On the Discrete Orlicz Electrostatic q -Capacitary Minkowski Problem

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

It is well known that the L 1 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 the L p Minkowski problem in the L p Brunn-Minkowski theory, which has been extensively studied (see, e.g., . 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][52][53][54][55][56][57][58][59][60]). The present paper is aimed at dealing with the Orlicz capacitary Minkowski problem.
The electrostatic q-capacitary measure μ q ðΩ, ⋅Þ (see [61]) of a bounded open convex set Ω in ℝ n is the measure on the unit sphere S n−1 defined for ω ⊂ S n−1 and 1 < q < n by where g −1 : S n−1 → ∂Ω (the boundary of Ω) denotes the inverse Gauss map, H n−1 the ðn − 1Þ-dimensional Hausdorff measure, and U the q-equilibrium potential of Ω. A convex body K is a compact convex set with nonempty interior in the n-dimensional Euclidean space ℝ n . Let K n denote the set of convex bodies in ℝ n , and let K n o denote the set of convex bodies with the origin in their interiors. The support function (see [62,63]) of K ∈ K n is defined for u ∈ S n−1 by where x ⋅ u denotes the standard inner product of x and u. Note that hðcK, uÞ = chðK, uÞ for c > 0. Let φ : ð0,∞Þ → ð0,∞Þ be a given continuous function. For 1 < q < n and K ∈ K n o , the Orlicz electrostatic q -capacitary measure, μ φ,q ðK, ⋅ Þ, of K is defined by When φðsÞ = s 1−p with p ∈ ℝ, the Orlicz electrostatic q -capacitary measure becomes the following L p electrostatic q-capacitary measure introduced by Zou and Xiong [64]: The Minkowski problem characterizing the Orlicz electrostatic q-capacitary measure, proposed in [65], is the following.
Let φ 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].
Let φðsÞ = s 1−p with p ∈ ℝ. Then, the Orlicz Minkowskitype problem is the L p electrostatic q-capacitary Minkowski problem introduced by Zou and Xiong [64]. In [64], they completely solved the L p electrostatic q-capacitary Minkowski problem for the case p > 1 and 1 < q < n. It is generally known that when p < 1, the L p Minkowski problem becomes much harder. Actually, the L p 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 electrostatic q-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 L p 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 the L p 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 set E of S n−1 is said to be in general position if E is not contained in a closed hemisphere of S n−1 and any n elements of E are linearly independent.
A polytope in ℝ n is the convex hull of a finite set of points in ℝ n 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 ðn − 1Þ-dimensional volume.
Our main theorem is stated as follows.
Theorem 1. Suppose φ : ð0,∞Þ → ð0,∞Þ is continuously differentiable and strictly increasing with φðsÞ → ∞ as s → ∞ such that ϕðtÞ = Ð ∞ t ð1/φðsÞÞds exists for t > 0 and lim , the unit vectors u 1 , ⋯, u N ∈ S n−1 are in general position, and δ u i is the Dirac delta. Then, for 1 < q < n , there exist a polytope P and constant c > 0 such that When φðsÞ = s 1−p with p < 0, and ϕðtÞ = −ð1/pÞt p , which satisfy the assumptions of Theorem 1, we obtain the following.

Corollary 2.
Let p < 0 and 1 < q < n. Suppose μ is a discrete measure on S n−1 , and its supports are in general position. If p + q ≠ n, then there exists a polytope P 0 such that μ = μ p,q ð P 0 , ⋅ Þ; if p + q = n, then there exist a polytope P and constant c > 0 such that μ = cμ p,q ðP, ⋅ Þ.
Obviously, this corollary makes up for the existing results for the L p electrostatic q-capacitary Minkowski problem, to some extent.
The rest of this paper is organized as follows. In Section 2, 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.

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 of K ∈ K n will be denoted by ∂K and int K, respectively. B = fx ∈ ℝ n : ffiffiffiffiffiffiffiffi x ⋅ x p ≤ 1g denotes the unit ball. The volume, the n-dimensional Lebesgue measure, of a convex body K ∈ K n is denoted by VðKÞ, and the volume of B is denoted by ω n . We will write CðS n−1 Þ for the set of continuous functions on S n−1 and C + ðS n−1 Þ for the set of positive functions in CðS n−1 Þ.
For x ∈ ∂K with K ∈ K n , g K ðxÞ is the Gauss map of K which is the family of all unit exterior normal vectors at x. In particular, g K ðxÞ consists of a unique vector for H n−1 2 Journal of Function Spaces -almost all x ∈ ∂K. The surface area measure of K is a Borel measure on S n−1 defined for a Borel set ω ⊂ S n−1 by For f ∈ C + ðS n−1 Þ, the Aleksandrov body associated with f , denoted by ½ f , is the convex body defined by It is easy to see that h ½ f ≤ f and ½h K = K for K ∈ K n o . The Hausdorff distance of two convex bodies K, L ∈ K n is defined by For a sequence of convex bodies K i ∈ K n and a convex body K ∈ K n , we have lim as i → ∞.
For K ∈ K n and u ∈ S n−1 , the support hyperplane HðK, uÞ of K at u is defined by the half-space H − ðK, uÞ at u is defined by and the support set FðK, uÞ at u is defined by Suppose that P is the set of polytopes in ℝ n and the unit vectors u 1 , ⋯, u N are in general position. Let Pðu 1 , ⋯, u N Þ be the subset of P. If P ∈ P with then P ∈ Pðu 1 , ⋯, u N Þ. Obviously, if P i ∈ Pðu 1 , ⋯, u N Þ and P i converges to a polytope P, then P ∈ Pðu 1 , ⋯, u N Þ. Let P N ðu 1 , ⋯, u N Þ be the subset of Pðu 1 , ⋯, u N Þ that any polytope in P N ðu 1 , ⋯, u N Þ has exactly N facets.

Electrostatic q-Capacity and q-Capacitary
Measure. Here, we collect some notion and basic facts on electrostatic q -capacity and q-capacitary measure (see [61,64,70]). Let E be a compact set in n-dimensional Euclidean space ℝ n . For 1 < q < n, the electrostatic q-capacity, C q ðEÞ, of E is defined (see [61]) by where C ∞ c ðℝ n Þ is the set of smooth functions with compact supports. When q = 2, the electrostatic q-capacity becomes the classical electrostatic capacity C 2 ðEÞ.
For K ∈ K n and 1 < q < n, we need the following isocapacitary inequality which is due to Maźya [71]: The following lemma (see [64,70]) gives some basic properties of the electrostatic q-capacity. Lemma 3. Let E and F be two compact sets in ℝ n and 1 < q < n.
(iv) The functional C q ð⋅Þ is continuous on K n with respect to the Hausdorff metric The following lemma is some basic properties of the electrostatic q-capacitary measure (compare [64,70]). Lemma 4. Let K ∈ K n and 1 < q < n.

Journal of Function Spaces
weakly as j → +∞ (iv) The measure μ q ðK, ⋅Þ is absolutely continuous with respect to the surface area measure SðK, ⋅Þ The following variational formula given in [61] of electrostatic q-capacity is critical.

Lemma 5.
Let I ⊂ ℝ be an interval containing 0 in its interior, and let h t ðuÞ = hðt, uÞ: I × S n−1 → ð0,∞Þ be continuous such that the convergence in is uniform on S n−1 . Then, 3

. An Associated Maximization Problem
In this section, we solve a maximization problem, and its solution is exactly the solution in Theorem 1. Suppose ϕ satisfies the assumptions of Theorem 1 and the unit vectors u 1 , ⋯, u N are in general position. For α 1 , ⋯, α N > 0 and P ∈ Pðu 1 , ⋯, u N Þ, define the function, Φ P : int P → ℝ, by Let 1 < q < n. We consider the following maximization problem: The solution to problem (25) is given in Theorem 9. Its proof requires the following three lemmas which are similar to those in [58]. For α 1 , ⋯, α N > 0, if the unit vectors u 1 , ⋯, u N ∈ S n−1 are in general position, then there exists a unique ξ ϕ ðPÞ ∈ int P such that Proof. Since φ : ð0,∞Þ → ð0,∞Þ is continuously differentiable and strictly increasing, we have for t > 0, Therefore, ϕ is strictly convex on ð0, ∞Þ. Let 0 < λ < 1 and ξ 1 , ξ 2 ∈ int P. Then, Equality holds if and only if ξ 1 ⋅ u k = ξ 2 ⋅ u k for all k = 1, ⋯, N. Since u 1 , ⋯, u N are in general position, ℝ n = linfu 1 , ⋯, u N g which is the smallest linear subspace of ℝ n containing fu 1 , ⋯, u N g. Thus, ξ 1 = ξ 2 . Namely, Φ P is strictly convex on int P.
We are ready to show the existence of a maximizer to problem (25). Theorem 9. Suppose α 1 , ⋯, α N > 0 and the unit vectors u 1 , ⋯, u N ∈ S n−1 are in general position. Let φ : ð0,∞Þ → ð0,∞Þ be continuously differentiable and strictly increasing with φð sÞ → ∞ as s → ∞ such that ϕðtÞ = Ð ∞ t ð1/φðsÞÞds exists for t > 0 and lim t→0 ϕðtÞ = ∞. Then, there exists a polytope P ∈ P N ð u 1 , ⋯, u N Þ such that ξ ϕ ðPÞ = o, C q ðPÞ = 1, and Proof. For x ∈ ℝ n and P ∈ Pðu 1 , ⋯, u N Þ, we first show From Lemma 6 and definition (24), we have Therefore, by (49) and (iii) of Lemma 2.1, we can choose a sequence P i ∈ Pðu 1 , ⋯, u N Þ with ξ ϕ ðP i Þ = o and C q ðP i Þ = 1 such that We next prove that P i is bounded. Assume that P i is not bounded. Since the unit vectors u 1 , ⋯, u N are in general position, from the proof of ( [45], Theorem 4.3), we see VðP i Þ is not bounded. However, from (15), and noting that C q ðP i Þ = 1, we have which is a contradiction. Therefore, P i is bounded.