Condenser capacity and hyperbolic diameter

Given a compact connected set $E$ in the unit disk $\mathbb{B}^{2}$, we give a new upper bound for the conformal capacity of the condenser $(\mathbb{B}^{2}, E)$ in terms of the hyperbolic diameter $t$ of $E$. Moreover, for $t>0$, we construct a set of hyperbolic diameter $t$ and apply novel numerical methods to show that it has larger capacity than a hyperbolic disk with the same diameter. The set we construct is called a Reuleaux triangle in hyperbolic geometry and it has constant hyperbolic width equal to $t$.

where the infimum is taken over the set of all C ∞ 0 (G) functions u : G → [0, ∞) with u(x) ≥ 1 for all x ∈ E and dm is the n-dimensional Lebesgue measure. Below, we often choose n = 2 and focus on the special case where G R 2 is simply connected and E is a continuum.
By classical results the capacity decreases under a geometric transformation called symmetrization [1, Ch 6, p.215], [5], [7,Thm 5.3.11], where (G s , E s ) is the condenser obtained by one of the well-known symmetrization procedures, such as the spherical symmetrization or the Steiner symmetrization. While finding the explicit formula for the capacity (1.1) is usually impossible, the lower bound (1.2) can be often estimated or given explicitly [5], [7, pp.180-181], [9,Chapter 9].
Our aim in this paper is to find upper bounds for the condenser capacity, when n = 2, G is a simply connected domain, and E is a connected compact set. Numerous bounds are given in the literature in terms of domain functionals, such as the area of G, the diameter of E and the distance from E to the boundary ∂G [5,7,9,15,17,21,22]. While these kinds of bounds have numerous applications as shown in the cited sources, these bounds do not reflect the conformal invariance of cap(G, E). We apply the conformally invariant hyperbolic metric in this paper and therefore our main results are conformally invariant.
By the conformal invariance of the capacity and the hyperbolic metric, we may assume without loss of generality that the domain G is the unit disk B 2 in the two-dimensional plane C = R 2 . Naturally, we use here the Riemann mapping theorem [2]. After this preliminary reduction, we look for upper bounds for the condenser capacity cap(B 2 , E) in terms of the hyperbolic metric ρ B 2 of the unit disk, when the hyperbolic diameter ρ B 2 (E) of the compact set E is fixed.
A first guess might be that for a compact set E ⊂ B 2 , a majorant for cap(B 2 , E) would be the capacity of a hyperbolic disk with the hyperbolic diameter equal to that of E. This guess is motivated by a measure-theoretic isodiametric inequality, see Remark 3.14. However, the main result of this paper is to show that this guess is wrong. For this purpose, we introduce the so-called hyperbolic Reuleaux triangle, which is a set of constant hyperbolic width, and then compute its conformal capacity with novel computational methods [13,16,18,19] to confirm our claim. A valid upper bound for cap (B 2 , E) in terms of the hyperbolic diameter is instead naturally given in terms of the capacity of the minimal hyperbolic disk containing the set E and here we apply the work of B.V. Dekster in [4] who found this minimal radius. Theorem 1.3. For a continuum E ⊂ B 2 with the hyperbolic diameter equal to t > 0, the inequality Given a number t > 0 , the hyperbolic Jung radius is the smallest number r > 0 such that every set E ⊂ B 2 with the hyperbolic diameter equal to t is contained in some hyperbolic disk with the radius equal to r . Originally, the Jung radius was found in the context of the Euclidean geometry [14,p. 33,Thm 2.8.4] and its hyperbolic counterpart was found for dimensions n ≥ 2 by B.V. Dekster in [4]. His result is formulated below as Theorem 3.3 and Theorem 1.3 is based on the special case n = 2 of his work. It should be noticed that, by the Riemann mapping theorem, (1.4) directly applies to the case of planar simply connected domains. The sharp upper bound in Theorem 1.3 is not known and this motivates the following open problem.
1.5. Open problem. Given t > 0, identify all connected compact sets E ⊂ B 2 with the hyperbolic diameter t, which maximize the capacity cap(B 2 , E). that we will analyse further, in order to find a lower bound for it. To this end we have to apply numerical methods. Our first step is to write an algorithm for computing the hyperbolic diameter of a set in a simply connected domain. The boundary integral equation method developed in a series of recent papers [13,16,18,19] is used. Using this method, we can compute the hyperbolic diameter and the capacity of a subset bounded by piecewise smooth curves in a polygonal domain or in the unit disk. We show that the capacity of a hyperbolic disk with diameter t, denoted b 1 (t) , is a minorant for the above function b(t) , i.e. b(t) ≥ b 1 (t) , see (3.12). For this purpose we introduce the aforementioned hyperbolic Reuleaux triangle and our numerical work shows that its capacity majorizes the capacity of a disk with the same hyperbolic diameter. A delicate point here is the essential role of the hyperbolic geometry: the hyperbolic Reuleaux triangle cannot be replaced by the Euclidean Reuleaux triangle with the same hyperbolic diameter, for its capacity is not a majorant for b 1 (t) for t > 2 . The numerical algorithm is of independent interest, because it enables one to experimentally study the hyperbolic geometry of planar simply connected polygonal domains. We apply our result to quasiconformal maps and prove the following result.
be a K-quasiconformal homeomorphism between two simply connected domains G 1 and G 2 in R 2 , and let E ⊂ G 1 be a continuum. Then where ρ G 1 and ρ G 2 refer to the hyperbolic metrics of G 1 and G 2 ,resp., and h(2, t) stands for the hyperbolic Jung radius of a set with the hyperbolic diameter equal to t defined in Theorem 3.3 due to B.V. Dekster [4].
For a large class of simply connected plane domains, so called ϕ-uniform domains, we give explicit bounds for the hyperbolic Jung radius of a compact set in the domain.
We are indebted to Prof. Alex Solynin for pointing out the above open problem.

Preliminaries
An open ball defined with the Euclidean metric is B n (x, r) = {y ∈ R n : |x − y| < r} and the corresponding closed ball is B n (x, r) = {y ∈ R n : |x − y| ≤ r}. The sphere of these balls is Note that if the center x or the radius r is not otherwise specified in these notations, it means that x = 0 and r = 1. In a metric space (X, d) , a ball centered at x and with radius r > 0 is B d (x, r) and the diameter of a non-empty set A ⊂ X is d(A).
In the Poincaré unit ball B n = {x ∈ R n :|x| < 1}, the hyperbolic metric is defined as [2, (2.8) p. 15] The hyperbolic segment between the points x, y is denoted by J[x, y]. Furthermore, the hyperbolic balls B ρ (q, R) are Euclidean balls with the center and the radius given by the following lemma.
For a given simply connected planar domain G , by means of the Riemann mapping theorem, one can define a conformal map of G onto the unit disk B 2 , f : G → B 2 = f (G), and thus define the hyperbolic metric ρ G in G by [2] As an example, consider the simply connected domain G inside the polygon with the vertices 0, 3, 3 + i, 2 + i, 2 + 0.2i, 1 + 0.2i, 1 + i, and i. The hyperbolic diameter of a compact set E ⊂ G, denoted by ρ G (E) , is defined by For the polygonal domain G in Figure 1 (left), let E ⊂ G be the closure of the square with the vertices 0.5 ± h + (0.5 ± h)i for 0 < h < 0.5 (see Figure 1 (right) for h = 2). The approximate values of the hyperbolic diameter of the set E for several values of h, computed by the method described in Appendix A.1 with α = 0.5 + 0.5i and n = 2 13 , are given in Table 1. Table 1 also presents the values of the capacity of the condenser (G, E), which are computed using the method described in Appendix A.4 with α = 1.5 + 0.1i, z 2 = 0.5 + 0.5i, and n = 2 13 .  Note that while we already defined the condenser capacity in (1.1), its definition can be also written as We often use the fact that the capacity is, in the same way as the modulus, conformally invariant.
(2) The value of the left hand side is independent of x by the Möbius invariance of the modulus and of the hyperbolic metric and hence we may assume that x = 0 . By Lemma 2.1, B ρ (x, R) = B ρ (0, R) = B n (0, th(R/2)) and hence the proof follows from part (1).
(2) By the Riemann mapping theorem, we may assume without loss of generality that G = B 2 and hence the proof follows from part (1). [14]. Let E ⊂ R n be a compact set with diameter equal to t > 0 . We say that E is a set of constant width if for every z ∈ ∂E, t = sup{|z − x| : x ∈ E} .  Let D 1 = B 2 (y, h) and let D 2 , D 3 be the disks obtained from D 1 by rotation around the origin with angles 2π/3 and 4π/3 , resp. Now, the hyperbolic Reuleaux triangle with vertices at the above points is D 1 ∩ D 2 ∩ D 3 .

Capacity and Jung radius
For a compact subset E of a metric space X, the Jung radius is the least number r > 0 such that, for some x ∈ X, E is a subset of the closed ball centered at x with the radius r [14]. The metric space in our work will be the hyperbolic disk and we denote the hyperbolic Jung radius of the set E by r Jung (E). Clearly, it follows from the conformal invariance of the hyperbolic metric that the Jung radius is conformally invariant. Because of the same reason, for every simply connected domain G R 2 and all compact sets E ⊂ G, there exists z ∈ G with cap(G, E) ≤ cap(G, B ρ (z, r Jung (E))). (3.1) By Lemma 2.1, B ρ (z, r Jung (E)) is conformally equivalent to B 2 (0, th(r Jung (E)/2)) and thus it follows from (3.1) and Riemann's mapping theorem that (3.2) cap(G, E) ≤ 2π log(1/th(r Jung (E)/2)) .  , u = 2n/(n + 1) , t > 0 .
For Lemma 3.7, which provides bounds for the function h(n, t), we first prove some preliminary results. (2) For all x ≥ 1, k > 0, the inequality x shk ≤ sh(kx) holds. Proof.
(2) If E is a compact subset of a simply connected domain G R 2 , then Proof.
(1) follows immediately from Theorem 3.3, Lemma 3.7 and Lemma 2.3 and some basic properties of the modulus.
3.11. Comparison of the bounds. As we have seen above, b(t) ≥ 2π/ log(1/th(t/4)) ≡ b 1 (t) . Define b 2 (t) as in Corollary 3.8 (2). Now, we know that for a hyperbolic Reuleaux triangle T of hyperbolic diameter equal to t we have  3.13. Capacity comparison: the Euclidean vs hyperbolic Reuleaux triangle. We have shown above that the hyperbolic Reuleaux triangle of diameter t has a larger capacity than b 1 (t) , the capacity of a hyperbolic disk with the same diameter. A natural question is: Why do we use for this purpose the hyperbolic Reuleaux triangle, not the Euclidean one? It follows easily from Lemma 2.1 that, as a point set, the hyperbolic triangle contains the Euclidean one and thus has a larger capacity. The key point now is that the capacity of the Euclidean Reuleaux triangle is smaller than b 1 (t) for t > 2 . In Figure 5 (left) we demonstrate this fact by graphing, as a function of t , the four quotients (1) Jung bound Corollary 3.8(2) divided by b 1 (t), (2) the capacity of the hyperbolic Reuleaux triangle/b 1 (t), (3) b 1 (t)/b 1 (t) (horizontal line), (4) the capacity of the Euclidean Reuleaux triangle/b 1 (t) . Figure 5 (right) displays three sets of equal hyperbolic diameter: a disk, a hyperbolic Reuleaux triangle (solid line) and a Euclidean Reuleaux triangle (dashed line).  Table 2 we have cap(B 2 , E) < cap(B 2 , D) < cap(B 2 , T ) for all large enough t .
Remark 3.14. For the Lebesgue measure of a measurable set E ⊂ R n , the well-known isodiametric inequality states that m(E) ≤ m(B n (0, r)) where the Euclidean diameter of E is 2r [12, p.548, Thm C.10]. A similar result was proven very recently by K.J. Böröczky and Á. Sagmeister in [3] for the balls in the hyperbolic geometry. As the above computational results demonstrate, for the condenser capacity there is no similar result.

Upper bounds for the hyperbolic Jung radius
In view of Corollary 3.8, it is natural to look for bounds of the hyperbolic Jung radius of a compact set in a simply connected plane domain G. Perhaps a first question to study is whether we can find an upper bound in terms of the domain functional d(E)/d(E, ∂G). As Example 4.1 demonstrates, this is not true in general simply connected domains, but by (4.4) such a majorant is valid for ϕ-uniform domains.  for all x, y ∈ G.
The class of ϕ-uniform domains [9, pp. 84-85] contains many types of domains, including, for instance, all convex domains and so called quasidisks, which are images of the unit disk under quasiconformal maps of the plane [6]. Now, we observe that if E is a compact subset of a simply connected ϕ-uniform domain G, then by Theorem 3.3, Finally, we give a simple sufficient condition for a domain G ⊂ R 2 to be ϕ-uniform: There exists c ≥ 1 such that every pair of points x, y in G can be joined by a curve γ with length at most c|x − y| so that d(γ, ∂G) ≥ (1/c) min{d G (x), d G (y)}.
Remark 4.5. Recall that in every plane domain G , the hyperbolic diameter of a continuum E ⊂ G is bounded in terms of d(E)/d(E, ∂G) [9, 6.32] and hence so is its hyperbolic Jung radius by Theorem 3.3.
The values of the parameters in the functions hypdist and annq are chosen as in [18,19]. The codes for all presented computations in this paper are available in the link https://github. com/mmsnasser/hypdiam.