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Yibin Feng, Yanping Zhou, "On the Discrete Orlicz Electrostatic -Capacitary Minkowski Problem", Journal of Function Spaces, vol. 2020, Article ID 3067985, 10 pages, 2020. https://doi.org/10.1155/2020/3067985
On the Discrete Orlicz Electrostatic -Capacitary Minkowski Problem
We establish the existence of solutions to the Orlicz electrostatic -capacitary Minkowski problem for polytopes. This contains a new result of the discrete electrostatic -capacitary Minkowski problem for and .
The Orlicz Brunn-Minkowski theory was originated from the works of Ludwig , Ludwig and Reitzner , 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 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 the 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 . The complete solution for arbitrary measures was given by Aleksandrov  and Fenchel and Jessen . The regularity was studied by, e.g., Lewy , Nirenberg , Pogorelov , Cheng and Yau , and Caffarelli et al. .
A generalization of the Minkowski problem is the Minkowski problem in the 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.  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 electrostatic -capacitary measure (see ) of a bounded open convex set in is the measure on the unit sphere defined for and bywhere (the boundary of ) denotes the inverse Gauss map, the -dimensional Hausdorff measure, and the -equilibrium potential of .
A convex body is a compact convex set with nonempty interior in the -dimensional Euclidean space . Let denote the set of convex bodies in , and let denote the set of convex bodies with the origin in their interiors. The support function (see [62, 63]) of is defined for bywhere denotes the standard inner product of and . Note that for .
Let be a given continuous function. For and , the Orlicz electrostatic -capacitary measure, , of is defined by
When with , the Orlicz electrostatic -capacitary measure becomes the following electrostatic -capacitary measure introduced by Zou and Xiong :
The Minkowski problem characterizing the Orlicz electrostatic -capacitary measure, proposed in , is the following.
1.1. The Orlicz Electrostatic -Capacitary Minkowski Problem
Let . Given a continuous function and a finite Borel measure on , what are the necessary and sufficient conditions so that for some convex body and constant ?
Let be a constant function. When , the Orlicz Minkowski-type problem is the classical electrostatic capacitary Minkowski problem. In the paper , Jerison established the existence of a solution to the electrostatic capacitary Minkowski problem. In a subsequent paper , he gave a new proof of the existence using a variational approach. The uniqueness was proved by Caffarelli et al. , and the regularity was given in . When , the Orlicz Minkowski-type problem is the electrostatic -capacitary Minkowski problem posed in . The existence and regularity for and the uniqueness for of its solutions were proved in , and the existence for was very recently solved in .
Let with . Then, the Orlicz Minkowski-type problem is the electrostatic -capacitary Minkowski problem introduced by Zou and Xiong . In , they completely solved the electrostatic -capacitary Minkowski problem for the case and . It is generally known that when , the Minkowski problem becomes much harder. Actually, the electrostatic -capacitary Minkowski problem for the case and is also very difficult. Therefore, it is worth mentioning that an important breakthrough of the problem for the case and was made by Xiong et al.  for discrete measures.
The existence of the Orlicz electrostatic -capacitary Minkowski problem was first investigated by Hong et al. . As a consequence, in , they obtained a complete solution (including both existence and uniqueness) to the electrostatic -capacitary Minkowski problem for the case and , which was independently solved by Zou and Xiong .
We observe the statement above. At present, there is no result about the electrostatic -capacitary Minkowski problem for the case and . In this paper, we study the Orlicz electrostatic -capacitary Minkowski problem including it.
A finite set of is said to be in general position if is not contained in a closed hemisphere of and any elements of are linearly independent.
A polytope in is the convex hull of a finite set of points in provided that it has positive -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 -dimensional volume.
Our main theorem is stated as follows.
Theorem 1. Supposeis continuously differentiable and strictly increasing withassuch thatexists forand. Let, where, the unit vectorsare in general position, andis the Dirac delta. Then, for, there exist a polytopeand constantsuch that
When with , and , which satisfy the assumptions of Theorem 1, we obtain the following.
Corollary 2. Let and . Suppose is a discrete measure on , and its supports are in general position. If , then there exists a polytope such that ; if , then there exist a polytope and constant such that .
Obviously, this corollary makes up for the existing results for the electrostatic -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 -capacitary Minkowski problem is considered and its corresponding solution is given. In Section 4, we give the proofs of Theorem 1 and Corollary 2.
2.1. Basics regarding Convex Bodies
The boundary and interior of will be denoted by and , respectively. denotes the unit ball. The volume, the -dimensional Lebesgue measure, of a convex body is denoted by , and the volume of is denoted by . We will write for the set of continuous functions on and for the set of positive functions in .
For with , is the Gauss map of which is the family of all unit exterior normal vectors at . In particular, consists of a unique vector for -almost all . The surface area measure of is a Borel measure on defined for a Borel set by
For , the Aleksandrov body associated with , denoted by , is the convex body defined by
It is easy to see that and for .
The Hausdorff distance of two convex bodies is defined by
For a sequence of convex bodies and a convex body , we have provided thatas .
For and , the support hyperplane of at is defined bythe half-space at is defined byand the support set at is defined by
Suppose that is the set of polytopes in and the unit vectors are in general position. Let be the subset of . If withthen . Obviously, if and converges to a polytope , then . Let be the subset of that any polytope in has exactly facets.
2.2. Electrostatic -Capacity and -Capacitary Measure
Let be a compact set in -dimensional Euclidean space . For , the electrostatic -capacity, , of is defined (see ) bywhere is the set of smooth functions with compact supports. When , the electrostatic -capacity becomes the classical electrostatic capacity .
For and , we need the following isocapacitary inequality which is due to Maźya :
Lemma 3. Let and be two compact sets in and .(i)If , then(ii)For ,(iii)For ,(iv)The functional is continuous on with respect to the Hausdorff metric
Lemma 4. Let and .(i)For ,(ii)For ,(iii)For , if , thenweakly as (iv)The measure is absolutely continuous with respect to the surface area measure
The following variational formula given in  of electrostatic -capacity is critical.
Lemma 5. Let be an interval containing in its interior, and let be continuous such that the convergence inis uniform on . 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 satisfies the assumptions of Theorem 1 and the unit vectors are in general position. For and , define the function, , by
Let . We consider the following maximization problem:
Lemma 6. Suppose is continuously differentiable and strictly increasing with as such that exists for and . For , if the unit vectors are in general position, then there exists a unique such that
Proof. Since is continuously differentiable and strictly increasing, we have for ,
Therefore, is strictly convex on .
Let and . Then,
Equality holds if and only if for all . Since are in general position, which is the smallest linear subspace of containing . Thus, . Namely, is strictly convex on .
Since , it follows that for any , there exists a such that
Note that is strictly decreasing on and . Then, whenever and . This together with the strict convexity of means that there exists a unique interior point of such that
Lemma 7. Suppose , the unit vectors are in general position, and is continuously differentiable and strictly increasing with as such that exists for and . If converges to a polytope , then and
Proof. Since and , it follows that is bounded. Let be a subsequence of with . We first show that by contradiction.
Assume . Then, , which contradicts the fact that
We next show that . Let . Then,
This contradicts the fact that
This means that and
Lemma 8. Suppose , the unit vectors are in general position, and is continuously differentiable and strictly increasing with as such that exists for and . Let and be small enough such that for
If the continuous function is continuously differentiable on and exists, then has a right derivative, denoted by , at .
Proof. The proof is based on the ideas of Wu et al. . Let be small enough and
From this and the fact that is an interior point of , it follows that for ,where .
Letfor , where . Then,
Let . Then,where is an matrix.
Since are in general position, . Thus, for any with , there exists a such that . Note that . Then, we have
Therefore, is negative definite. Thus,
From this, the fact that for , follows by (38), the fact that is continuous on and for all , and the implicit function theorem, it follows that is continuously differentiable on a neighbourhood of small enough. Thus, is continuously differentiable for small enough , and
From the Lagrange mean value theorem, we obtain that for and , there exists a with such that
Thus,exists. Namely, exists.
We are ready to show the existence of a maximizer to problem (25).
Theorem 9. Suppose and the unit vectors are in general position. Let be continuously differentiable and strictly increasing with as such that exists for and . Then, there exists a polytope such that , , and
Proof. For and , we first showFrom Lemma 6 and definition (24), we haveTherefore, by (49) and (iii) of Lemma 2.1, we can choose a sequence with and such thatWe next prove that is bounded. Assume that is not bounded. Since the unit vectors are in general position, from the proof of (, Theorem 4.3), we see is not bounded. However, from (15), and noting that , we havewhich is a contradiction. Therefore, is bounded.
We now prove that , i.e., are facets for all . If not, there exists an such that is not a facet of . Choose small enough so that the polytope
Letfor . Then, from Lemma 5, we have for small enough ,
Thus, from (iii) and (iv) of Lemma 4, it follows that is continuously differentiable for every , and
These imply thatis continuous for every , and
Therefore, satisfies the conditions of Lemma 8. Noting that , we see exists.
From this and (38), we have
Therefore, there exists a small enough such that
This together with (61) has
Note that . Let . Then, , , , and
This contradicts (53). Thus, .
4. Solving the Orlicz Electrostatic -Capacitary Minkowski Problem
Proof of Theorem 1. By Theorem 9, there exists a polytope with and such thatFor , choose small enough so that the polytope defined byhas exactly facets. Then, for .LetThen, and . By Lemma 5 and (iv) of Lemma 4, we obtainDefine