© Wuhan University 2025

0 Introduction

The setting for this paper will be $n$-dimensional Euclidean space, ${\mathbb{R}}^n$. A compact convex set with nonempty interior 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 Minkowski problem for electrostatic $q$-capacity is a very significant variant among Minkowski problems, regarding the research and applications of the Minkowski problems, we can refer to Refs. [1-18]. For the $L_p$ Minkowski problem for electrostatic $q$-capacity, we refer to Refs. [19-32] and reference therein. When $q=n$, the $n$-capacity is also known as the logarithmic capacity. The logarithmic capacity in the Euclidean plane ${\mathbb{R}}^{\mathrm{2}}$ has been researched systemically because of its function in two-dimensional potential theory, which is the study of planar harmonic functions in mathematics and mathematical physics (see e.g., Refs. [33-39]); Nevertheless, as a result of its nonlinear nature, the planar logarithmic capacity in higher dimensional extension has gotten comparatively little attention (see Refs. [20,37]).

The collection of convex bodies will be denoted by ${\mathcal{K}}^n$. For $\mathrm{1}<q<n,$ the electrostatic $q$-capacity of a compact set $K$ is defined by (see Ref. [21])

$C_q(K)=\mathrm{i}\mathrm{n}\mathrm{f}\{\underset{\mathit{ℝ}}{\int }|\nabla \mathit{v}{|}^q\mathrm{d}\mathit{x}:\mathit{x}\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)\mathrm{a}\mathrm{n}\mathrm{d}\mathit{v}\ge {\chi }_K\},$(1)

where $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ denotes the set of smooth functions with compact supports, and ${\chi }_K$ is the characteristic function of $K$. For $K\in {\mathcal{K}}^n$, its electrostatic $q$-capacitary measure ${\mu }_q(K,\cdot )$ is a finite Borel measure on ${\mathbb{S}}^{n-\mathrm{1}}$, defined for Borel $\omega \subset {\mathbb{S}}^{n-\mathrm{1}}$ and $q\in (\mathrm{1},n)$ by

${\mu }_q(K,\omega )=\underset{\mathit{x}\in g^{-\mathrm{1}}(\omega )}{\int }|\nabla V(\mathit{x}){|}^q\mathrm{d}{\mathrm{\mathscr{H}}}^{n-\mathrm{1}}(\mathit{x}),$(2)

where $g^{-\mathrm{1}}:{\mathbb{S}}^{n-\mathrm{1}}\to \partial K$ denotes the inverse Gauss map and $V$ is the $q$-equilibrium potential of $K$, see Ref. [38] for details.

Jerison ^[34] solved the Minkowski problem for classical capacity ($q=\mathrm{2}$) and established the necessary and sufficient condition for existence of its solution. It is surprising to note that these conditions are the same as those in the classical Minkowski problem. Caffarelli et al^[39] settled the uniqueness of its solutions, while the regularity depends on the regularity of solutions to the Monge-Ampère equation (see Ref. [40]). Colesanti et al^[21] demonstrated the existence and regularity of the solution for $q\in (\mathrm{1},\mathrm{2})$ as well as the uniqueness of the solution when $q\in (\mathrm{1},n)$. Akman et al^[41] solved the existence of the solution for $q\in (\mathrm{2},n)$.

For $K\in {\mathcal{K}}^n$, the support function of $K$ is defined by $h_K(\mathit{v})=\mathrm{m}\mathrm{a}\mathrm{x}\{\mathit{x}\cdot \mathit{v}:\mathit{x}\in K\}$, where $\mathit{x}\cdot \mathit{v}$ denotes the standard inner product in ${\mathbb{R}}^n$. The support function is positively homogeneous of degree 1, and is usually restricted to ${\mathbb{S}}^{n-\mathrm{1}}$. The collection of convex bodies containing the origin $o$ in their interiors will be denoted by ${\mathcal{K}}_o^n$. Firey^[42] first proposed the concept of the $L_p$ Minkowski sum of convex bodies in the 1962. The $L_p$ $q$-capacitary measure was introduced by Du and Xiong^[28], and they also began to study the $L_p$ Minkowski problem for electrostatic $q$-capacity.

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

${\mu }_{p,q}(K,\omega )=\underset{\omega }{\int }{h}_K{(v)}^{\mathrm{1}-p}\mathrm{d}{\mu }_q(K,v).$(3)

The $L_p$ q-capacitary measure is resulted from the variation of the $q$-capacity functional of the $L_p$ Minkowski sum of convex bodies. For $p\in [\mathrm{1},\mathrm{\infty })$ and $K,L\in {\mathcal{K}}_o^n$ (see Refs. [26,28]),

$\frac{\mathrm{d}}{\mathrm{d}t}{C}_q((\mathrm{1}-t)\cdot K{+}_{p}t\cdot L){|}_{t={\mathrm{0}}^{+}}=\frac{q-\mathrm{1}}{p}\underset{{\mathbb{S}}^{n-\mathrm{1}}}{\int }{h}_L^p(v)\mathrm{d}{\mu }_{p,q}(K,v).$(4)

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

Wang et al ^[25] and Xiong et al ^[36] studied the discrete logarithmic Minkowski problem for $q$-capacity for $p=\mathrm{0}$ and $q\in (\mathrm{1},n)$. Zou et al^[28] solved the $L_p$ Minkowski problem for $q$-capacity for $p\in (\mathrm{1},\mathrm{\infty })$ and $q\in (\mathrm{1},n)$. Xiong et al^[26] solved the $L_p$ discrete Minkowski problem for $q$-capacity for $p\in (\mathrm{0},\mathrm{1})$ and $q\in (\mathrm{1},\mathrm{2})$.This conclusion was augmented for general measures by Feng et al^[23].

For $K\in {\mathcal{K}}^n$, Akman et al^[43] proved that there exists a unique solution $V=V_K$ to the boundary value problem of $n$-Laplace equation:

$\{\begin{array}{ll}\nabla \cdot (|\nabla v{|}^{n-\mathrm{2}}\nabla v)=\mathrm{0}& \mathrm{i}\mathrm{n}{\mathbb{R}}^n\backslash K,\\ v=\mathrm{0}& \mathrm{o}\mathrm{n}\partial K,\\ v(x)=F(x)+a+o(\mathrm{1})& \mathrm{a}\mathrm{s}|x|\to \mathrm{\infty },\end{array}$

where $a\in \mathbb{R}\backslash \{\mathrm{0}\}$ is uniquely determined by $K$, and

$F(x)=(n{\omega }_n{)}^{\frac{\mathrm{1}}{\mathrm{1}-n}}\mathrm{l}\mathrm{o}\mathrm{g}|x|+\mathrm{1}$

is the functional solution to $n$-Laplace equation, ${\omega }_n$ denotes the volume of unit ball $B^n$ in ${\mathbb{R}}^n$. Then, the logarithmic capacity $C_n(K)$ of $K$ is defined by

$C_n(K)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{a}{(n{\omega }_n{)}^{\frac{\mathrm{1}}{\mathrm{1}-n}}}\right).$(5)

Furthermore, the Hadamard variational formula is established:

$\frac{\mathrm{d}{C}_n(K+tL)}{\mathrm{d}t}{|}_{t={\mathrm{0}}^{+}}=(n{\omega }_n{)}^{\frac{\mathrm{1}}{n-\mathrm{1}}}{C}_n(K)\underset{{\mathbb{S}}^{n-\mathrm{1}}}{\int }{h}_L(v)\mathrm{d}{\mu }_n(K,v).$(6)

Henceforth, the logarithmic capacitary measure ${\mu }_n(K,\cdot )$ of $K$ has emerged. By (6) and the positive homogeneity of $C_n(K)$, it follows that

$(n{\omega }_n{)}^{\frac{\mathrm{1}}{\mathrm{1}-n}}=\underset{{\mathbb{S}}^{n-\mathrm{1}}}{\int }{h}_K(v)\mathrm{d}{\mu }_n(K,v).$(7)

Following Xiao^[44], for $K\in {\mathcal{K}}^n$, the logarithmic capacitary measure ${\mu }_n(K,\cdot )$ of $K$ is a finite Borel measure on ${\mathbb{S}}^{n-\mathrm{1}}$, defined for Borel $\omega \subset {\mathbb{S}}^{n-\mathrm{1}}$ by

${\mu }_n(K,\omega )=\frac{\mathrm{1}}{{\omega }_n}\underset{g^{-\mathrm{1}}(\omega )}{\int }|\nabla V(x){|}^n\mathrm{d}{\mathrm{\mathscr{H}}}^{n-\mathrm{1}}(x).$(8)

The Minkowski problem for the logarithmic capacity was introduced by Akman et al^[43], who also solved its existence and uniqueness. For $p\in (\mathrm{0},\mathrm{1})$, Lu et al^[24] extended the $q$-index of the $L_p$ $q$-capacitary measure to $q=n$, and solved its associated $L_p$ Minkowski problem. For more other works related to the $L_p$ Minkowski problem for the logarithmic capacity, we refer to Chen^[5,19].

The study of the Orlicz Minkowski problem for $q$-capacity is of great importance, especially in light of the classical Minkowski problem and its numerous extensions. The Orlicz additions were proposed by Gardner et al^[45] and independently by Xi et al^[46], in order to provide the foundation of the newly initiated Orlicz-Brunn-Minkowski theory for convex bodies starting from the works of Lutwak et al^[47-48]. The Orlicz theory is in great demand and is rapidly developing (see e.g., Refs. [49-54] ). Hong et al^[55] combined the $q$-capacity for $q\in (\mathrm{1},n)$ with the Orlicz addition of convex domains to develop the $q$-capacitary Orlicz-Brunn-Minkowski theory. Ji et al^[56] demonstrated the existence part of the discrete Orlicz Minkowski problem for $q$-capacity when $q\in (\mathrm{1},\mathrm{2})$. Xiong et al^[57] solved the existence of solutions to the Orlicz Minkowski problem for $q$-capacity for $q\in (n,\mathrm{\infty })$. Hu and Li^[58] proved the existence of the Orlicz Minkowski problem for logarithmic capacity $(q=n)$.

The Orlicz Minkowski problem for the logarithmic capacity. Given a function $\varphi$ and finite Borel measure $\mu$ on ${\mathbb{S}}^{n-\mathrm{1}}$, what are the necessary and sufficient conditions on $\mu$ such that there exists a convex body $K$, satisfying

$\alpha \mathrm{d}{\mu }_n(K,\cdot )=\frac{\mathrm{d}\mu }{\phi (h_K)},$

where $\alpha >\mathrm{0}$ is a constant?

Our main goal is to solve the Orlicz Minkowski problem for the logarithmic capacity. It is worth noting that Hu and Li ^[58] first studied the Orlicz Minkowski problem for the logarithmic capacity. However, we consider a different extremal problem, which may contribute to a more concise proof process. Meanwhile, for discrete measures, we remove their restrictive condition that the measure $\mu$ does not have any pair of antipodal point masses, without using an approximation scheme. Additionally, we generalize the Orlicz Minkowski problem for the logarithmic capacity to general measures by applying an approximation scheme.

Theorem 1   Suppose $\mu$ is a finite Borel measure on ${\mathbb{S}}^{n-\mathrm{1}}$ which is not concentrated on any closed hemisphere and that $\phi :(\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty })$ is continuous decreasing and $\phi (s)\to \mathrm{\infty }\mathrm{a}\mathrm{s}s\to {\mathrm{0}}^{+}$. Then there exists a convex body $K$ containing the origin and a constant $\alpha >\mathrm{0}$, such that

$\alpha \mathrm{d}{\mu }_n(K,\cdot )=\frac{\mathrm{d}\mu }{\phi (h_K)}.$

This article is structured as follows. In Section 1, we introduce some basic facts about convex body and the logarithmic capacity. In Section 2, we investigate an extreme problem and elucidate the connection between its solution and the solution to the Orlicz Minkowski problem for the logarithmic capacity, and complete the proof of the main result.

1 Preliminaries

For basic facts on convex bodies, we recommend the books by Gardner^[59], Gruber^[60], Schneider^[61].

Denote $B^n$ is the standard unit ball of ${\mathbb{R}}^n$ and denote its surface by ${\mathbb{S}}^{n-\mathrm{1}}$. For a set $A\subset {\mathbb{R}}^n$, $\mathrm{i}\mathrm{n}\mathrm{t}A$ and $\partial A$ denote the interior and boundary of $A$, respectively. For a compact convex set $K$,

$d(K)=\mathrm{s}\mathrm{u}\mathrm{p}\{|x-y|:x,y\in K\}$

is the diameter of $K$. Write $S(K,·)$ for the classical surface area measure of $K$. For $K\in {\mathcal{K}}^n$, the support hyperplane $H(K,\mathit{v})$ in the direction $\mathit{v}\in {\mathbb{S}}^{n-\mathrm{1}}$ is defined by

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

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

$F(K,\mathit{v})=K\bigcap H(K,\mathit{v}).$

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

$[f]=\underset{u\in {\mathbb{S}}^{n-\mathrm{1}}}{\cap }\{\mathit{x}\in {\mathit{ℝ}}^n:\mathit{x}\cdot \mathit{v}\le f(\mathit{v})\}.$

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

Suppose the unit vectors ${\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N$ are not concentrated on any closed hemi-sphere. Let $P({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N)$ be the set with $P\in P({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N)$ such that for fixed ${\alpha }_{\mathrm{1}},\cdots ,{\alpha }_N\ge \mathrm{0}$,

$P={\displaystyle \underset{j=\mathrm{1}}{\overset{N}{\cap }}}\{\mathit{x}\in {\mathit{ℝ}}^n:\mathit{x}\cdot {\mathit{v}}_j\le {\alpha }_j\}.$

If $P\in P({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N)$, then $P$ has at most $N$ facets (i.e., $(n-\mathrm{1})$-dimensional faces) and the collection of the unit outer normals of $P$ is a subset of $\{{\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N\}$. Let $P_N({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N)$ be the subset of $P({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N)$ such that a polytope $P\in P_N({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N)$ if $P\in P({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N)$ and $P$ has exactly $N$ facets. A finite subset $U$ of ${\mathbb{S}}^{n-\mathrm{1}}$ is said to be in general position if $U$ is not concentrated on any closed hemisphere of ${\mathbb{S}}^{n-\mathrm{1}}$ and any $k$ elements of $V$, $\mathrm{1}\le k\le n$, are linearly independent. It is of great importance that the outer unit normals of the polytope are in general position, since any convex body can be approximated by these polytopes.

In the following, we list some properties about the logarithmic capacity. For more details, see, e.g., Refs. [5,19,24,43-44,62].

Let $K\in {\mathcal{K}}^n$. First, by the definition of the logarithmic capacity, $C_n(K)$ is positively homogeneous of degree 1, that is, $C_n(tk)=t{C}_n(k)$, for $t>\mathrm{0}$. Second, the functional $C_n$ is translation invariant, namely, $C_n(K+\mathit{x})=C_n(K)$, for $\mathit{x}\in {\mathbb{R}}^n$. Third, if a sequence of compact convex sets $\{K_i{\}}_{(i=\mathrm{1})}^{(\mathrm{\infty })}$ converges to a compact set $K$, then either $K$ is a single point or $\underset{i\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{C}_n(K_i)=C_n(K)>\mathrm{0}$.

Since ${\mu }_q(K,\cdot )$ is absolutely continuous with respect to the surface measure $S(K,\cdot )$. The logarithmic capacitary measure ${\mu }_n$ is positively homogeneous of degree $-\mathrm{1}$, that is, ${\mu }_n(tK,\cdot )=t^{-\mathrm{1}}{\mu }_n(K,\cdot ),\mathrm{f}\mathrm{o}\mathrm{r}t>\mathrm{0}$.

The following Lemma will be used. For its proof, we can refer to the Theorem 3.2 in Ref. [24].

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

$h^{\text{'}}(\mathrm{0},\mathit{v})=\underset{t\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{h(t,\mathit{v})-h(\mathrm{0},\mathit{v})}{t}$

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

$\frac{\mathrm{d}{C}_n([h_t])}{\mathrm{d}t}{|}_{t=\mathrm{0}}=(n{\omega }_n{)}^{\frac{\mathrm{1}}{n-\mathrm{1}}}{C}_n([h_{\mathrm{0}}])\underset{{\mathbb{S}}^{n-\mathrm{1}}}{\int }{h}^{\text{'}}(\mathrm{0},\mathit{v})\mathrm{d}{\mu }_n([h_{\mathrm{0}}],\mathit{v}).$(9)

2 Proof of Main Result

In this section, we first study the Orlicz Minkowski problem for logarithmic capacity for discrete measures and then study the general measures by means of approximation methods.

Suppose that $\phi :(\mathrm{0},\mathrm{\infty })\to (\mathrm{0},\mathrm{\infty })$ be a continuous, decreasing function and satisfying $\phi (s)\to \mathrm{\infty }$, as $s\to {\mathrm{0}}^{+}$. Define the function $\varphi$ by

$\varphi (t)={\int }_{\mathrm{0}}^t\frac{\mathrm{1}}{\phi (s)}\mathrm{d}s,t>\mathrm{0}.$(10)

Since $\phi$ is decreasing, it follows that the derivative of $\varphi$ is increasing and therefore $\varphi$ is convex.

Suppose $\mu$ is a discrete Borel measure on ${\mathbb{S}}^{n-\mathrm{1}}$ such that $\mu ={\displaystyle \sum _{k=\mathrm{1}}^N}{c}_k{\delta }_{{\mathit{v}}_k}$, where $c_{\mathrm{1}},\cdots ,c_N\in (\mathrm{0},\mathrm{\infty })$ and ${\delta }_{{\mathit{v}}_k}$ denotes the delta measure that is concentrated at the point ${\mathit{v}}_k$. For a polytope $Q$ containing the origin, define

${\Psi }_{\mu }(Q)=\underset{{\mathbb{S}}^{n-\mathrm{1}}}{\int }\varphi (h_Q(\mathit{v}))\mathrm{d}\mu (\mathit{v})={\displaystyle \sum _{k=\mathrm{1}}^N}{c}_k\varphi (h_Q({\mathit{v}}_k)).$(11)

In the following, we study the extremal problem

$\mathrm{i}\mathrm{n}\mathrm{f}\{{\Psi }_{\mu }(Q):Q\in P({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N),C_n(Q)=\mathrm{1}\},$(12)

and show the relationship between the extreme problem and the Orlicz Minkowski problem for logarithmic capacity for the discrete measure $\mu$.

We focus on the existence of the extreme problem for ${\Psi }_{\mu }$, and demonstrate that there is a solution to the Orlicz Minkowski problem for logarithmic capacity for the discrete measure $\mu$. To complete the proof of existence, Lemma 2 will be required, for its proof can refer to the Theorem 4.3 in Ref. [13].

Lemma 2   If the unit vectors ${\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N$ are in general position,$P_i\in P({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N)$ with $o\in P$ and $d(P_i)$ is not bounded, then $V(P_i)$ is not bounded.

Lemma 3   Suppose $\mu$ is a finite discrete Borel measure on ${\mathbb{S}}^{n-\mathrm{1}}$. If the support set of $\mu$ is in general position, then there exists an $n$-dimensional polytope $P$ solving the extremal problem (12).

Proof   Let

$\beta =\mathrm{i}\mathrm{n}\mathrm{f}\{{\Psi }_{\mu }(Q):Q\in P({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N),C_n(Q)=\mathrm{1}\}.$

Take a minimizing sequence $\{P_i{\}}_i$ for the extremal problem (12), such that $P_i\in P({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N),C_n(P_i)=\mathrm{1}$ and

$\underset{i\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\Phi }_{\mu }(P_i)=\beta .$

We show that the sequence $\{P_i{\}}_i$ is bounded. Assume that $\{P_i{\}}_i$ is not bounded, by Lemma 2, we have $V(P_i)$ is not bounded. On the other hand, from the fact that $C_n(P_i)=\mathrm{1}$ and the isocapacitary inequality for logarithmic capacity (Theorem 4.1 in Ref. [44]), i.e.,

${(\frac{V(P_i)}{{\omega }_n})}^{\frac{\mathrm{1}}{n}}\le C_n(P_i),$

we have $V(P_i)\le {\omega }_n$, which is a contradiction. Thus, $\{P_i{\}}_i$ is bounded.

According to the Blaschke selection theorem, $\{P_{i_j}{\}}_j$ is a convergent subsequence of $\{P_i{\}}_i$ that converges to $P$.

In the following, we show $\mathrm{d}\mathrm{i}\mathrm{m}P=n$.

If $\mathrm{d}\mathrm{i}\mathrm{m}P=n-\mathrm{1}$, then there exists a ${\mathit{v}}_{i_{\mathrm{0}}}\in \{{\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N\}$ such that $P\subseteq {\mathit{v}}_{i_{\mathrm{0}}}^{\perp }$ and thus,

$h_P({\mathit{v}}_{i_{\mathrm{0}}})=h_P(-{\mathit{v}}_{i_{\mathrm{0}}})=\mathrm{0}.$

Then the surface measure $S(P,\cdot )$ is concentrated at the points $\{\pm {\mathit{v}}_{i_{\mathrm{0}}}\}$. Since the logarithmic capacitary measure ${\mu }_n(P,\cdot )$ is absolutely continuous with respect to $S(P,\cdot )$, so ${\mu }_n(P,\cdot )$ is also concentrated at the points $\{\pm {\mathit{v}}_{i_{\mathrm{0}}}\}$. By (5) and (7), we get $C_n(P)=\mathrm{0}\mathrm{o}\mathrm{r}{C}_n(P)=\mathrm{\infty }$, which contradicts that $C_n(P)=\mathrm{1}$.

If $\mathrm{d}\mathrm{i}\mathrm{m}P\le n-\mathrm{2}$, then $S(P,\cdot )\equiv \mathrm{0}$. Since the logarithmic capacitary measure ${\mu }_n(P,\cdot )$ is absolutely continuous with respect to $S(P,\cdot )$, we have ${\mu }_n(P,\cdot )\equiv \mathrm{0}$. By (5) and (7), we get $C_n(P)=\mathrm{0}\mathrm{o}\mathrm{r}{C}_n(P)=\mathrm{\infty }$, which is a contradiction.

Therefore, $\mathrm{d}\mathrm{i}\mathrm{m}P=n$. This completes the proof.

Lemma 4   Let $K$ be a convex body in ${\mathbb{R}}^n$ with $o\in K\subseteq B(o,r)$. Then there exists a constant $c>\mathrm{0}$, such that for Borel set $\omega \subseteq {\mathbb{S}}^{n-\mathrm{1}},{\mu }_n(K,\omega )\ge cS(K,\omega )$.

Proof   According to Theorem 3.2 of Xiao^[62], there exists a constant $d>\mathrm{0}$ depending only on $r$ and $n$, such that $|\nabla V(x)|\ge d$ almost everywhere on $\partial K$ with respect to $\mathrm{d}{\mathrm{\mathscr{H}}}^{n-\mathrm{1}}$. Then, by (8) we get

$\begin{array}{l}{\mu }_n(K,\omega )=\frac{\mathrm{1}}{{\omega }_n}\underset{g^{-\mathrm{1}}(\omega )}{\int }|\nabla V(x){|}^n\mathrm{d}{\mathrm{\mathscr{H}}}^{n-\mathrm{1}}(x)\ge \frac{d^n}{{\omega }_n}S(K,\omega )\\ =cS(K,\omega ),\end{array}$

where $c=\frac{d^n}{{\omega }_n}$.

Lemma 5   The solution $P$ contains the origin in its interior.

Proof   Suppose that $o\in \partial P$. Without loss of generality, let $F(P,{\mathit{v}}_{i_{\mathrm{0}}})$ be a facet of $P$, where ${\mathit{v}}_{i_{\mathrm{0}}}\in \{{\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N\}$. Thus $h_P({\mathit{v}}_{i_{\mathrm{0}}})=\mathrm{0}$. For sufficiently small $t>\mathrm{0}$, we write

$P_t={\displaystyle \underset{k=\mathrm{1}}{\overset{N}{\cap }}}\{\mathit{x}\in {\mathit{ℝ}}^n:\mathit{x}\cdot {\mathit{v}}_k\le h_P({\mathit{v}}_k)+t{\delta }_{{\mathit{v}}_{i_{\mathrm{0}}}}\},$

$\tau (t)P_t=\frac{P_t}{C_n(P_t)}.$

Then $C_n(\tau (t)P_t)=\mathrm{1},\tau (t)P_t\to P$, as $t\to {\mathrm{0}}^{+}$, and $h_{P_t}({\mathit{v}}_k)=h_P({\mathit{v}}_k)+t{\delta }_{{\mathit{v}}_{i_{\mathrm{0}}}},k=\mathrm{1},\cdots ,N$.

From (10), (11), Lemma 1, the fact that $h_P({\mathit{v}}_{i_{\mathrm{0}}})=\mathrm{0}$ and ${\mu }_n(P,\{{\mathit{v}}_{i_{\mathrm{0}}}\})\ge cS(P,\{{\mathit{v}}_{i_{\mathrm{0}}}\})>\mathrm{0}$ by Lemma 4, we have

$\begin{array}{l}\frac{\partial }{\partial t}{|}_{t={\mathrm{0}}^{+}}{\Psi }_{\mu }(\tau (t)P_t)\\ ={\displaystyle \sum _{k=\mathrm{1}}^N}{c}_k{\varphi }^{\text{'}}(h_P({\mathit{v}}_k))({\tau }^{\text{'}}(\mathrm{0})h_P({\mathit{v}}_k)+{\delta }_{{\mathit{v}}_{i_{\mathrm{0}}}})\\ ={\displaystyle \sum _{k=\mathrm{1}}^N}{c}_k\frac{\mathrm{1}}{\phi (h_P({\mathit{v}}_k))}\left(-(n{\omega }_n{)}^{\frac{\mathrm{1}}{n-\mathrm{1}}}{\displaystyle \sum _{j=\mathrm{1}}^N}{\delta }_{{\mathit{v}}_{i_{\mathrm{0}}}}{\mu }_n(P,\{{\mathit{v}}_j\})h_P({\mathit{v}}_k)+{\delta }_{{\mathit{v}}_{i_{\mathrm{0}}}}\right)\\ =\frac{c_{i_{\mathrm{0}}}}{\phi (h_P({\mathit{v}}_{i_{\mathrm{0}}}))}-(n{\omega }_n{)}^{\frac{\mathrm{1}}{n-\mathrm{1}}}\left({\displaystyle \sum _{k=\mathrm{1}}^N}\frac{c_k{h}_P({\mathit{v}}_k)}{\phi (h_P({\mathit{v}}_k))}\right){\mu }_n(P,\{{\mathit{v}}_{i_{\mathrm{0}}}\})<\mathrm{0}.\end{array}$

Then there exists a $t_{\mathrm{0}}>\mathrm{0}$ such that ${\Psi }_{\mu }(\tau (t_{\mathrm{0}})P_{t_{\mathrm{0}}})<{\Psi }_{\mu }(P)$, which contradicts ${\Psi }_{\mu }(P)$ is the minimum. Therefore, the origin must be inside the solution $P$.

Lemma 6   The solution $P$ has exactly $N$ facets whose outer normal vectors are ${\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N$.

Proof   We employ the method of proof by contradiction. Suppose ${\mathit{v}}_{i_{\mathrm{0}}}\in \{{\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N\}$ such that the support set $F(P,{\mathit{v}}_{i_{\mathrm{0}}})=P\bigcap H(P,{\mathit{v}}_{i_{\mathrm{0}}})$ is not a facet of $P$. For sufficiently small $t>\mathrm{0}$, let

$P_t=P\bigcap \{\mathit{x}\in {\mathbb{R}}^n:\mathit{x}\cdot {\mathit{v}}_{i_{\mathrm{0}}}\le h_P({\mathit{v}}_{i_{\mathrm{0}}})-t\}$

and

$\tau P_t=\tau (t)P_t=C_n(P_t{)}^{-\mathrm{1}}{P}_t.$

Thus, $C_n(\tau P_t)=\mathrm{1}$ and $\tau P_t\to P$, as $t\to {\mathrm{0}}^{+}$ and $h_{P_t}({\mathit{v}}_k)=h_P({\mathit{v}}_k)-t{\delta }_{{\mathit{v}}_{i_{\mathrm{0}}}},k=\mathrm{1},\cdots ,N$.

From (10), (11) and Lemma 1, we get $\frac{\partial }{\partial t}{|}_{t={\mathrm{0}}^{+}}{\Psi }_{\mu }(\tau (t)P_t)$

$={\displaystyle \sum _{k=\mathrm{1}}^N}{c}_k{\varphi }^{\text{'}}(h_P({\mathit{v}}_k))({\tau }^{\text{'}}(\mathrm{0})h_P({\mathit{v}}_k)-{\delta }_{{\mathit{v}}_{i_{\mathrm{0}}}})$

$={\displaystyle \sum _{k=\mathrm{1}}^N}{c}_k\frac{\mathrm{1}}{\phi (h_P({\mathit{v}}_k))}\left((n{\omega }_n{)}^{\frac{\mathrm{1}}{n-\mathrm{1}}}{\displaystyle \sum _{j=\mathrm{1}}^N}{\delta }_{{\mathit{v}}_{i_{\mathrm{0}}}}{\mu }_n(P,\{{\mathit{v}}_j\})h_P({\mathit{v}}_k)-{\delta }_{{\mathit{v}}_{i_{\mathrm{0}}}}\right)$

$=-\frac{c_{i_{\mathrm{0}}}}{\phi (h_P({\mathit{v}}_{i_{\mathrm{0}}}))}+(n{\omega }_n{)}^{\frac{\mathrm{1}}{n-\mathrm{1}}}\left({\displaystyle \sum _{k=\mathrm{1}}^N}\frac{c_k{h}_P({\mathit{v}}_k)}{\phi (h_P({\mathit{v}}_k))}\right){\mu }_n(P,\{{\mathit{v}}_{i_{\mathrm{0}}}\}).$

Since $F(P,{\mathit{v}}_{i_{\mathrm{0}}})$ is not a facet of $P$, we calculate that ${\mu }_n(P,\{{\mathit{v}}_{i_{\mathrm{0}}}\})=\mathrm{0}$. Thus

$\frac{\partial }{\partial t}{|}_{t={\mathrm{0}}^{+}}{\Phi }_{\mu }(\tau (t)P_t)<\mathrm{0}.$

Then there exists a $t_{\mathrm{0}}>\mathrm{0}$ such that ${\Psi }_{\mu }(\tau (t_{\mathrm{0}})P_{t_{\mathrm{0}}})<{\Psi }_{\mu }(P)$, which contradicts ${\Psi }_{\mu }(P)$ is the minimum. Thus, $P$ has exactly $N$ facets.

Lemma 7   If the polytope $P\in P_N({\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N)$ and $o\in \mathrm{i}\mathrm{n}\mathrm{t}P$ is the solution to the extremal problem (10), then

$\alpha \mathrm{d}{\mu }_n(P,\cdot )=\frac{\mathrm{d}\mu }{\phi (h_P)},$

where $\alpha =(n{\omega }_n{)}^{\frac{\mathrm{1}}{n-\mathrm{1}}}\underset{{\mathbb{S}}^{n-\mathrm{1}}}{\int }\frac{h_P}{\phi (h_P)}\mathrm{d}\mu .$

Proof   For $a_{\mathrm{1}},\cdots ,a_N\in \mathbb{R}$ and small |t|, let polytope

$P_t={\displaystyle \underset{k=\mathrm{1}}{\overset{N}{\cap }}}\{\mathit{x}\in {\mathit{ℝ}}^n:\mathit{x}\cdot {\mathit{v}}_k\le h_P({\mathit{v}}_k)+t{a}_k\}.$

Since ${\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N$ are normal vectors of $P$, there exists an $\epsilon >\mathrm{0}$, such that for any $|t|<\epsilon ,{\mathit{v}}_{\mathrm{1}},\cdots ,{\mathit{v}}_N$ are normals vectors of $P_t$. Then $h_{P_t}({\mathit{v}}_k)=h_P({\mathit{v}}_k)+t{a}_k,k=\mathrm{1},\cdots ,N,|t|<\epsilon .$

Let

$\tau (t)P_t=\frac{P_t}{C_n(P_t)}.$

Since $C_n(\tau (t)P_t)=\mathrm{1},\tau (t)P_t\to P$ as $t\to \mathrm{0}$ and $P$ is the solution to the extremal problem, which gives

$\begin{array}{l}\mathrm{0}=\frac{\partial }{\partial t}{|}_{t=\mathrm{0}}{\Psi }_{\mu }(\tau (t)P_t)\\ ={\displaystyle \sum _{k=\mathrm{1}}^N}{c}_k\frac{\partial }{\partial t}{|}_{t=\mathrm{0}}\varphi (\tau (t)h_{P_t}({\mathit{v}}_k))\\ ={\displaystyle \sum _{k=\mathrm{1}}^N}{c}_k{\varphi }^{\text{'}}(\tau (t)h_{P_t}({\mathit{v}}_k)){|}_{t=\mathrm{0}}[{\tau }^{\text{'}}(t)h_{P_t}({\mathit{v}}_k){|}_{t=\mathrm{0}}+\tau (t)a_k{|}_{t=\mathrm{0}}]\\ ={\displaystyle \sum _{k=\mathrm{1}}^N}{c}_k{\varphi }^{\text{'}}(h_{P_t}({\mathit{v}}_k)){|}_{t=\mathrm{0}}[{\tau }^{\text{'}}(\mathrm{0})h_P({\mathit{v}}_k)+a_k].\end{array}$