# On the Discrete Orlicz Electrostaticq-Capacitary Minkowski Problem

Authors: Yibin Feng; Yanping Zhou
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We establish the existence of solutions to the Orlicz electrostaticq-capacitary Minkowski problem for polytopes. This contains a new result of the discrete Lp electrostatic q-capacitary Minkowski problem for p<0and 1<q<n.


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## Body

## 1. Introduction

The Orlicz Brunn-Minkowski theory was originated from the works of Ludwig [1], Ludwig and Reitzner [2], and Lutwak et al. [3, 4]. Hereafter, the new theory has quickly become an important branch of convex geometry (see, e.g., [5–10]). A special case of the theory is the Lp Brunn-Minkowski theory which is credited to Lutwak [11, 12] and attracted increasing interest in recent years (see, e.g., [13–20].It is well known that theL1 Brunn-Minkowski theory is the classical Brunn-Minkowski theory. One of the cornerstones of the classical Brunn-Minkowski theory is the Minkowski problem. More than a century ago, Minkowski himself solved the Minkowski problem for discrete measures [21]. The complete solution for arbitrary measures was given by Aleksandrov [22] and Fenchel and Jessen [23]. The regularity was studied by, e.g., Lewy [24], Nirenberg [25], Pogorelov [26], Cheng and Yau [27], and Caffarelli et al. [28].A generalization of the Minkowski problem is theLp Minkowski problem in the Lp Brunn-Minkowski theory, which has been extensively studied (see, e.g., [29–49]. Naturally, the corresponding Minkowski problem in the Orlicz Brunn-Minkowski theory is called the Orlicz Minkowski problem which was first investigated by Haberl et al. [50] for even measures. Today, great progress has been made on it (see, e.g., [51–60]). The present paper is aimed at dealing with the Orlicz capacitary Minkowski problem.The electrostaticq-capacitary measure μqΩ,⋅ (see [61]) of a bounded open convex set Ω in ℝn is the measure on the unit sphere Sn−1 defined for ω⊂Sn−1 and 1<q<n by(1)μqΩ,ω=∫g−1ω∇UqdHn−1,where g−1:Sn−1→∂Ω (the boundary of Ω) denotes the inverse Gauss map, Hn−1 the n−1-dimensional Hausdorff measure, and U the q-equilibrium potential of Ω.A convex bodyK is a compact convex set with nonempty interior in the n-dimensional Euclidean space ℝn. Let Kn denote the set of convex bodies in ℝn, and let Kon denote the set of convex bodies with the origin in their interiors. The support function (see [62, 63]) of K∈Kn is defined for u∈Sn−1 by(2)hKu=hK,u=maxx⋅u:x∈K,where x⋅u denotes the standard inner product of x and u. Note that hcK,u=chK,u for c>0.Letφ:0,∞→0,∞ be a given continuous function. For 1<q<n and K∈Kon, the Orlicz electrostatic q-capacitary measure, μφ,qK,⋅, of K is defined by(3)dμφ,qK,⋅=φhKdμqK,⋅.Whenφs=s1−p with p∈ℝ, the Orlicz electrostatic q-capacitary measure becomes the following Lp electrostatic q-capacitary measure introduced by Zou and Xiong [64]:(4)dμp,qK,⋅=hK1−pdμqK,⋅.The Minkowski problem characterizing the Orlicz electrostaticq-capacitary measure, proposed in [65], is the following.


### 1.1. The Orlicz Electrostaticq-Capacitary Minkowski Problem

Let1<q<n. Given a continuous function φ:0,∞→0,∞ and a finite Borel measure μ on Sn−1, what are the necessary and sufficient conditions so that μ=cμφ,qK,⋅ for some convex body K and constant c>0?Letφ be a constant function. When q=2, the Orlicz Minkowski-type problem is the classical electrostatic capacitary Minkowski problem. In the paper [66], Jerison established the existence of a solution to the electrostatic capacitary Minkowski problem. In a subsequent paper [67], he gave a new proof of the existence using a variational approach. The uniqueness was proved by Caffarelli et al. [68], and the regularity was given in [66]. When 1<q<n, the Orlicz Minkowski-type problem is the electrostatic q-capacitary Minkowski problem posed in [61]. The existence and regularity for 1<q<2 and the uniqueness for 1<q<n of its solutions were proved in [61], and the existence for 2<q<n was very recently solved in [69].Letφs=s1−p with p∈ℝ. Then, the Orlicz Minkowski-type problem is the Lp electrostatic q-capacitary Minkowski problem introduced by Zou and Xiong [64]. In [64], they completely solved the Lp electrostatic q-capacitary Minkowski problem for the case p>1 and 1<q<n. It is generally known that when p<1, the Lp Minkowski problem becomes much harder. Actually, the Lp electrostatic q-capacitary Minkowski problem for the case p<1 and 1<q<n is also very difficult. Therefore, it is worth mentioning that an important breakthrough of the problem for the case 0<p<1 and 1<q<2 was made by Xiong et al. [70] for discrete measures.The existence of the Orlicz electrostaticq-capacitary Minkowski problem was first investigated by Hong et al. [65]. As a consequence, in [65], they obtained a complete solution (including both existence and uniqueness) to the Lp electrostatic q-capacitary Minkowski problem for the case p>1 and 1<q<n, which was independently solved by Zou and Xiong [64].We observe the statement above. At present, there is no result about theLp electrostatic q-capacitary Minkowski problem for the case p<0 and 1<q<n. In this paper, we study the Orlicz electrostatic q-capacitary Minkowski problem including it.A finite setE of Sn−1 is said to be in general position if E is not contained in a closed hemisphere of Sn−1 and any n elements of E are linearly independent.A polytope inℝn is the convex hull of a finite set of points in ℝn provided that it has positive n-dimensional volume. The convex hull of a subset of these points is called a facet of the polytope if it lies entirely on the boundary of the polytope and has positive n−1-dimensional volume.Our main theorem is stated as follows.Theorem 1.
Supposeφ:0,∞→0,∞is continuously differentiable and strictly increasing withφs→∞ass→∞such thatϕt=∫t∞1/φsdsexists fort>0andlimt→0+ϕt=∞. Letμ=∑i=1Nαiδui, whereα1,⋯,αN>0, the unit vectorsu1,⋯,uN∈Sn−1are in general position, andδuiis the Dirac delta. Then, for1<q<n, there exist a polytopePand constantc>0such that(5)μ=cμφ,qP,⋅.Whenφs=s1−p with p<0, and ϕt=−1/ptp, which satisfy the assumptions of Theorem 1, we obtain the following.Corollary 2.
Letp<0 and 1<q<n. Suppose μ is a discrete measure on Sn−1, and its supports are in general position. If p+q≠n, then there exists a polytope P0 such that μ=μp,qP0,⋅; if p+q=n, then there exist a polytope P and constant c>0 such that μ=cμp,qP,⋅.Obviously, this corollary makes up for the existing results for theLp electrostatic q-capacitary Minkowski problem, to some extent.The rest of this paper is organized as follows. In Section2, some of the necessary facts about convex bodies and capacity are presented. In Section 3, a maximizing problem related to the Orlicz electrostatic q-capacitary Minkowski problem is considered and its corresponding solution is given. In Section 4, we give the proofs of Theorem 1 and Corollary 2.


## 1.1. The Orlicz Electrostaticq-Capacitary Minkowski Problem

Let1<q<n. Given a continuous function φ:0,∞→0,∞ and a finite Borel measure μ on Sn−1, what are the necessary and sufficient conditions so that μ=cμφ,qK,⋅ for some convex body K and constant c>0?Letφ be a constant function. When q=2, the Orlicz Minkowski-type problem is the classical electrostatic capacitary Minkowski problem. In the paper [66], Jerison established the existence of a solution to the electrostatic capacitary Minkowski problem. In a subsequent paper [67], he gave a new proof of the existence using a variational approach. The uniqueness was proved by Caffarelli et al. [68], and the regularity was given in [66]. When 1<q<n, the Orlicz Minkowski-type problem is the electrostatic q-capacitary Minkowski problem posed in [61]. The existence and regularity for 1<q<2 and the uniqueness for 1<q<n of its solutions were proved in [61], and the existence for 2<q<n was very recently solved in [69].Letφs=s1−p with p∈ℝ. Then, the Orlicz Minkowski-type problem is the Lp electrostatic q-capacitary Minkowski problem introduced by Zou and Xiong [64]. In [64], they completely solved the Lp electrostatic q-capacitary Minkowski problem for the case p>1 and 1<q<n. It is generally known that when p<1, the Lp Minkowski problem becomes much harder. Actually, the Lp electrostatic q-capacitary Minkowski problem for the case p<1 and 1<q<n is also very difficult. Therefore, it is worth mentioning that an important breakthrough of the problem for the case 0<p<1 and 1<q<2 was made by Xiong et al. [70] for discrete measures.The existence of the Orlicz electrostaticq-capacitary Minkowski problem was first investigated by Hong et al. [65]. As a consequence, in [65], they obtained a complete solution (including both existence and uniqueness) to the Lp electrostatic q-capacitary Minkowski problem for the case p>1 and 1<q<n, which was independently solved by Zou and Xiong [64].We observe the statement above. At present, there is no result about theLp electrostatic q-capacitary Minkowski problem for the case p<0 and 1<q<n. In this paper, we study the Orlicz electrostatic q-capacitary Minkowski problem including it.A finite setE of Sn−1 is said to be in general position if E is not contained in a closed hemisphere of Sn−1 and any n elements of E are linearly independent.A polytope inℝn is the convex hull of a finite set of points in ℝn provided that it has positive n-dimensional volume. The convex hull of a subset of these points is called a facet of the polytope if it lies entirely on the boundary of the polytope and has positive n−1-dimensional volume.Our main theorem is stated as follows.Theorem 1.
Supposeφ:0,∞→0,∞is continuously differentiable and strictly increasing withφs→∞ass→∞such thatϕt=∫t∞1/φsdsexists fort>0andlimt→0+ϕt=∞. Letμ=∑i=1Nαiδui, whereα1,⋯,αN>0, the unit vectorsu1,⋯,uN∈Sn−1are in general position, andδuiis the Dirac delta. Then, for1<q<n, there exist a polytopePand constantc>0such that(5)μ=cμφ,qP,⋅.Whenφs=s1−p with p<0, and ϕt=−1/ptp, which satisfy the assumptions of Theorem 1, we obtain the following.Corollary 2.
Letp<0 and 1<q<n. Suppose μ is a discrete measure on Sn−1, and its supports are in general position. If p+q≠n, then there exists a polytope P0 such that μ=μp,qP0,⋅; if p+q=n, then there exist a polytope P and constant c>0 such that μ=cμp,qP,⋅.Obviously, this corollary makes up for the existing results for theLp electrostatic q-capacitary Minkowski problem, to some extent.The rest of this paper is organized as follows. In Section2, some of the necessary facts about convex bodies and capacity are presented. In Section 3, a maximizing problem related to the Orlicz electrostatic q-capacitary Minkowski problem is considered and its corresponding solution is given. In Section 4, we give the proofs of Theorem 1 and Corollary 2.


## 2. Preliminaries

### 2.1. Basics regarding Convex Bodies

For quick later reference, we list some basic facts about convex bodies. Good general references are the books of Gardner [62] and Schneider [63].The boundary and interior ofK∈Kn will be denoted by ∂K and intK, respectively. B=x∈ℝn:x⋅x≤1 denotes the unit ball. The volume, the n-dimensional Lebesgue measure, of a convex body K∈Kn is denoted by VK, and the volume of B is denoted by ωn. We will write CSn−1 for the set of continuous functions on Sn−1 and C+Sn−1 for the set of positive functions in CSn−1.Forx∈∂K with K∈Kn, gKx is the Gauss map of K which is the family of all unit exterior normal vectors at x. In particular, gKx consists of a unique vector for Hn−1-almost all x∈∂K. The surface area measure of K is a Borel measure on Sn−1 defined for a Borel set ω⊂Sn−1 by(6)SK,ω=∫x∈gK−1ωdHn−1x.Forf∈C+Sn−1, the Aleksandrov body associated with f, denoted by f, is the convex body defined by(7)f=∩u∈Sn−1ξ∈ℝn:ξ⋅u≤fu.It is easy to see thathf≤f and hK=K for K∈Kon.The Hausdorff distance of two convex bodiesK,L∈Kn is defined by(8)δK,L=maxu∈Sn−1hKu−hLu.For a sequence of convex bodiesKi∈Kn and a convex body K∈Kn, we have limi→∞Ki=K provided that(9)δKi,K→0,as i→∞.ForK∈Kn and u∈Sn−1, the support hyperplane HK,u of K at u is defined by(10)HK,u=x∈ℝn:x⋅u=hK,u,the half-space H−K,u at u is defined by(11)H−K,u=x∈ℝn:x⋅u≤hK,u,and the support set FK,u at u is defined by(12)FK,u=K∩HK,u.Suppose thatP is the set of polytopes in ℝn and the unit vectors u1,⋯,uN are in general position. Let Pu1,⋯,uN be the subset of P. If P∈P with(13)P=∩k=1NH−P,uk,then P∈Pu1,⋯,uN. Obviously, if Pi∈Pu1,⋯,uN and Pi converges to a polytope P, then P∈Pu1,⋯,uN. Let PNu1,⋯,uN be the subset of Pu1,⋯,uN that any polytope in PNu1,⋯,uN has exactly N facets.


### 2.2. Electrostaticq-Capacity and q-Capacitary Measure

Here, we collect some notion and basic facts on electrostaticq-capacity and q-capacitary measure (see [61, 64, 70]).LetE be a compact set in n-dimensional Euclidean space ℝn. For 1<q<n, the electrostatic q-capacity, CqE, of E is defined (see [61]) by(14)CqE=inf∫ℝn∇uqdx:u∈Cc∞ℝnandu≥1onE,where Cc∞ℝn is the set of smooth functions with compact supports. When q=2, the electrostatic q-capacity becomes the classical electrostatic capacity C2E.ForK∈Kn and 1<q<n, we need the following isocapacitary inequality which is due to Maźya [71]:(15)VKn−q/n≤q−1n−qq−1nωnq/n−1CqK.The following lemma (see [64, 70]) gives some basic properties of the electrostatic q-capacity.Lemma 3.
LetE and F be two compact sets in ℝn and 1<q<n.(i)
IfE⊂F, then(16)CqE≤CqF(ii)
Forλ>0,(17)CqλE=λn−qCqE(iii)
Forx0∈ℝn,(18)CqE+x0=CqE(iv)
The functionalCq⋅ is continuous on Kn with respect to the Hausdorff metricThe following lemma is some basic properties of the electrostaticq-capacitary measure (compare [64, 70]).Lemma 4.
LetK∈Kn and 1<q<n.(i)
Forλ>0,(19)μqλK,⋅=λn−q−1μqK,⋅(ii)
Forx0∈ℝn,(20)μqK+x0,⋅=μqK,⋅(iii)
ForKj,K∈Kn, if Kj→K, then(21)μqKj,⋅→μqK,⋅weakly as j→+∞(iv)
The measureμqK,⋅ is absolutely continuous with respect to the surface area measure SK,⋅The following variational formula given in [61] of electrostatic q-capacity is critical.Lemma 5.
LetI⊂ℝ be an interval containing 0 in its interior, and let htu=ht,u:I×Sn−1→0,∞ be continuous such that the convergence in(22)h′0,u=limt→0ht,u−h0,utis uniform on Sn−1. Then,(23)dCqhtdtt=0=q−1∫Sn−1h′0,udμqh0,u.


## 2.1. Basics regarding Convex Bodies

For quick later reference, we list some basic facts about convex bodies. Good general references are the books of Gardner [62] and Schneider [63].The boundary and interior ofK∈Kn will be denoted by ∂K and intK, respectively. B=x∈ℝn:x⋅x≤1 denotes the unit ball. The volume, the n-dimensional Lebesgue measure, of a convex body K∈Kn is denoted by VK, and the volume of B is denoted by ωn. We will write CSn−1 for the set of continuous functions on Sn−1 and C+Sn−1 for the set of positive functions in CSn−1.Forx∈∂K with K∈Kn, gKx is the Gauss map of K which is the family of all unit exterior normal vectors at x. In particular, gKx consists of a unique vector for Hn−1-almost all x∈∂K. The surface area measure of K is a Borel measure on Sn−1 defined for a Borel set ω⊂Sn−1 by(6)SK,ω=∫x∈gK−1ωdHn−1x.Forf∈C+Sn−1, the Aleksandrov body associated with f, denoted by f, is the convex body defined by(7)f=∩u∈Sn−1ξ∈ℝn:ξ⋅u≤fu.It is easy to see thathf≤f and hK=K for K∈Kon.The Hausdorff distance of two convex bodiesK,L∈Kn is defined by(8)δK,L=maxu∈Sn−1hKu−hLu.For a sequence of convex bodiesKi∈Kn and a convex body K∈Kn, we have limi→∞Ki=K provided that(9)δKi,K→0,as i→∞.ForK∈Kn and u∈Sn−1, the support hyperplane HK,u of K at u is defined by(10)HK,u=x∈ℝn:x⋅u=hK,u,the half-space H−K,u at u is defined by(11)H−K,u=x∈ℝn:x⋅u≤hK,u,and the support set FK,u at u is defined by(12)FK,u=K∩HK,u.Suppose thatP is the set of polytopes in ℝn and the unit vectors u1,⋯,uN are in general position. Let Pu1,⋯,uN be the subset of P. If P∈P with(13)P=∩k=1NH−P,uk,then P∈Pu1,⋯,uN. Obviously, if Pi∈Pu1,⋯,uN and Pi converges to a polytope P, then P∈Pu1,⋯,uN. Let PNu1,⋯,uN be the subset of Pu1,⋯,uN that any polytope in PNu1,⋯,uN has exactly N facets.


## 2.2. Electrostaticq-Capacity and q-Capacitary Measure

Here, we collect some notion and basic facts on electrostaticq-capacity and q-capacitary measure (see [61, 64, 70]).LetE be a compact set in n-dimensional Euclidean space ℝn. For 1<q<n, the electrostatic q-capacity, CqE, of E is defined (see [61]) by(14)CqE=inf∫ℝn∇uqdx:u∈Cc∞ℝnandu≥1onE,where Cc∞ℝn is the set of smooth functions with compact supports. When q=2, the electrostatic q-capacity becomes the classical electrostatic capacity C2E.ForK∈Kn and 1<q<n, we need the following isocapacitary inequality which is due to Maźya [71]:(15)VKn−q/n≤q−1n−qq−1nωnq/n−1CqK.The following lemma (see [64, 70]) gives some basic properties of the electrostatic q-capacity.Lemma 3.
LetE and F be two compact sets in ℝn and 1<q<n.(i)
IfE⊂F, then(16)CqE≤CqF(ii)
Forλ>0,(17)CqλE=λn−qCqE(iii)
Forx0∈ℝn,(18)CqE+x0=CqE(iv)
The functionalCq⋅ is continuous on Kn with respect to the Hausdorff metricThe following lemma is some basic properties of the electrostaticq-capacitary measure (compare [64, 70]).Lemma 4.
LetK∈Kn and 1<q<n.(i)
Forλ>0,(19)μqλK,⋅=λn−q−1μqK,⋅(ii)
Forx0∈ℝn,(20)μqK+x0,⋅=μqK,⋅(iii)
ForKj,K∈Kn, if Kj→K, then(21)μqKj,⋅→μqK,⋅weakly as j→+∞(iv)
The measureμqK,⋅ is absolutely continuous with respect to the surface area measure SK,⋅The following variational formula given in [61] of electrostatic q-capacity is critical.Lemma 5.
LetI⊂ℝ be an interval containing 0 in its interior, and let htu=ht,u:I×Sn−1→0,∞ be continuous such that the convergence in(22)h′0,u=limt→0ht,u−h0,utis uniform on Sn−1. Then,(23)dCqhtdtt=0=q−1∫Sn−1h′0,udμqh0,u.


## 3. An Associated Maximization Problem

In this section, we solve a maximization problem, and its solution is exactly the solution in Theorem1.Supposeϕ satisfies the assumptions of Theorem 1 and the unit vectors u1,⋯,uN are in general position. For α1,⋯,αN>0 and P∈Pu1,⋯,uN, define the function, ΦP:intP→ℝ, by(24)ΦPξ=∑k=1NαkϕhP,uk−ξ⋅uk.Let1<q<n. We consider the following maximization problem:(25)supminξ∈intQΦQξ:CqQ=1,Q∈Pu1,⋯,uN.The solution to problem (25) is given in Theorem 9. Its proof requires the following three lemmas which are similar to those in [58].Lemma 6.
Supposeφ:0,∞→0,∞ is continuously differentiable and strictly increasing with φs→∞ as s→∞ such that ϕt=∫t∞1/φsds exists for t>0 and limt→0ϕt=∞. For α1,⋯,αN>0, if the unit vectors u1,⋯,uN∈Sn−1 are in general position, then there exists a unique ξϕP∈intP such that(26)ΦPξϕP=minξ∈intPΦPξ.Proof.
Sinceφ:0,∞→0,∞ is continuously differentiable and strictly increasing, we have for t>0,(27)ϕ′′t=φ′tφ2t>0.Therefore,ϕ is strictly convex on 0,∞.Let0<λ<1 and ξ1,ξ2∈intP. Then,(28)λΦPξ1+1−λΦPξ2=λ∑k=1NαkϕhP,uk−ξ1⋅uk+1−λ∑k=1NαkϕhP,uk−ξ2⋅uk=∑k=1NαkλϕhP,uk−ξ1⋅uk+1−λϕhP,uk−ξ2⋅uk≥∑k=1NαkϕhP,uk−λξ1+1−λξ2⋅uk=ΦPλξ1+1−λξ2.Equality holds if and only ifξ1⋅uk=ξ2⋅uk for all k=1,⋯,N. Since u1,⋯,uN are in general position, ℝn=linu1,⋯,uN which is the smallest linear subspace of ℝn containing u1,⋯,uN. Thus, ξ1=ξ2. Namely, ΦP is strictly convex on intP.SinceP∈Pu1,⋯,uN, it follows that for any x∈∂P, there exists a ui0∈u1,⋯,uN such that(29)hP,ui0=x⋅ui0.Note thatϕ is strictly decreasing on 0,∞ and limt→0ϕt=∞. Then, ΦPξ→∞ whenever ξ∈intP and ξ→x. This together with the strict convexity of Φp means that there exists a unique interior point ξϕP of P such that(30)ΦPξϕP=minξ∈intPΦPξ.Lemma 7.
Supposeα1,⋯,αN>0, the unit vectors u1,⋯,uN∈Sn−1 are in general position, and φ:0,∞→0,∞ is continuously differentiable and strictly increasing with φs→∞ as s→∞ such that ϕt=∫t∞1/φsds exists for t>0 and limt→0ϕt=∞. If Pi∈Pu1,⋯,uN converges to a polytope P, then limi→∞ξϕPi=ξϕP and(31)limi→∞ΦPiξϕPi=ΦPξϕP.Proof.
SincePi→P and ξϕPi∈intPi, it follows that ξϕPi is bounded. Let ξϕPij be a subsequence of ξϕPi with limj→∞ξϕPij=ξ0. We first show that ξ0∈intP by contradiction.Assumeξ0∈∂P. Then, limj→∞ΦPijξϕPij=∞, which contradicts the fact that(32)limj→∞ΦPijξϕPij≤limj→∞ΦPijξϕP=ΦPξϕP<∞.We next show thatξ0=ξϕP. Let ξ0≠ξϕP. Then,(33)limj→∞ΦPijξϕPij=ΦPξ0>ΦPξϕP=limj→∞ΦPijξϕP.This contradicts the fact that(34)limj→∞ΦPijξϕPij≤limj→∞ΦPijξϕP.This means thatlimi→∞ξϕPi=ξϕP and(35)limi→∞ΦPiξϕPi=ΦPξϕP.Lemma 8.
Supposeα1,⋯,αN>0, the unit vectors u1,⋯,uN∈Sn−1 are in general position, and φ:0,∞→0,∞ is continuously differentiable and strictly increasing with φs→∞ as s→∞ such that ϕt=∫t∞1/φsds exists for t>0 and limt→0ϕt=∞. Let P∈Pu1,⋯,uN and δ≥0 be small enough such that for k0∈1,⋯,N,(36)Pδ=P∩x:x⋅uk0≤hP,uk0−δ∈Pu1,⋯,uN.If the continuous functionλ:0,∞→0,∞ is continuously differentiable on 0,∞ and limδ→0λ′δ exists, then ξδ≔ξϕλδPδ has a right derivative, denoted by ξ+′0, at 0.Proof.
The proof is based on the ideas of Wu et al. [58]. Let δ≥0 be small enough and(37)Φδ=minξ∈intλδPδ∑k=1NαkϕλδhPδ,uk−ξ⋅uk=∑k=1NαkϕλδhPδ,uk−ξδ⋅uk.From this and the fact thatξδ is an interior point of λδPδ, it follows that for i=1,⋯,n,(38)∑k=1Nαkϕ′λδhPδ,uk−ξδ⋅ukuk,i=0,where uk=uk,1,⋯,uk,nT.Let(39)Fiδ,ξ1,⋯,ξn=∑k=1Nαkϕ′λδhPδ,uk−ξ⋅ukuk,ifor i=1,⋯,n, where ξ=ξ1,⋯,ξn. Then,(40)∂Fi∂ξj=−∑k=1Nαkϕ′′λδhPδ,uk−ξ⋅ukuk,iuk,j,(41)∂Fi∂δ=∑k=1Nαkϕ′′λδhPδ,uk−ξ⋅ukuk,iλ′δhPδ,uk−αk0ϕ′′λδhP,uk0−δλδ−ξ⋅uk0uk0,iλδ.LetF=F1,⋯,Fn. Then,(42)∂F∂ξδ,ξ1δ,⋯,ξnδn×n=−∑k=1Nαkϕ′′λδhPδ,uk−ξδ⋅ukukukT,where ukukT is an n×n matrix.Sinceu1,⋯,uN are in general position, ℝn=linu1,⋯,uN. Thus, for any x∈ℝn with x≠0, there exists a ui0∈u1,⋯,uN such that ui0⋅x≠0. Note that ϕ′′>0. Then, we have(43)xT−∑k=1Nαkϕ′′λδhPδ,uk−ξδ⋅ukukukTx=−∑k=1Nαkϕ′′λδhPδ,uk−ξδ⋅ukx⋅uk2≤−αi0ϕ′′λδhPδ,ui0−ξδ⋅ui0x⋅ui02<0.Therefore,∂F/∂ξδ,ξ1δ,⋯,ξnδ is negative definite. Thus,(44)det∂F∂ξδ,ξ1δ,⋯,ξnδ≠0.From this, the fact that fori=1,⋯,n, Fiδ,ξ1δ,⋯,ξnδ=0 follows by (38), the fact that ∂Fi/∂ξj is continuous on ξ and δ for all 1≤i,j≤n, and the implicit function theorem, it follows that ξδ=ξϕλδPδ is continuously differentiable on a neighbourhood of δ small enough. Thus, ξδ is continuously differentiable for small enough δ>0, and(45)dξ1dδdξ2dδ⋮dξndδ=−∂F1∂ξ1∂F1∂ξ2⋯∂F1∂ξn∂F2∂ξ1∂F2∂ξ2⋯∂F2∂ξn⋮⋮⋯⋮∂Fn∂ξ1∂Fn∂ξ2⋯∂Fn∂ξn−1∂F1∂δ∂F2∂δ⋮∂Fn∂δ.This together with (40) and (41) implies limδ→0+ξ′δ exists.From the Lagrange mean value theorem, we obtain that forδ>0 and i=1,⋯,n, there exists a δiδ with 0<δiδ<δ such that(46)ξiδ−ξi0δ=ξi′δiδ.Thus,(47)dξiδdδδ=0+=limδ→0+ξiδ−ξi0δ=limδ→0+ξi′δiδexists. Namely, ξ+′0 exists.We are ready to show the existence of a maximizer to problem (25).Theorem 9.
Supposeα1,⋯,αN>0 and the unit vectors u1,⋯,uN∈Sn−1 are in general position. Let φ:0,∞→0,∞ be continuously differentiable and strictly increasing with φs→∞ as s→∞ such that ϕt=∫t∞1/φsds exists for t>0 and limt→0ϕt=∞. Then, there exists a polytope P∈PNu1,⋯,uN such that ξϕP=o, CqP=1, and(48)ΦPo=supminξ∈intQΦQξ:CqQ=1,Q∈Pu1,⋯,uN.Proof.
Forx∈ℝn and P∈Pu1,⋯,uN, we first show(49)ΦP+xξϕP+x=ΦPξϕP.
From Lemma6 and definition (24), we have(50)ΦP+xξϕP+x=minξ∈intP+xΦP+xξ=minξ∈intP+x∑k=1NαkϕhP+x,uk−ξ⋅uk=minξ−x∈intP∑k=1NαkϕhP,uk−ξ−x⋅uk=minξ′∈intP∑k=1NαkϕhP,uk−ξ′⋅uk=ΦPξϕP.
Therefore, by (49) and (iii) of Lemma 2.1, we can choose a sequence Pi∈Pu1,⋯,uN with ξϕPi=o and CqPi=1 such that(51)limi→∞ΦPio=supminξ∈intQΦQξ:CqQ=1,Q∈Pu1,⋯,uN.
We next prove thatPi is bounded. Assume that Pi is not bounded. Since the unit vectors u1,⋯,uN are in general position, from the proof of ([45], Theorem 4.3), we see VPi is not bounded. However, from (15), and noting that CqPi=1, we have(52)VPi≤q−1n−qnq−1/n−qnωnq/n−n/n−q,which is a contradiction. Therefore, Pi is bounded.By the Blaschke selection theorem, we can assume that a subsequence ofPi converges to a polytope P∈Pu1,⋯,uN. Thus, from (iv) of Lemma 3 and Lemma 7, it follows that CqP=1, ξϕP=o, and(53)ΦPo=supminξ∈intQΦQξ:CqQ=1,Q∈Pu1,⋯,uN.We now prove thatP∈PNu1,⋯,uN, i.e., FP,ui are facets for all i=1,⋯,N. If not, there exists an i0∈1,⋯,N such that FP,ui0 is not a facet of P. Choose δ≥0 small enough so that the polytope(54)Pδ=P∩x:x⋅ui0≤hP,ui0−δ∈Pu1,⋯,uN.Letλδ=CqPδ−1/n−q. Then, λδPδ∈Pu1,⋯,uN, CqλδPδ=1 follows by (ii) of Lemma 3, and λδ is continuous in 0,∞. Since for any δi→0, there is that λδiPδi→P, it follows from Lemma 7 that ξϕλδiPδi→ξϕP=o. This implies(55)limδ→0ξϕλδPδ=o.Let(56)ri0u=1,u=ui0,0,u≠ui0for u∈Sn−1. Then, from Lemma 5, we have for small enough δ≥0,(57)dCqPδdδ=dCqhPδdδ=limt→0CqhPδ+tri0−CqhPδt=q−1∫Sn−1ri0udμqPδ,u=q−1μqPδ,ui0.Thus, from (iii) and (iv) of Lemma4, it follows that CqPδ is continuously differentiable for every δ>0, and(58)limδ→0+dCqPδdδ=0.These imply that(59)λ′δ=−1n−qCqPδ−1/n−q−1dCqPδdδis continuous for every δ>0, and(60)limδ→0+λ′δ=0.Therefore,λδ=CqPδ−1/n−q satisfies the conditions of Lemma 8. Noting that ξδ=ξϕλδPδ, we see ξ+′0 exists.Recall(61)Φδ=∑k=1NαkϕλδhPδ,uk−ξδ⋅uk.From this and (38), we have(62)∑k=1Nαkϕ′hP,ukuk=0.Thus, it follows from (60), (61), and (62) that(63)dΦδdδδ=0+=−αi0ϕ′hP,ui0−∑k=1Nαkϕ′hP,ukξ+′0⋅uk=−αi0ϕ′hP,ui0−ξ+′0⋅∑k=1Nαkϕ′hP,ukuk=−αi0ϕ′hP,ui0>0.This means(64)limδ→0+Φδ−Φ0δ>0.Therefore, there exists aδ0>0 small enough such that(65)Φδ0>Φ0.This together with (61) has(66)Φλδ0Pδ0ξϕλδ0Pδ0>ΦPξϕP=ΦPo.Note thatλδ0=CqPδ0−1/n−q. Let P0=λδ0Pδ0−ξϕλδ0Pδ0. Then, P0∈Pu1,⋯,uN, CqP0=1, ξϕP0=o, and(67)ΦP0o>ΦPo.This contradicts (53). Thus, P∈PNu1,⋯,uN.


## 4. Solving the Orlicz Electrostaticq-Capacitary Minkowski Problem

Proof of Theorem 1.
By Theorem9, there exists a polytope P∈PNu1,⋯,uN with ξϕP=o and CqP=1 such that(68)ΦPo=supminξ∈intQΦQξ:CqQ=1,Q∈Pu1,⋯,uN.
Forγ1,⋯,γN∈ℝ, choose ∣t∣ small enough so that the polytope Pt defined by(69)Pt=∩i=1Nx:x⋅ui≤hP,ui+tγihas exactly N facets. Then, hPt,ui=hP,ui+tγi for i=1,⋯,N.Let(70)βt=CqPt−1/n−q.
Then,βtPt∈PNu1,⋯,uN and CqβtPt=1. By Lemma 5 and (iv) of Lemma 4, we obtain(71)β′0=−1n−qdCqPtdtt=0=−q−1n−q∑i=1NγiμqP,ui.
Defineξt=ξϕβtPt and(72)Φt=minξ∈intβtPt∑k=1NαkϕβthPt,uk−ξ⋅uk=∑k=1NαkϕβthPt,uk−ξt⋅uk.
Sinceξt is an interior point of βtPt, this has(73)∑k=1Nαkϕ′βthPt,uk−ξt⋅ukuk,i=0,for i=1,⋯,n, where uk=uk,1,⋯,uk,nT. Note that ξ0 is the origin. Then, letting t=0 in (73), we have(74)∑k=1Nαkϕ′hP,ukuk,i=0,for i=1,⋯,n. Hence,(75)∑k=1Nαkϕ′hP,ukuk=0.
Let(76)Fit,ξ1,⋯,ξn=∑k=1Nαkϕ′βthPt,uk−ξ1uk,1+⋯+ξnuk,nuk,ifor i=1,⋯,n. Then,(77)∂Fi∂tt,ξ1,⋯,ξn=∑k=1Nαkϕ′′βthPt,uk−ξ1uk,1+⋯+ξnuk,nβ′thPt,uk+βtγkuk,i,∂Fi∂ξjt,ξ1,⋯,ξn=−∑k=1Nαkϕ′′βthPt,uk−ξ1uk,1+⋯+ξnuk,nuk,iuk,j.
Thus,(78)∂F∂ξ0,⋯,0n×n=−∑k=1Nαkϕ′′hP,ukukukT,where ukukT is an n×n matrix.
Sinceu1,⋯,uN are in general position, ℝn=linu1,⋯,uN. Thus, for any x∈ℝn with x≠0, there exists a ui0∈u1,⋯,uN such that ui0⋅x≠0. Note that ϕ′′>0. Then, we have(79)xT−∑k=1Nαkϕ′′hP,ukukukTx=−∑k=1Nαkϕ′′hP,ukx⋅uk2≤−αi0ϕ′′hP,ui0x⋅ui02<0.
Hence,∂F/∂ξ0,⋯,0 is negative definite. This implies det∂F/∂ξ0,⋯,0≠0. By this, the facts that for all i=1,⋯,n, Fi0,⋯,0=0 follows by (74) and ∂Fi/∂ξj is continuous on t and ξ for all 1≤i,j≤n, and for the implicit function theorem, it follows that(80)ξ′0=ξ1′0,⋯,ξn′0exists.
SinceΦ0 is a maximizer of Φt, from (71), (72), and (75), we get(81)0=Φ′0=∑k=1Nαkϕ′hP,ukβ′0hP,uk+γk−ξ′0⋅uk=∑k=1Nαkϕ′hP,uk−q−1n−q∑i=1NγiμqP,uihP,uk+γk−ξ′0⋅∑k=1Nαkϕ′hP,ukuk=∑k=1Nαkϕ′hP,ukγk−q−1n−q∑i=1NγiμqP,ui∑k=1Nαkϕ′hP,ukhP,uk=∑k=1Nαkϕ′hP,uk−q−1n−q∑j=1Nαjϕ′hP,ujhP,ujμqP,ukγk.
Sinceγ1,⋯,γN are arbitrary,(82)αk=−q−1n−q∑j=1Nαjϕ′hP,ujhP,uj1ϕ′hP,ukμqP,uk,for k=1,⋯,N. Let(83)c=q−1n−q∑j=1Nαjϕ′hP,ujhP,uj.
Then, fork=1,⋯,N,(84)αk=cφhP,ukμqP,uk,
i.e.,(85)μ=cμφ,qP,⋅.
This completes the proof.Proof of Corollary 2.
Letφs=s1−p with p<0 in Theorem 1. Then, ϕt=−1/ptp for t>0 and limt→0+ϕt=∞. Therefore, we see φ and ϕ satisfy the conditions of Theorem 1. Thus, from the theorem, (3), and (4), we obtain(86)μ=chP1−p⋅μqP,⋅=cμp,qP,⋅.
Ifp+q≠n, then from (i) of Lemma 4, we have(87)μ=μp,qc1/n−p−qP,⋅.
LetP0=c1/n−p−qP. Then, our desired result is given. If p+q=n, then (86) is just the desired result.



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*Source: 3067985-2020-10-06.xml*