Copyright © 2010 Peter Hästö et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show that the equivalence of the Apollonian metric and its inner metric remains unchanged by the removal of a point from the domain. For this we need to assume that the complement of the domain is not contained in a hyperplane. This improves a result of the authors wherein the same conclusion was reached under the stronger assumption that the domain contains an exterior point.

1. Introduction and the Main Result

The Apollonian metric was first introduced by Barbilian [1] in 1934-35 and then rediscovered by Beardon [2] in 1995. This metric has also been considered in [3–14]. It should also be noted that the same metric has been studied from a different perspective under the name of the Barbilian metric, for instance, in [1, 15–20]; compare, for example, [21] for a historical overview and more references. One interesting historical point, made in [21], is that Barbilian himself proposed the name “Apollonian metric” in 1959, which was later independently coined by Beardon [2]. Recently, the Apollonian metric has also been studied with certain group structures [22].

In this paper we mainly study the equivalence of the Apollonian metric and its inner metric proving a result which is a generalization of Theorem in [12]. In addition, we also consider the metric and its inner metric, namely, the quasihyperbolic metric. Inequalities among these metrics (see Table 1) and the geometric characterization of these inequalities in certain domains have been studied in [12, 13]. We start by defining the above metrics and stating our main result. The notation used mostly is from the standard books by Beardon [23] and Vuorinen [24].

Table 1: Inequalities between the metrics , and . The subscripts are omitted for clarity with the understanding that every metric is defined in the same domain. The A-column refers to whether the inequality can occur in simply connected planar domains, the B-column refers to whether it can occur in proper subdomains of .

We will be considering domains (open connected nonempty sets) in the Mbius space . The “Apollonian metric” is defined for by the formula(1.1) (with the understanding that ) where denotes the boundary of . It is in fact a metric if and only if the complement of is not contained in a hyperplane and a pseudometric otherwise, as was noted in [2, Theorem ]. Some of the main reasons for the interest in the metric are that (i)the formula has a very nice geometric interpretation, see Section 2.2,(ii)it is invariant under Mbius map, (iii)it equals the hyperbolic metric in balls and half-spaces.

Now we define the inner metric as follows. Let be a path, that is, a continuous function. If is a metric in , then the -length of is defined by (1.2) where the supremum is taken over all and all sequences satisfying . All the paths in this paper are assumed to be rectifiable, that is, to have finite Euclidean length. The inner metric of the metric is defined by the formula (1.3) where the infimum is taken over all paths connecting and in . We denote the inner metric of the Apollonian metric by and call it the “Apollonian inner metric”. Strictly speaking, the Apollonian inner metric is only a pseudometric in a general domain ; it is a metric if and only if the complement of is not contained in an -dimensional plane [10, Theorem ]. We say that a path joining is a geodesic (of the metric ) if ; there always exists a geodesic path for the Apollonian inner metric connecting and in such that [10].

Let be a domain and . The metric [25], which is a modification of a metric from [26], is defined by (1.4) where denotes the shortest Euclidean distance from to the boundary of . The quasihyperbolic metric from [27] is defined by (1.5) where the infimum is taken over all paths joining and in . Note that the quasihyperbolic metric is the inner metric of the metric.

We now recall some relations on the set of metrics in for an overview of our previous work in [12].

Definition 1.1. Let and be metrics on (i)We write if there exists a constant such that , similarly for the relation . (ii)We write if and . (iii)We write if and .

Let us first of all note that the following inequalities hold in every domain (1.6) The first two are from [2, Theorem ] and the second two are from [7, Remark , Corollary ]. We see that, of the four metrics to be considered, the Apollonian is the smallest and the quasihyperbolic is the largest.

In this paper we are especially concerned with the relation , that is, the question whether or not the Apollonian metric is quasiconvex. We note that this always holds in simply connected uniform planar domains [7, Theorem , Lemma ]. Also, in convex uniform domains this relation always holds: from [6, Theorem ] we know that in convex domains; additionally, if is uniform; hence . On the other hand, there are also domains in which , for example, the infinite strip. Finally, we note that in [13, Corollary ] it was shown that implies that is uniform.

In [12], we have undertaken a systematic study of which of the inequalities in (1.6) can hold in the strong form with and which of the relations , and can hold. Thus we are led to twelve inequalities, which are given along with the results in Table 1, where we have indicated in column A whether the inequality can hold in simply connected planar domains and in column B whether it can hold in arbitrary proper subdomains of . Two entries, 11B and 12B, could not be dealt with at that time, but they have meanwhile been resolved in [13]. From the table we see that most of the cases cannot occur, which means that there are many restrictions on which inequalities can occur together.

One ingredient in the proofs of some of the inequalities in [12] was the following result, which shows that removing a point from the domain (i.e., adding a boundary point) does not affect the inequality .

Theorem 1.2. Let be a domain with an exterior point. Let and . If , then as well.

Note that by Mbius invariance, one may assume that the exterior point is in fact , in which case the domain is bounded, as was the assumption in the original source. This assumption was of a technical nature, and in this article we show that indeed it can be replaced by a much weaker assumption that the complement of is not contained in a hyperplane. Note that this is a minimal assumption for to be a metric in the first place, as noted above.

Theorem 1.3 (Main Theorem). Let be a domain whose boundary is not contained in a hyperplane. Let and . If , then as well.

The structure of the rest of this paper is as follows. We start by reviewing the notation and terminology. These tools will be applied in later sections to prove the new results of this article. The main problem in this paper is the inequality where the integral representation [10, Theorem ] of the Apollonian inner metric plays a crucial rule. The main result shows that if the boundary of the domain contains points which form extreme points of an -simplex, then the equivalence of the Apollonian metric and its inner metric will remain unchanged even if we remove a point from the original domain.

2. Background

2.1. Notation

The notation used conforms largely to that in [23, 24], as was mentioned in Section 1.

We denote by the standard basis of and by the dimension of the Euclidean space under consideration and assume that . For we denote by its th coordinate. The following notation is used for Euclidean balls and spheres: (2.1) For we denote by the smallest angle between the vectors and at .

We use the notation for the one-point compactification of , equipped with the chordal metric. Thus an open ball of is an open Euclidean ball, an open half-space, or the complement of a closed Euclidean ball. We denote by , and the boundary, complement, and closure of , respectively, all with respect to .

We also need some notation for quantities depending on the underlying Euclidean metric. For we write (2.2) For a path in we denote by its Euclidean length.

2.2. The Apollonian Balls Approach

In this subsection we present the Apollonian balls approach which gives a geometric interpretation of the Apollonian metric.

For we define (2.3) The numbers and are called the Apollonian parameters of and (with respect to ) and by the definition (2.4) The balls (in !), (2.5) are called the Apollonian balls about and , respectively. We collect some immediate results regarding these balls; similar results obviously hold with and interchanged.(1) and . (2)If and denote the inversions in the spheres and , then (2.6)(3)If , we have . If, moreover, , then .

2.3. Uniformity

Uniform domains were introduced by Martio and Sarvas in [28, ], but the following definition is an equivalent form from [26, equation ]. In the paper in [29] there is a survey of characterizations and implications of uniformity; as an example we mention that a Sobolev mapping can be extended from to the whole space if is uniform; see [30].

Definition 2.1. A domain is said to be uniform with constant if for every there exists a path , parameterized by arc-length, connecting and in , such that (1), (2).

The relevance of uniformity to our investigation comes from [26, Corollary ] which states that a domain is uniform if and only if . This condition is also equivalent to ; see [13, Theorem ]. Thus we have a geometric characterization of domains satisfying these inequalities as well.

2.4. Directed Density and the Apollonian Inner Metric

We start by introducing some concepts which allow us to calculate the Apollonian inner metric. First we define a directed density of the Apollonian metric as follows: (2.7) where . If is independent of in every point of , then the Apollonian metric is isotropic and we may denote and call this function the density of at . In order to present an integral formula for the Apollonian inner metric we need to relate the density of the Apollonian metric with the limiting concept of the Apollonian balls, which we call the Apollonian spheres.

Definition 2.2. Let and (i)If for every and , then let . (ii)If for every and , then let to be the largest negative real number such that . (iii)Otherwise let to be the largest real number such that .

Define in the same way but using the vector instead of . We define the Apollonian spheres through in direction by (2.8)for finite radii and by the limiting half-space for infinite radii.

Using these spheres, we can present a useful result from [7].

Lemma 2.3 (see [7, Lemma ]). Let be open, and . Let be the radii of the Apollonian spheres at in the direction . Then (2.9) where one understands .

The following result shows that we can find the Apollonian inner metric by integrating over the directed density, as should be expected. This is also used as a main tool for proving our main result. Piecewise continuously differentiable means continuously differentiable except at a finite number of points.

Lemma 2.4 (see [10, Theorem ]). If , then (2.10) where the infimum is taken over all paths connecting and in that are piecewise continuously differentiable (with the understanding that for all , even though is not defined).

3. The Proof of the Main Theorem

Proof. In this proof we denote by the distance to the boundary of , not of . It is enough to prove the inequality , because other way inequality always holds. Let and denote . Let be a path connecting and such that ; note that such a length-minimizing path exists by [10, Theorem ].Case 1. (a) and .

Let be such that . Let be the collection of boundary points of where they form the vertices of an -simplex. Denote by the largest ball with radius and centered at such that is inside the -simplex ; see Figure 1. Define . Denote by the ball with radius and centered at . Define . Let be a ball tangent to with maximal radius, denoted by .

Choose . Consider the ball centred at with radius and denote it by . Then we see that . Since (3.1) we see that (3.2) for .

We now estimate the density of the Apollonian spheres (see Definition 2.2) in passing through and in the direction . In order to compare the density with the densities and , we consider two possibilities of the choice of w.r.t. .

We first assume that . Denote by the ray from along . Consider a sphere with radius and centered at such that is tangent to . Denote . Construction of gives that, for , the Apollonian spheres passing through and in the direction are smaller in size than the sphere .

This gives (3.3) where and is obtained using the cosine formula in the triangle .

Now the sphere with radius and centre at passing through and gives (3.4) where and . If the Apollonian spheres (passing through and in the direction ) are affected by the boundary point , then by Lemma 2.3 we have (3.5) where denotes the radius of the smaller Apollonian sphere which touches . Denote . Since , using the sine formula in the triangle we get (3.6) Then we see that (3.7) Thus, from (3.5) we get (3.8) Since , we notice that the Euclidean triangle inequalities of the triangle give . We then obtain (3.9)

We next assume that . It is clear that if then is contained in a hyperplane, which contradicts our assumption. Thus if , then , and since the density function is continuous it has a greatest lower bound; namely, there exists a constant such that for we have (3.10) Therefore, (3.3) and (3.10) together give (3.11)

Since , we note that for all . Thus, if the Apollonian spheres passing through and in the direction are affected by the boundary point , then by Lemma 2.3(3.12) hold, where denotes the radius of the smaller Apollonian sphere which touches . Then (3.2), (3.9), (3.11), and (3.12) together give (3.13) Thus, by the definition of the inner metric and Lemma 2.4, we get the relation (3.14) for some constant . This gives (3.15) where the second inequality holds by assumption and the third holds trivially, as is a subdomain of .

(b) and intersects .

Let be an intersecting part of from to (if there are more intersecting parts, we proceed similarly). Let be the shortest circular arc on from to , as shown in Figure 2.

Using the density bounds (3.2) and (3.10), we get for every . Then we see that the inequalities (3.16) hold. But since and , we have . This shows that , where the path is obtained from by modifying with the circular arc joining to . Since , (3.13) implies that . So we get (3.17) Thus we have shown that holds for all .Case 2 (). Without loss of generality we assume that . Since , it is clear by the definition and the monotonicity property of the Apollonian metric that (3.18) Let , where is the path which is circular about the point from to and is the radial part from to , as shown in Figure 3.

Since the Apollonian spheres are not affected by the boundary point in the circular part, we have (3.19) where the first equality holds since the Apollonian metric equals the hyperbolic metric in a ball. For , by monotonicity in the domain of definition, we see that (3.20) Hence, by Lemma 2.4 we have (3.21) Since is increasing for and we have (3.22) for the choice , the inequality (3.23) holds. This inequality is equivalent to (3.24) Using , we easily get . We have thus shown that (3.25) for some constant .Case 3. and .

Let be such that (3.26) Let , where and is a path connecting and such that (3.27) As we discussed in the previous case, we have (3.28) Since , it follows by Case 1 that (3.29) It is now sufficient to see that .

If , then the triangle inequality and the fact together give (3.30) where the equality holds due to the fact that . But for , we have . For the choice , the inequality (3.30) reduces to (3.31) where the first inequality holds since and the last holds by the definition of the Apollonian metric.

We next move on to the case . If , we see (by the triangle inequality ) that (3.32) holds. Using (3.32) and the fact that , we get (3.33) Since , we get the following upper bound for (3.34) where the first inequality follows by the triangle inequality and the fact that . We see that the function is increasing for , so . Thus, for the choice , we get (3.35) where the last inequality holds by (3.32). On the other hand, if , then is bounded above by and is bounded below by , so the inequality is clear. Thus for the choice of , and we obtain (3.36) which concludes that (3.37)

We have now verified the inequality in all of the possible cases, so the proof is complete.

Figure 1: The largest ball tangent to and contained in , where .

Figure 2: The geodesic path (w.r.t. the Apollonian inner metric ) connecting and intersects , and its modification from to along the circular part.

Figure 3: A short path connecting and in .

Of course, we can iterate the main result, to remove any finite set of points from our domain. Like in [12], we get the following.

Corollary 3.1. Let be a domain whose boundary does not lie in a hyperplane. Suppose that is a finite nonempty sequence of points in and define . Assume that and . Then Inequality ( ) in Table 1, , holds.

Acknowledgments

The authors thank the referees for their careful reading of this paper and their comments. The work of the second and third authors was supported by National Board for Higher Mathematics, DAE, India.

References

1. D. Barbilian, “Einordnung von Lobayschewskys Massenbestimmung in einer gewissen allgemeinen Metrik der Jordansche Bereiche,” Casopsis Mathematiky a Fysiky, vol. 64, pp. 182–183, 1935.
2. A. F. Beardon, “The Apollonian metric of a domain in ${ℝ}^n$,” in Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), pp. 91–108, Springer, New York, NY, USA, 1998. View at Zentralblatt MATH · View at MathSciNet
3. M. Borovikova and Z. Ibragimov, “Convex bodies of constant width and the Apollonian metric,” Bulletin of the Malaysian Mathematical Sciences Society. Second Series, vol. 31, no. 2, pp. 117–128, 2008. View at Zentralblatt MATH · View at MathSciNet
4. F. W. Gehring and K. Hag, “The Apollonian metric and quasiconformal mappings,” in In the Tradition of Ahlfors and Bers (Stony Brook, NY, 1998), vol. 256 of Contemporary Mathematics, pp. 143–163, American Mathematical Society, Providence, RI, USA, 2000. View at Zentralblatt MATH · View at MathSciNet
5. A. D. Rhodes, “An upper bound for the hyperbolic metric of a convex domain,” The Bulletin of the London Mathematical Society, vol. 29, no. 5, pp. 592–594, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
6. P. Seittenranta, “Möbius-invariant metrics,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 125, no. 3, pp. 511–533, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
7. P. A. Hästö, “The Apollonian metric: uniformity and quasiconvexity,” Annales Academiæ Scientiarium Fennicæ. Mathematica, vol. 28, no. 2, pp. 385–414, 2003. View at Zentralblatt MATH · View at MathSciNet
8. P. A. Hästö, “The Apollonian metric: limits of the comparison and bilipschitz properties,” Abstract and Applied Analysis, no. 20, pp. 1141–1158, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
9. P. A. Hästö, “The Apollonian metric: quasi-isotropy and Seittenranta's metric,” Computational Methods and Function Theory, vol. 4, no. 2, pp. 249–273, 2004. View at Zentralblatt MATH · View at MathSciNet
10. P. A. Hästö, “The Apollonian inner metric,” Communications in Analysis and Geometry, vol. 12, no. 4, pp. 927–947, 2004. View at Zentralblatt MATH · View at MathSciNet
11. P. A. Hästö, “The Apollonian metric: the comparison property, bilipschitz mappings and thick sets,” Journal of Applied Analysis, vol. 12, no. 2, pp. 209–232, 2006. View at Zentralblatt MATH · View at MathSciNet
12. P. Hästö, S. Ponnusamy, and S. K. Sahoo, “Inequalities and geometry of the Apollonian and related metrics,” Romanian Journal of Pure and Applied Mathematics, vol. 51, no. 4, pp. 433–452, 2006. View at Zentralblatt MATH · View at MathSciNet
13. M. Huang, S. Ponnusamy, X. Wang, and S. K. Sahoo, “The Apollonian inner metric and uniform domains,” to appear in Mathematische Nachrichten.
14. Z. Ibragimov, “Conformality of the Apollonian metric,” Computational Methods and Function Theory, vol. 3, no. 1-2, pp. 397–411, 2003. View at Zentralblatt MATH · View at MathSciNet
15. D. Barbilian, “Asupra unui principiu de metrizare,” Studii şi Cercetări Matematice, vol. 10, pp. 68–116, 1959.
16. D. Barbilian and N. Radu, “Les $J$-métriques finslériennes naturelles et la fonction de représentation de Riemann,” Studii şi Cercetări Matematice, vol. 13, pp. 21–36, 1962. View at MathSciNet
17. L. M. Blumenthal, Distance Geometry. A Study of the Development of Abstract Metrics. With an Introduction by Karl Menger, vol. 13 of University of Missouri Studies, University of Missouri, Columbia, SC, USA, 1938.
18. W.-G. Boskoff, Hyperbolic Geometry and Barbilian Spaces, Istituto per la Ricerca di Base. Series of Monographs in Advanced Mathematics, Hadronic Press, Palm Harbor, Fla, USA, 1996. View at MathSciNet
19. W. G. Boskoff, M. G. Ciucă, and B. D. Suceavă, “Distances induced by Barbilian's metrization procedure,” Houston Journal of Mathematics, vol. 33, no. 3, pp. 709–717, 2007. View at Zentralblatt MATH · View at MathSciNet
20. P. J. Kelly, “Barbilian geometry and the Poincaré model,” The American Mathematical Monthly, vol. 61, pp. 311–319, 1954. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
21. W. G. Boskoff and B. D. Suceavă, “Barbilian spaces: the history of a geometric idea,” Historia Mathematica, vol. 34, no. 2, pp. 221–224, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
22. C. A. Nolder, “The Apollonian metric in Iwasawa groups,” preprint.
23. A. F. Beardon, The Geometry of Discrete Groups, vol. 91 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1995. View at MathSciNet
24. M. Vuorinen, Conformal Geometry and Quasiregular Mappings, vol. 1319 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1988. View at MathSciNet
25. M. Vuorinen, “Conformal invariants and quasiregular mappings,” Journal d'Analyse Mathématique, vol. 45, pp. 69–115, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
26. F. W. Gehring and B. G. Osgood, “Uniform domains and the quasihyperbolic metric,” Journal d'Analyse Mathématique, vol. 36, pp. 50–74, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
27. F. W. Gehring and B. P. Palka, “Quasiconformally homogeneous domains,” Journal d'Analyse Mathématique, vol. 30, pp. 172–199, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
28. O. Martio and J. Sarvas, “Injectivity theorems in plane and space,” Annales Academiae Scientiarum Fennicæ. Series A, vol. 4, no. 2, pp. 383–401, 1979. View at Zentralblatt MATH · View at MathSciNet
29. F. W. Gehring, “Uniform domains and the ubiquitous quasidisk,” Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 89, no. 2, pp. 88–103, 1987. View at Zentralblatt MATH · View at MathSciNet
30. P. W. Jones, “Quasiconformal mappings and extendability of functions in Sobolev spaces,” Acta Mathematica, vol. 147, no. 1-2, pp. 71–88, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet