Condenser capacity and hyperbolic perimeter

We study the conformal capacity by using novel computational algorithms based on implementations of the fast multipole method, and analytic techniques. Especially, we apply domain functionals to study the capacities of condensers $(G,E)$ where $G$ is a simply connected domain in the complex plane and $E$ is a compact subset of $G$. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of $E$. Our computational experiments demonstrate, for instance, sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed.


Introduction
A condenser is a pair (G, E), where G ⊂ R n is a domain and E is a compact non-empty subset of G. The conformal capacity of this condenser is defined as [16,22,25] cap(G, E) = inf u∈A G |∇u| n dm, (1.1) where A is the class of C ∞ 0 (G) functions u : G → [0, ∞) with u(x) ≥ 1 for all x ∈ E and dm is the n-dimensional Lebesgue measure. The conformal capacity of a condenser is one of the key notions in potential theory of elliptic partial differential equations [25,34] and it has numerous applications to geometric function theory, both in the plane and in higher dimensions, [12,16,17,22].
The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length, or, perimeter. Constrained extremal problems of this type, where the constraints involve geometric or physical quantities, have been studied in several thousands of papers, see e.g. [4, p. 151], [5,10,13,33]. Motivated by the fact that many domain functionals of mathematical physics such as capacity, moment of inertia, principal frequency, or torsional rigidity have analytic formulas only in rare exceptional cases, Pólya and Szegő developed [41] a systematic theory to prove upper and lower bounds for these functionals in terms of simpler domain functionals such as area, perimeter, inradius, and circumradius. In addition to these domain functionals, they used the method of symmetrization as a method to transform a condenser (G, E) onto another, symmetrized condenser (G * , E * ) . The key fact here is that the integral in (1.1) decreases under symmetrization [4, p. 96] and hence we will have a lower bound cap (G, E) ≥ cap (G * , E * ). (1.2) There are several variants of the symmetrization method and depending on which one is applied, the sets G * and E * exhibit various symmetry properties with respect to spheres or hyperplanes [4, p. 253]. Due to this symmetry, the capacity of the symmetrized condenser is often easier to estimate than the original one.
Note that, while the lower bound of (1.2) is clearly sharp if (G, E) = (G * , E * ), in most cases the symmetrization method only yields crude estimates. The domain functionals of Pólya and Szegő [41], such as volume, area, perimeter, inradius, and circumradius, expressed in terms of Euclidean geometry, have numerous applications and they behave well under symmetrization, but they do not seem to be natural in the study of conformal capacity. The reason is that Euclidean geometry does not reflect optimally the conformal invariance of the conformal capacity. This observation led us to use hyperbolic geometry, which is available in the planar case n = 2, when the domain G is the unit disk B 2 or, more generally by Riemann's mapping theorem, a simply connected plane domain. For dimensions n ≥ 3, Liouville's theorem states that conformal mappings are Möbius transformations and hence Riemann's theorem does not apply. For generalized versions of Liouville's theorem, see Yu.G. Reshetnyak [ Many authors have proved upper and lower bounds for several kinds of capacities, including the conformal capacity that we are focusing here on, see for instance V. Maz´ya [34]. In spite of all this work, there does not seem to exist bounds in the form with a quantitative upper bound for the deviation U − L, even in the simplest case G = B 2 . In particular, there is no quantitative two-sided variant for the symmetrization inequality (1.2). Here, a fundamental difficulty is that the value of cap (G, E) is unknown. For the isoperimetric inequality, quantitative variants have been proved by N. Fusco [13] and, in the framework of the hyperbolic geometry, by V. Bögelein, F. Duzaar, Ch. Scheven [10]. Inequalities for the p-capacity were proved very recently by J. Xiao [48] and E. Mukoseeva [35]. In a series of papers [8,9,19,20,21], the third author with several coauthors has studied numerical computation of condenser capacities using the finite element method.
Despite the extensive literature dealing with condenser capacity [17,22,34], the actual values of cap (G, E) have remained rather elusive quantities. This is largely due to the unavailability of computational tools that can be used for wide ranges of domains. In fact, we have not seen a systematic compilation of concrete bounds for the capacity published since the pioneering work of Pólya and Szegő [41].
In this paper, our goal is to combine analytic methods with efficient numerical techniques and with extensive experiments to demonstrate the precision of the methods and the behavior of the numerical algorithms. To find new upper and lower bounds for the condenser capacity, we introduce new kinds of domain functionals expressed in terms of hyperbolic geometry of the unit disk B 2 . The numerical methods of these experiments are based on the boundary integral method developed by the first author and his coauthors in a series of recent papers [32,36,37,38,39,40].
The first question we study is whether the symmetrization lower bound (1.2) for cap (B 2 , E), where the interior of E is a simply connected domain with a piecewise smooth boundary ∂E, could be improved by using the hyperbolic perimeter of E. We will give examples to demonstrate that this is not true in general. For convex sets E, we have a positive result. Our experiments led us to an experimental verification of the next two theorems. We learned afterwards that both results are well-known. where I = [0, r] is a segment with the same hyperbolic perimeter as ∂E.
Note that a spherical symmetrization argument, see Lemma 3.11(1) below, shows that where the segment [0, s] ⊂ B 2 has the same hyperbolic diameter as the set E and therefore s ≤ r and hence Theorem 1.4 gives a better lower bound than the symmetrization method.
where F is a disk with hyperbolic perimeter equal to that one of E .
For some observations about Theorem 1.5 see Remark 3.6. The remaining part of this paper is organized as follows: Sections 2 and 3 present preliminary materials about hyperbolic geometry, quadrilateral and its modulus, and conformal capacity, which will be used in the following sections. Analytical results for simple condensers are presented in Section 4 and numerical methods for computation of the capacity of condensers and the modulus of quadrilaterals are presented in Section 5. In these two sections, we compare various lower bounds for the capacity to the symmetrization inequality (1.2). We also give a lower bound for cap (G, E) in the case when G \ E is a polygonal ring domain. This lower bound is sharp in the case when the polygonal ring domain has certain regularity properties. The results of this computational experiment are presented in the form of tables and graphics. In the final section of this paper, Section 6, we point out that finding connections between the geometric domain functional d(E)/d(E, ∂G) and cap (G, E) seems to offer problems for further investigations.
Acknowledgements. The second author was financially supported by the University of Turku Graduate School UTUGS. We are indebted to D. Betsakos and R. Kühnau for informing us about the literature [15,Corollary 6] and [30, p. 99, Thm 9.8], resp., related to Theorems 1.5 and 1.4. We are also thankful to the referees for their useful and constructive comments.

Preliminaries
Consider first the notations for the Euclidean metric. Let d G (x) be the Euclidean distance between a point x in a domain G and the boundary ∂G. Denote the Euclidean diameter of a nonempty set F by d(F ) and the Euclidean distance between two non-empty sets E, F by d(E, F ). Denote the Euclidean open ball with a center x ∈ R n and a radius r > 0 by B n (x, r), the corresponding closed ball by B n (x, r) and its boundary sphere by S n−1 (x, r). Suppose that x = 0 and r = 1 here, if they are not otherwise specified.
2.1. Hyperbolic geometry. Define now the hyperbolic metric in the Poincaré unit ball B n as in [6], [7, (2.8) Here and below, sh, ch and th stand for the hyperbolic sine, cosine and tangent functions, respectively. Let J[x, y] be the hyperbolic segment between the points x, y and [x, y] its Euclidean counterpart. Note that, for any simply connected domain G R 2 , one can choose a conformal map f : G → B 2 = f (G) by means of Riemann's mapping theorem and thus define the hyperbolic metric ρ G in G by [7] 2.4. Hyperbolic disks. We use the notation for the hyperbolic disk centered at x ∈ B 2 with radius M > 0 . It is a basic fact that they are Euclidean disks with the center and radius given by [22, p. 56, (4.20)] Note the special case x = 0, Lemma 2.7. [6, Thm 7.2.2, p. 132] The area of a hyperbolic disc of radius r is 4πsh 2 (r/2) and the length (or the perimeter) of a hyperbolic circle of radius r is 2πsh(r).

Quasihyperbolic metric.
For a domain G R n , define the weight function as in [22, (5 By [22, (5.2), p. 68], the quasihyperbolic distance between x, y ∈ G is now where Γ xy is the family of all rectifiable curves in G joining x and y. Note that if G is a simply connected domain in the plane, then the quasihyperbolic metric fulfills the inequality for all points x, y ∈ G [14, p. 21, (4.15)].
The next lemma is based on a standard covering lemma. Note that here the connectedness of the set F is essential as shown in [46].
2.11. Quadrilateral and its modulus. A bounded Jordan curve in the complex plane divides the extended complex plane C ∞ = C ∪ {∞} into two domains D 1 and D 2 so that the common boundary of these domains is the curve in question. One of these domains is bounded and the other one is unbounded. If D 1 is the bounded domain and z 1 , z 2 , z 3 , z 4 ∈ ∂D 1 are distinct points occurring in this order when traversing ∂D 1 in the positive direction, then (D 1 ; z 1 , z 2 , z 3 , z 4 ) is a quadrilateral [12].

Capacity
Let G ⊂ R n be a domain and E ⊂ G a compact non-empty set. For k = 1, 2, . . ., choose domains G k and compact sets E k such that Then it is well-known that lim k→∞ cap (G k , E k ) = cap (G, E), (3.1) see [16, p. 167]. Unfortunately, there does not seem to exist a quantitative estimate for the speed of convergence in (3.1).
Numerous variants of the definition (1.1) of capacity are given in [16]. First, the family A may be replaced by several other families by [16,Lemma 5.21,p. 161]. Furthermore, where ∆(E, F ; G) is the family of all curves joining nonempty sets E and F in the closure of the domain G and M stands for the modulus of a curve family [16,Thm 5.23,p. 164]. For the basic facts about capacities and moduli, the reader is referred to [16,22,44]. (2) If R > 0, then for x ∈ B n , M(∆(S n−1 , B ρ (x, R); B n )) = ω n−1 (log(1/th(R/2))) 1−n .
The desired formula now follows using the half angle formula for th .  Here, e 1 , . . . , e n are the unit vectors of R n . These capacity functions fulfill γ n (s) = 2 n−1 τ n (s 2 −1) for s > 1 and several estimates are given in [22,Chapter 9] for n ≥ 3 . In the case n = 2, r ∈ (0, 1), the following explicit formulas are well-known [22, (7.18), p. 122], 3.9. Quadrilateral modulus and curve families. The modulus of a quadrilateral (D; z 1 , z 2 , z 3 , z 4 ) defined in 2.11 is connected with the modulus of the family of all curves in D, joining the opposite boundary arcs (z 2 , z 3 ) and (z 4 , z 1 ), in a very simple way, as follows The next lemma is based on the symmetrization method. (1) If x, y ∈ B n , x = y , and E ⊂ B n is a continuum with x, y ∈ E, then Equality holds here if E is the geodesic segment J[x, y] of the hyperbolic metric joining x and y .
(2) If G is a simply connected domain in R 2 , E ⊂ G is a continuum, and x, y ∈ G, x = y , then 3.12. Jung radius in quasihyperbolic geometry. For a domain G ⊂ R n and a compact non-empty set E ⊂ G, define the Jung radius of E in the quasihyperbolic metric as From the monotonicity property of the capacity, we immediately get the upper bound for some z ∈ G and T = r k−Jung (E) ≤ k G (E). In particular, by Lemmas 2.10 and 3.3 and the subadditivity of the modulus [22,Ch. 7], for λ ∈ (0, 1), for a continuum E ⊂ G where m is as in Lemma 2.10.

Analytical results for simple condensers
In this section, we study cap (G, E) and relate its values to various domain functionals. In particular, we focus on the symmetric condenser of Lemma 4.2 and show that the capacity cap (G, E) cannot be estimated from below in the same way as in Theorem 1.4 because here E is nonconvex. We also consider the case when G \ E is a polygonal ring domain defined in Subsection 4.15. We apply the Schwarz-Christoffel transformation to give an algorithm for a lower bound of its capacity. Note that here it is not required that E is convex. This algorithm will be implemented in the next section.

Symmetric segments.
We consider here condensers of the form For these condensers the capacity can be explicitly given.
Proof. Let θ = 2π/m andD The domainD 1 can be mapped by the conformal mapping onto the upper half of the unit disk and the two segments from s to 0 and from 0 to s e θi are mapped onto the segment [−s m/2 , s m/2 ]. Let∆ be the family of curves in the upper half of the unit disk connecting the segment [−s m/2 , s m/2 ] to the upper half of the unit circle. Then by symmetry, By symmetry, it also follows from [22, 7.12, 9.20], Using the formula (2.2), we have and hence M(∆) = π µ (2s m/2 /(s m + 1)) .

By [22, (7.20)]
µ(s m ) = 2µ 2s m/2 1 + s m which together with the previous equality yields The proof then follows from (4.3) and (4.4). Proof. The hyperbolic perimeter of a segment [0, r] is equal to twice of its hyperbolic diameter, so the value of the perimeter is by Lemma 2.7 4 arthr = 2 log 1 + r 1 − r , 0 < r < 1 and its capacity is by (3.8) and the perimeter of set E in Lemma 4.2 is For t > 0, choose now r and s such that Then for these values of r and s, the capacities are .
Now, we claim that for some values of the parameters s and m, equivalently t and m, such that and numerical computation shows that we can choose, for instance, m = 5 and s = 0.5.
The Euler integral representation [1,29] (4.10) links the hypergeometric function with the conformal Schwarz-Christoffel transformation. As shown in [24], this transformation delivers a conformal map of the upper half plane onto a polygonal quadrilateral.
(1) The hypotheses in Theorem 4.11 imposed on the triple a, b, c imply that the quadrilateral Q is convex.
(2) The algorithm of Theorem 4.11 will be implemented and applied for numerical computation in the following sections.

Polygonal ring domains.
A domain G ⊂ R n is called a ring domain if it is homeomorphic to the spherical annulus B n (t)\B n (s) for some numbers 0 < s < t. We consider here planar domains characterized as follows: There exist closed convex quadrilaterals P j , j = 1, . . . , m, with intP j ∩ intP k = ∅ for all j = k such that P = int ∪ m j=1 P j is a ring domain and its both boundary components are polygonal with m vertices. Assume, moreover, that the inner boundary component of P can be written as where I j and E j are opposite sides of P j . The set P is now called a polygonal ring domain (see Figure 1). The degenerate case when the inner polygon is a segment, will be also studied in Section 5, see Figure 19.

Numerical algorithms
In this section, we describe numerical methods for computation of the capacity of condensers and the modulus of quadrilaterals.
5.1. Computation of hyperbolic perimeter in the unit disk. If E ⊂ B 2 is a continuum with piecewise smooth boundary, then the hyperbolic perimeter of E is [6] h The integrand is 2π-periodic and hence it can be accurately approximated by the trapezoidal rule [11] to obtain and n is an even integer.

5.3.
Computation of hyperbolic perimeter in simply connected domains. As above, let E ⊂ G be a continuum in a simply connected domain G and f : G → B 2 = f (G) be a conformal map. Then f maps the connected set E onto a connected setÊ ⊂ B 2 . If h-perim G (E) is the hyperbolic perimeter of E with respect to the hyperbolic metric ρ G in G, is computed by approximating numerically the conformal mapping f , which is done here by the numerical method presented in [37,39]. The derivative ζ (t) is computed by approximating the real and imaginary parts of ζ(t) by trigonometric interpolating polynomials and then differentiating the interpolating polynomials. These polynomials can be computed using FFT [47]. In terms of ζ(t) and ζ (t), we can compute h-perim G (E) by 5.4. Algorithm for the capacity of a polygonal ring domain. Consider a bounded simply connected domain G in the complex plane and a compact set E ⊂ G such that D = G \ E is a doubly connected domain. In this paper, the capacity of the condenser (G, E) will be computed by the MATLAB function annq from [40]. In this function annq, the capacity is computed by a fast method based on an implementation of the Fast Multipole Method toolbox [18] in solving the boundary integral equation with the generalized Neumann kernel [36].
We assume that the boundary components of D = G\E are piecewise smooth Jordan curves. Let Γ 1 be the external boundary component and Γ 2 be the inner boundary component such that Γ 1 is oriented counterclockwise and Γ 2 is oriented clockwise. We parametrize each boundary component Γ j by a 2π-periodic function η j (δ j (t)), 0 ≤ t ≤ 2π, where δ j : [0, 2π] → [0, 2π] is a bijective strictly monotonically increasing function and η j is a 2π-periodic parametrization of Γ j , which is assumed to be smooth except at the corner points. The function δ j is introduced to remove the singularity in the solution of the integral equation at the corner points [28]. When Γ j is smooth, we assume δ j (t) = t, 0 ≤ t ≤ 2π. If Γ j has corners, we choose the function δ j as in [32, p. 697]. Then, we define the vectors et and etp in MATLAB by The values of the parameters in the function annq are chosen as in [40].
The readers are referred to [32,36,40] for more details. The values of µ(r) are then computed as described in [40]. This method for computing QM (A, B) is implemented in MATLAB as in the following code, which is based on the Mathematica code presented in [24]. The function u should be identically zero by (5.7). The surface plot of this function u is presented in Figure 3. The figure shows that the maximum value of u(x, y) over the rectangle is of the order 10 −14 . The MATLAB function QM is also tested for several values of A and B as in Table 1 where, in view of (5.7), the error is computed by    To compute M(Γ 1 ), we use the linear transformation z → z − 1 e 2πi/m − 1 to map the quadrilateral P 1 onto the quadrilateral Q 1 with the vertices 0, 1, A, and B, where (see Figure 4 (right)) Then M(Γ 1 ) = 1/M(Q 1 ), and hence cap P, .
The values of M(Q 1 ) are computed using the above MATLAB function QM and these values are considered as exact values. The values of cap (P, ∪ m j=1 I j ) are computed numerically using the MATLAB function annq. The absolute error between the values of m/M(Q 1 ) and the approximate values of cap (P, ∪ m j=1 I j ) for several values of m and λ are presented in Figure 5. Table 2 presents the values of cap (P, ∪ m j=1 I j ) obtained with the function annq for several values of m and λ. For m = ∞, the outer boundary of P reduces to the unit circle |z| = 1 and the inner boundary reduces to the circle |z| = λ and hence the capacity is 2π/ log(1/λ).
the quadrilateral P j is mapped onto the quadrilateral Q j with the vertices 0, 1, A j , and B j where . . , m (see Figure 1). Let Γ j = ∆(I j , E j ; P j ) so that Hence, by Lemma 4.18, we have .
For several values of m, the values of M(Q j ) are computed using the MATLAB function QM and the values of cap (P, ∪ m j=1 I j ) are computed numerically using the MATLAB function annq. The obtained numerical results are presented in Figure 6. These results validate the inequality (5.10).

Convex Euclidean polygons.
Assume that E ⊂ B 2 is a convex Euclidean polygonal closed region such that ∂E is a polygon with m vertices v 1 , v 2 , . . ., v m where m is an integer chosen randomly such that 3 ≤ m ≤ 12. We choose a real number s randomly such that 0.05 < s < 0.95, then we assume the vertices are given by where τ j is a random number on (0, 1) (see Figure 7 for The capacity cap (B 2 , E) is calculated using the MATLAB function annq.

Hyperbolically convex polygons.
Assume that E ⊂ B 2 is a hyperbolically convex polygonal closed region such that ∂E is a hyperbolic polygon with m vertices v 1 , v 2 , . . ., v m chosen as in (5.12) (see Figure 9 for m = 10 and s = 0.9308).

Nonconvex Euclidean polygons.
Assume that E ⊂ B 2 is a nonconvex Euclidean polygonal closed region such that ∂E is a polygon with m vertices v 1 , v 2 , . . . , v m , where m is an integer chosen randomly such that 3 ≤ m ≤ 12 and θ 1 , θ 2 , . . . , θ m are as in (5.12). Then, we assume that the vertices are given by where s j is chosen randomly such that 0.5 < s 2j−1 < 0.95 and 0.05 < s 2j < 0.5 (see Figure 11 for m = 12).
The boundary of E(t) consists of two circular arcs and two hyperbolic half-circles. Let the radius of the two circular arcs be u and v where 0 < v < r < u and ρ B 2 (r, v) = ρ B 2 (r, u) = t. Letv = 2 arth (v),r = 2 arth (r),û = 2 arth (u), thenû −r =r −v = t, and henceû =r + t andv =r − t. Thus The two circular arcs have the hyperbolic center 0 and the hyperbolic radiiû andv. Hence, by Lemma 2.7, the total hyperbolic length of these two circular arcs is which can be simplified as The two hyperbolic half-circles have hyperbolic centers r, r iθ and a hyperbolic radius t. Thus, by Lemma 2.7, the total hyperbolic length of these two hyperbolic half-circles is 2πsh(t). Hence, the hyperbolic perimeter of the set E(t) is Finally, let I(t) = [0, a(t)] where a(t) is a positive real number such that h-perim B 2 (I(t)) = f (t), i.e., a(t) = th(f (t)/4). Thus, as in (5.14), (5.19) cap (B 2 , I(t)) = 2π µ(th(f (t)/4)) .
The hyperbolic perimeter of E(t) with respect to the metric ρ G will be computed using the method described in Subsection 5.3. Let f (t) = h-perim G (E(t)) and let D(t) = B 2 (0, R(t)) be the Euclidean disk with h-perim B 2 (∂D(t)) = f (t). Then, it follows from (3.5) that cap (B 2 , D(t)) is given by cap (B 2 , D(t)) = 2π .
The values of the capacities cap (G, E(t)) and cap (B 2 , D(t)) for 0.01 ≤ t ≤ 2 are presented in Figure 16. The polygonal domain here is of the type considered in Subsection 5.9 with m = 8. Thus, a lower bound for the capacity cap (G, E(t)) can be obtained from the inequality (5.10), where the quadrilaterals Q j are defined as in Subsection 5.9. The computed values for this lower bound are presented in Figure 16. As we can see from these results, the values of this lower bound become close to the values of cap (G, E(t)) as t increases.  5.21. Capacity of a half disk. We consider here the capacity where 0 < x < 1 and t > 0 (see Figure 17). By Lemma 2.7, the hyperbolic perimeter of E is P ≡ π sht + 2t.  Table 3. The values of the capacity cap (B 2 , E) should not depend on the values x and this fact could be used to check the accuracy of the MATLAB function annq used to compute cap (B 2 , E). It is clear from Table 3 that the obtained values of cap (B 2 , E) for x = 0.5 and x = 0.75 are almost identical, where the absolute value of the differences between these values are: 1.0 × 10 −12 for t = 0.5, 1.1 × 10 −11 for t = 1, 2.1 × 10 −10 for t = 2, and 1.8 × 10 −9 for t = 3. For x = 0.75, the boundary of the inner half circle becomes even closer to the outer boundary compared to the case x = 0.5 especially for large t (see Figure 17). Hence, the results obtained for x = 0.75 will not be as accurate as for x = 0.5 and this could explain the increase in the absolute value of the differences between the obtained values of cap (B 2 , E) for x = 0.5 and x = 0.75.  As Table 3 indicates, the upper bound for the capacity is more accurate than the lower bound in Theorem 1.4 which is sharp when the set E is a segment. Thus it is natural to look for a better lower bound for a massive set such as the half disk. The next lemma provides such a bound (see Table 4 and Figure 18).  cap (B 2 , E) ≥ π/ log(1/th(t/2)) + π/µ(tht) .
Proof. Without loss of generality, we may assume that x = 0 . Then by (2.5), B ρ (0, t) = B 2 (0, th(t/2)) . Let H 2 = {z ∈ R 2 : Imz > 0} and Then Γ 1 and Γ 2 are separate subfamilies of Γ and hence by a symmetry property of the modulus [22,p. 127,Thm 4.3.3] cap (B 2 , E) ≥ M(Γ 1 ) + M(Γ 2 ) = π/ log(1/th(t/2)) + π/µ(tht), and therefore the lemma is proved.   where a, b, c, d are positive real numbers such that 0 < c < d < b (see Figure 19 (left)). The exact value of the capacity cap (F, E) can be obtained with the help of conformal mappings. Let where K(·) is defined by (3.8) and The function sn −1 (w; k) is a Schwarz-Christoffel transformation mapping H 2 conformally onto the rectangle with corners ± K(k), The values of the capacity cap (F, E) for several values of a, b, c, d are presented in Table 5. The ring domain F \E can be regarded as a ring domain of the type considered in Subsection 5.9 with m = 6 (see Figure 19 (right)). Thus, a lower bound for the capacity cap (F, E) can be obtained from the inequality (5.10) where the quadrilateral Q j are defined as in Subsection 5.9 (the vertices of the quadrilaterals are the black dots in Figure 19 (right)). By symmetry, we have M(Q 1 ) = M(Q 4 ), M(Q 2 ) = M(Q 3 ), and M(Q 5 ) = M(Q 6 ). Using linear transformation, the quadrilateral Q 1 can be mapped to a quadrilateral in the upper half-plane such that the vertex a is mapped to 0 and the vertex a + bi is mapped to 1. Then the value of M(Q 1 ) will be computed using the MATLAB function QM. Similarly, the quadrilaterals Q 2 and Q 5 can be mapped onto quadrilaterals of the form described in Theorem 4.21 and then the MATLAB function QMt is used to compute the values of M(Q 2 ) and M(Q 5 ). The values of the lower bound (5.27) are given in Table 5. , E) depends on the "size" or the "shape" of the set E. We now discuss some known results of this type and thereby point out ideas for further studies.
For a compact set E ⊂ R n , we consider its tubular neighbourhood, defined as and study the function c(E, t) ≡ cap (E(t), E).
It is a well-known fact that for every compact set E ⊂ R n , the boundary of its tubular neighborhood has a finite (n − 1)-dimensional measure. This fact was refined and further studied under various structure conditions on the set E in [26]. Note that the function c(E, t) here is decreasing with respect to t and its behaviour is closely related to the size of the set E when t → 0 + . As we will see below, this dependency is mutual: if E is "thick" or "big", there is a lower bound for c(E, t) tending to ∞, whereas, if the function c(E, t) converges to ∞ slowly, the set E is small.
We say that the compact set E is of capacity zero if c(E, t 0 ) = 0 for some t 0 > 0 and denote this by cap E = 0 , in the opposite case cap E > 0 . Theorem 6.2. (J. Väisälä [45]) If lim t→0 + c(E, t) < ∞, then E is of capacity zero.
The theorem above follows from the results in [45]. It should be observed that if E is of capacity zero, then c(E, t) = 0 for all t > 0. In his PhD thesis [23], V. Heikkala proved the next result and attributed the idea of its proof to J. Mály. Theorem 6.3. (V. Heikkala [23,Thm 4.6]) Let h : (0, ∞) → (0, ∞) be a decreasing homeomorphism, which satisfies h(t) → ∞ as t → 0 + . Then there exists a compact set E ⊂ R n with cap E > 0 satisfying c(E, t) < h(t) for all t ∈ (0, 1).
Furthermore, Heikkala studied the function c(E, t) in more detail under various measure theoretic thickness conditions. For instance, he proved a lower bound for c(E, t) if E is uniformly perfect and an upper bound if E satisfies the so called Ahlfors condition. For these results, see [23]. Heikkala's results were refined, extended and generalized by J. Lehrbäck [31] to the context of metric measure spaces and more general capacities.
The aforementioned results [23,31] dealing with uniformly perfect sets or sets satisfying the Ahlfors condition depend on the pertinent structure parameters of the set E and hence so do the obtained growth estimates for c(E, t). There are also results of other type where bounds such as c(E, t) ≤ a 1 t −n c(E, 1), t ∈ (0, 1), (6.4) were proved for a compact set E ⊂ B n with the constant a 1 only depending on n. See [22,Lemma 8.22], [42, Lemma 3.3, p. 60]. The proof makes use of the Whitney extension theorem and standard gradient estimates for mollifying functions, see also [31] and [34,Ch. 13]. The point here is that the growth rate of t −n is independent of E and the power −n is the best possible (independent of E) as shown in [22, p. 146].
We conclude by discussing the possible use of the domain functional d(E)/d(E, ∂G) in the estimation of the capacity cap (G, E) when G ⊂ R 2 is a bounded simply connected domain. It follows from Lemma 2.10 and (3.16) that such a bound exists if the set E is connected.
The class of simply connected domains is too general for our purpose; it contains many potential theoretic counterexample domains such as "rooms connected by narrow corridors", which we would have to exclude. Thus we consider a suitable subclass of domains [22, p. 84].
Simple examples of ϕ-domains are convex domains, which satisfy the condition above with ϕ(t) ≡ t. More generally, suppose that there exists a constant c ≥ 1 such that, for all x, y ∈ G, there exists a curve γ joining x and y so that (γ) ≤ c|x−y| and d G (z) ≥ (1/c) min{d G (x), d G (y)} for all z ∈ γ. In this case, the domain G is ϕ-uniform with ϕ(t) ≡ c 2 t [46, 2.19 (2)].
Suppose now that F is a compact set in a simply connected ϕ-uniform domain G. Then k G (F ) ≤ ϕ d(F ) d(F, ∂G) and, because clearly r k−Jung (F ) ≤ k G (F ), the inequality (3.15) yields cap (G, F ) ≤ cap (G, B k (ϕ(d(F )/d(F, ∂G)))). (6.6) Next, recall that in a simply connected plane domain G by (2.9) k G (F ) ≤ 2ρ G (F ) (6.7) and, finally, for a compact set F in a ϕ-uniform simply connected planar domain G we see by (2.5) that cap (G, F ) ≤ 2π/th(U/2), U = 2ϕ(d(F )/d(F, ∂G)). (6.8) Further study of the connection between the domain functional d(F )/d(F, ∂G) and cap (G, F ) seems to be worthwhile. For instance, sharp inequalities are unknown.