#### 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  in 1934-35 and then rediscovered by Beardon  in 1995. This metric has also been considered in . 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, 1520]; compare, for example,  for a historical overview and more references. One interesting historical point, made in , is that Barbilian himself proposed the name “Apollonian metric” in 1959, which was later independently coined by Beardon . Recently, the Apollonian metric has also been studied with certain group structures .

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 . 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  and Vuorinen .

We will be considering domains (open connected nonempty sets) in the Mbius space . The “Apollonian metric” is defined for by the formula (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 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 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 .

Let be a domain and . The metric , which is a modification of a metric from , is defined by where denotes the shortest Euclidean distance from to the boundary of . The quasihyperbolic metric from  is defined by 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 .

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 : 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 , 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 . 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  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: 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 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 The numbers and are called the Apollonian parameters of and (with respect to ) and by the definition The balls (in !), 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 (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  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 .

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: 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 for finite radii and by the limiting half-space for infinite radii.

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

Lemma 2.3 (see [7, Lemma ]). Let be open, and . Let be the radii of the Apollonian spheres at in the direction . Then 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 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 we see that 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 where and is obtained using the cosine formula in the triangle .
Now the sphere with radius and centre at passing through and gives 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 where denotes the radius of the smaller Apollonian sphere which touches . Denote . Since , using the sine formula in the triangle we get Then we see that Thus, from (3.5) we get Since , we notice that the Euclidean triangle inequalities of the triangle give . We then obtain
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 Therefore, (3.3) and (3.10) together give
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 hold, where denotes the radius of the smaller Apollonian sphere which touches . Then (3.2), (3.9), (3.11), and (3.12) together give Thus, by the definition of the inner metric and Lemma 2.4, we get the relation for some constant . This gives 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 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 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 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 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 Hence, by Lemma 2.4 we have Since is increasing for and we have for the choice , the inequality holds. This inequality is equivalent to Using , we easily get . We have thus shown that for some constant .
Case 3. and .
Let be such that Let , where and is a path connecting and such that As we discussed in the previous case, we have Since , it follows by Case 1 that It is now sufficient to see that .
If , then the triangle inequality and the fact together give where the equality holds due to the fact that . But for , we have . For the choice , the inequality (3.30) reduces to 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 holds. Using (3.32) and the fact that , we get Since , we get the following upper bound for : 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 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 which concludes that
We have now verified the inequality in all of the possible cases, so the proof is complete.

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