{n-q}{h}_P\left(v_k\right)\sum _{j=1}^N{\lambda }_j{\mu }_q(P,\left\{v_j\right\})+{\lambda }_k\right)}\hfill \\ \\ & {\displaystyle =\sum _{k=1}^N\frac{{\alpha }_k{\lambda }_k}{\phi \left(h_P\left(v_k\right)\right)}-\frac{q-1}{n-q}\left(\sum _{k=1}^N\frac{{\alpha }_k{h}_P\left(v_k\right)}{\phi \left(h_P\left(v_k\right)\right)}\right)\left(\sum _{j=1}^N{\lambda }_j{\mu }_q(P,\left\{v_j\right\})\right)}\hfill \\ \\ & {\displaystyle =\sum _{k=1}^N{\lambda }_k\left(\frac{{\alpha }_k}{\phi \left(h_P\left(v_k\right)\right)}-\frac{q-1}{n-q}\left(\sum _{j=1}^N\frac{{\alpha }_j{h}_P\left(v_j\right)}{\phi \left(h_P\left(v_j\right)\right)}\right){\mu }_q(P,\left\{v_k\right\})\right).}\hfill \end{array}$$

Since $a_k\in \mathbb{R},k=1,\dots ,N$ are arbitrary, we conclude that

$${\displaystyle c{\mu }_q(P,\left\{v_k\right\})=\frac{{\alpha }_k}{\phi \left(h_P\left(v_k\right)\right)},k=1,\dots ,N,}$$

where

$${\displaystyle c=\frac{q-1}{n-q}\sum _{j=1}^N\frac{{\alpha }_j{h}_P\left(v_j\right)}{\phi \left(h_P\left(v_j\right)\right)}=\frac{q-1}{n-q}{\int }_{{\mathbb{S}}^{n-1}}\frac{h_P}{\phi \left(h_P\right)}d\mu .}$$

Then, ${\displaystyle c{\mu }_q(P,·)=\frac{d\mu }{\phi \left(h_P\right)},}$ i.e., $\mu =c{\mu }_{\phi ,q}(P,·)$ as desired. □

Now, we focus on the Orlicz electrostatic q-capacity Minkowski problem for general measures.

Proof of Theorem 2.

From ([52], Lemma 4.1), for a given finite Borel measure $\mu$ on ${\mathbb{S}}^{n-1}$, which is not concentrated on any closed hemisphere, we can construct a sequence of finite discrete measures $\left\{{\mu }_i\right\}$ on ${\mathbb{S}}^{n-1}$ such that ${\mu }_i\left({\mathbb{S}}^{n-1}\right)=\mu \left({\mathbb{S}}^{n-1}\right)$ and ${\mu }_i\to \mu$ weakly as $i\to +\infty$. Moreover, $\mathrm{supp}{\mu }_i$ are in a general position. By Theorem 1, for the discrete measure ${\mu }_i$, there exists a polytope $P_i$ that contains the origin as an interior point and a constant $c_i>0$ such that

$${\mu }_i=c_i{\mu }_{\phi ,q}(P_i,·),$$

where

$${\displaystyle c_i=\frac{q-1}{n-q}{\int }_{{\mathbb{S}}^{n-1}}\frac{h_{P_i}}{\phi \left(h_{P_i}\right)}d{\mu }_i\left(v\right).}$$

Moreover, $P_i$ is the solution to the following extreme problem

$$inf\left\{{\int }_{{\mathbb{S}}^{n-1}}\varphi \left(h_P\left(v\right)\right)d{\mu }_i\left(v\right):\mathrm{supp}{S}_P\subseteq \mathrm{supp}{\mu }_i,dimP=n,C_q\left(P\right)=1\right\}.$$

For the subsequent proof, we refer to the methods and techniques in [52]. Firstly, we show that $\left\{P_i\right\}$ is bounded.

We study the following function:

$$x\mapsto {\int }_{{\mathbb{S}}^{n-1}}{(x·v)}_{+}d{\mu }_i\left(v\right).$$

From the fact that

$${((x+y)·v)}_{+}\le {(x·v)}_{+}+{(y·v)}_{+}$$

and the Minkowski integral inequality, we obtain that the function $x\mapsto {\int }_{{\mathbb{S}}^{n-1}}{(x·v)}_{+}d{\mu }_i\left(v\right)$ is convex. Since ${\mu }_i$ is not concentrated on any closed hemisphere, this function is strictly positive for any nonzero x. Therefore, it can be regarded as the support function of a unique convex body that contains the origin as an interior point, say $K_{{\mu }_i}\in {\mathcal{K}}_o^n$. Similarly, $K_{\mu }\in {\mathcal{K}}_o^n$. So, ${\displaystyle m=\underset{{\mathbb{S}}^{n-1}}{min}{h}_{K_{\mu }}>0}$. Since ${\mu }_i\to \mu$, weakly, it follows that $h_{K_{{\mu }_i}}\to h_{K_{\mu }}$ uniformly on ${\mathbb{S}}^{n-1}$. Thus, for sufficiently large i,

$${\int }_{{\mathbb{S}}^{n-1}}{(u·v)}_{+}d{\mu }_i\left(v\right)\ge {\displaystyle \frac{m}{2}}>0,u\in {\mathbb{S}}^{n-1}.$$

Let

$${\displaystyle C_i=\bigcap _{v\in \mathrm{supp}{\mu }_i}\{x\in {\mathbb{R}}^n:x·v\le 1\}}$$

and ${\tau }_i\in {\mathbb{R}}^n$ such that $C_q\left({\tau }_i{C}_i\right)=1$. Since $C_i\supseteq B$, it follows that

$${\tau }_i\le C_q{\left(B\right)}^{-{\displaystyle \frac{1}{n-q}}}:=M.$$

Since $P_i$ is the solution to the extreme problem; $h_{C_i}=1,v\in \mathrm{supp}{\mu }_i$; and $\varphi$ is increasing, we obtain that

$$\begin{array}{cc}\hfill {\displaystyle {\int }_{{\mathbb{S}}^{n-1}}\varphi \left(h_{P_i}\left(v\right)\right)d{\mu }_i\left(v\right)}& {\displaystyle \le {\int }_{{\mathbb{S}}^{n-1}}\varphi \left(h_{{\tau }_i{C}_i}\left(v\right)\right)d{\mu }_i\left(v\right)}\hfill \\ \\ & {\displaystyle =\varphi \left({\tau }_i\right){\mu }_i\left({\mathbb{S}}^{n-1}\right)}\hfill \\ \\ & {\displaystyle \le \varphi \left(M\right){\mu }_i\left({\mathbb{S}}^{n-1}\right).}\hfill \end{array}$$

That is ${\displaystyle \varphi \left(M\right)\ge \frac{1}{{\mu }_i\left({\mathbb{S}}^{n-1}\right)}{\int }_{{\mathbb{S}}^{n-1}}\varphi \left(h_{P_i}\left(v\right)\right)d{\mu }_i\left(v\right)}$. Let $R_i$ be the maximum radius of $P_i$, then we have

$$h_{P_i}\left(v\right)\ge R_i{(u_i·v)}_{+}.$$

Since $\varphi$ is increasing, convex, we obtain that

$$\begin{array}{cc}\hfill M& {\displaystyle \ge {\varphi }^{-1}\left(\frac{1}{{\mu }_i\left({\mathbb{S}}^{n-1}\right)}{\int }_{{\mathbb{S}}^{n-1}}\varphi \left(h_{P_i}\left(v\right)\right)d{\mu }_i\left(v\right)\right)}\hfill \\ \\ & {\displaystyle \ge {\varphi }^{-1}\left(\varphi \left(\frac{1}{{\mu }_i\left({\mathbb{S}}^{n-1}\right)}{\int }_{{\mathbb{S}}^{n-1}}{h}_{P_i}\left(v\right)d{\mu }_i\left(v\right)\right)\right)}\hfill \\ \\ & {\displaystyle =\frac{1}{{\mu }_i\left({\mathbb{S}}^{n-1}\right)}{\int }_{{\mathbb{S}}^{n-1}}{h}_{P_i}\left(v\right)d{\mu }_i\left(v\right)}\hfill \\ \\ & {\displaystyle \ge \frac{1}{{\mu }_i\left({\mathbb{S}}^{n-1}\right)}{\int }_{{\mathbb{S}}^{n-1}}{R}_i{(u_i·v)}_{+}d{\mu }_i\left(v\right).}\hfill \end{array}$$

Then, ${\displaystyle R_i\le \frac{2}{m}M\mu \left({\mathbb{S}}^{n-1}\right)}$. Thence, $\left\{P_i\right\}$ is bounded.

According to the Blaschke selection theorem, the sequence $\left\{P_i\right\}$ has a convergent subsequence, also written as $\left\{P_i\right\}$, such that $P_i\to K$. Then $o\in K$ and $C_q\left(K\right)=1$.

Secondly, we prove that $dimK=n$ by contradiction.

If $dimK=n-1$, which is similar to the proof of Lemma 3, we obtain $C_q\left(K\right)=0$, which is a contradiction.

If $dimK\le n-2<n-q$, it follows from ([58], p. 179) that $C_q\left(K\right)=0$, which is a contradiction.

Therefore, $dimK=n$.

Finally, we show that K is the solution to the Orlicz electrostatic q-capacity Minkowski problem. Since $P_i$ is the solution to this problem for the discrete measure ${\mu }_i$, we have

$${\displaystyle c{\mu }_q(P_i,·)=\frac{d\mu }{\phi \left(h_{P_i}\right)}.}$$

Based on this and $P_i\to K$ and ${\mu }_i\to \mu$ weakly, we obtain ${\mu }_q(P_i,·)\to {\mu }_q(K,·)$, $h_{P_i}\to h_K$ uniformly on ${\mathbb{S}}^{n-1}$ and $c_i\to c$. Then,

$${\displaystyle c{\mu }_q(K,·)=\frac{d\mu }{\phi \left(h_K\right)},}$$

i.e.,

$$\mu =c{\mu }_{\phi ,q}(K,·),$$

as desired. □