# Equivalence of the Apollonian and Its Inner Metric

Authors: Peter Hästö; S. Ponnusamy; S. K. Sahoo
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## 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.


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## Body

## 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 Theorem5.1 in [12]. In addition, we also consider the jD 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 metricsαD, jD, α̃D, and kD. 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 ℝn.
No.InequalityABNo.InequalityAB(1)α≈j≈α̃≈k++(7)α≈j≪α̃≪k−−(2)α≪j≈α̃≈k−−(8)α≪j≪α̃≪k−−(3)α≈j≈α̃≪k−−(9)α≈α̃≪j≈k−+(4)α≪j≈α̃≪k−−(10)α≪α̃≪j≈k−+(5)α≈j≪α̃≈k++(11)α≈α̃≪j≪k−−(6)α≪j≪α̃≈k++(12)α≪α̃≪j≪k−−We will be considering domains (open connected nonempty sets)D in the Möbius space ℝn¯:=ℝn∪{∞}. The “Apollonian metric” is defined for x,y∈Dℝn¯ by the formula(1.1)αD(x,y):=supa,b∈∂Dlog|a-yb-y||b-xb-y||a-xb-y||b-y|,
(with the understanding that |∞-x|/|∞-y|=1) where ∂D denotes the boundary of D. It is in fact a metric if and only if the complement of D is not contained in a hyperplane and a pseudometric otherwise, as was noted in [2, Theorem 1.1]. Some of the main reasons for the interest in the metric are that  (i)
the formula has a very nice geometric interpretation, see Section2.2,(ii)
it is invariant under Möbius map,(iii)
it equals the hyperbolic metric in balls and half-spaces.Now we define the inner metric as follows. Letγ:[0,1]→D⊂ℝn be a path, that is, a continuous function. If d is a metric in D, then the d-length of γ is defined by (1.2)d(γ):=sup∑i=0k-1d(γ(ti),γ(ti+1)),
where the supremum is taken over all k<∞ and all sequences {ti} satisfying 0=t0<t1<⋯<tk=1. All the paths in this paper are assumed to be rectifiable, that is, to have finite Euclidean length. The inner metric of the metric d is defined by the formula (1.3)d̃(x,y):=infγd(γ),
where the infimum is taken over all paths γ connecting x and y in D. We denote the inner metric of the Apollonian metric by α̃D and call it the “Apollonian inner metric”. Strictly speaking, the Apollonian inner metric is only a pseudometric in a general domain Dℝn; it is a metric if and only if the complement of D is not contained in an (n-2)-dimensional plane [10, Theorem 1.2]. We say that a path γ joining x,y is a geodesic (of the metric d) if d(x,y)=d(γ); there always exists a geodesic path γ for the Apollonian inner metric α̃D connecting x and y in D such that α̃D(γ)=α̃D(x,y) [10].LetDℝn be a domain and x,y∈D. The jD metric [25], which is a modification of a metric from [26], is defined by (1.4)jD(x,y):=log(1+|x-y|min{d(x,∂D),d(y,∂D)}),
where d(x,∂D) denotes the shortest Euclidean distance from x to the boundary ∂D of D. The quasihyperbolic metric from [27] is defined by (1.5)kD(x,y):=infγ∫|dz|d(z,∂D),
where the infimum is taken over all paths γ joining x and y in D. Note that the quasihyperbolic metric is the inner metric of the jD metric.We now recall some relations on the set of metrics inD for an overview of our previous work in [12].Definition 1.1.
Letd and d′ be metrics on D.  (i)
We writed≲d′ if there exists a constant K>0 such that d≤Kd′, similarly for the relation d≳d′.(ii)
We writed≈d′ if d≲d′ and d≳d′.(iii)
We writed≪d′ if d≲d′ and d≵d′.Let us first of all note that the following inequalities hold in every domainDℝn:  (1.6)αD≲jD≲kD,αD≲α̃D≲kD.
The first two are from [2, Theorem 3.2] and the second two are from [7, Remark 5.2, Corollary 5.4]. 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 relationαD≈α̃D, 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 1.10, Lemma 6.4]. Also, in convex uniform domains this relation always holds: from [6, Theorem 4.2] we know that αD≈jD in convex domains; additionally, jD≈kD if D is uniform; hence α̃D≲kD≲jD≲αD≲α̃D. On the other hand, there are also domains in which αD≪α̃D, for example, the infinite strip. Finally, we note that in [13, Corollary 1.4] it was shown that αD≈α̃D implies that D 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 jD≪α̃D, jD≈α̃D, and jD≫α̃D 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 ℝn. 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 αD≈α̃D.Theorem 1.2.
LetDℝn be a domain with an exterior point. Let p∈D and G:=D∖{p}. If αD≈α̃D, then αG≈α̃G as well.Note that by Möbius 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 D is not contained in a hyperplane. Note that this is a minimal assumption for αD to be a metric in the first place, as noted above.Theorem 1.3 (Main Theorem).
LetDℝn be a domain whose boundary is not contained in a hyperplane. Let p∈D and G:=D∖{p}. If αD≈α̃D, then αG≈α̃G 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 inequalityαG≳α̃G where the integral representation [10, Theorem 1.4] of the Apollonian inner metric plays a crucial rule. The main result shows that if the boundary of the domain contains n+1 points which form extreme points of an n-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{e1,e2,…,en} the standard basis of ℝn and by n the dimension of the Euclidean space under consideration and assume that n≥2. For x∈ℝn we denote by xi its ith coordinate. The following notation is used for Euclidean balls and spheres: (2.1)Bn(x,r):={y∈ℝn:|x-y|<r},Sn-1(x,r):={y∈ℝn:|x-y|=r},Bn:=Bn(0,1),Sn-1:=Sn-1(0,1).
For x,y,z∈ℝn we denote by xyẑ the smallest angle between the vectors x-y and z-y at 0.We use the notationℝn¯:=ℝn∪{∞} for the one-point compactification of ℝn, equipped with the chordal metric. Thus an open ball of ℝn¯ is an open Euclidean ball, an open half-space, or the complement of a closed Euclidean ball. We denote by ∂G, Gc, and G¯ the boundary, complement, and closure of G, respectively, all with respect to ℝn¯.We also need some notation for quantities depending on the underlying Euclidean metric. Forx∈Gℝn we write (2.2)δ(x):=d(x,∂G):=min{|x-z|:z∈∂G}.
For a path γ in ℝn we denote by ℓ(γ) 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.Forx,y∈Gℝn¯ we define (2.3)qx:=supa∈∂G|a-y||a-x|,qy:=supb∈∂G|b-x||b-y|.
The numbers qx and qy are called the Apollonian parameters of x and y (with respect to G) and by the definition (2.4)αG(x,y)=log(qxqy).
The balls (in ℝn¯!), (2.5)Bx:={z∈ℝn¯:|z-x||z-y|<1qx},By:={z∈ℝn¯:|z-y||z-x|<1qy},
are called the Apollonian balls about x and y, respectively. We collect some immediate results regarding these balls; similar results obviously hold with x and y interchanged.(1)
x∈Bx⊂G and Bx¯∩∂G≠∅.(2)
Ifix and iy denote the inversions in the spheres ∂Bx and ∂By, then
(2.6)y=ix(x)=iy(x).(3)
If∞∉G, we have qx≥1. If, moreover, ∞∉G¯, then qx>1.


### 2.3. Uniformity

Uniform domains were introduced by Martio and Sarvas in [28, 2.12], but the following definition is an equivalent form from [26, equation (1.1)]. 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 G to the whole space if G is uniform; see [30].Definition 2.1.
A domainGℝn is said to be uniform with constant K if for every x,y∈G there exists a path γ, parameterized by arc-length, connecting x and y in G, such that  (1)
ℓ(γ)≤K|x-y|,(2)
Kδ(γ(t))≥min{t,ℓ(γ)-t}.The relevance of uniformity to our investigation comes from [26, Corollary 1] which states that a domain is uniform if and only if kG≈jG. This condition is also equivalent to α̃G≲jG; see [13, Theorem 1.2]. 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)α¯G(x;r)=limt→01tαG(x,x+tr|r|),
where r∈ℝn∖{0}. If α¯G(x;r) is independent of r in every point of G, then the Apollonian metric is isotropic and we may denote α¯G(x):=α¯G(x;e1) and call this function the density of αG at x. 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.
LetGℝn¯, x∈G and θ∈Sn-1.  (i)
IfBn(x+sθ,s)⊂G for every s>0 and ∞∉G, then let r+=∞.(ii)
IfBn(x+sθ,s)⊂G for every s>0 and ∞∈G, then let r+ to be the largest negative real number such that G⊂Bn(x+r+θ,|r+|).(iii)
Otherwise letr+>0 to be the largest real number such that Bn(x+r+θ,r+)⊂G.Definer- in the same way but using the vector -θ instead of θ. We define the Apollonian spheres throughxin directionθ by (2.8)S+:=Sn-1(x+r+θ,r+),S-:=Sn-1(x-r-θ,r-)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 5.8]).
LetGℝn¯ be open, x∈G∖{∞} and θ∈Sn-1. Let r± be the radii of the Apollonian spheres S± at x in the direction θ. Then
(2.9)α¯G(x;θ)=12r++12r-,
where one understands 1/∞=0.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 1.4]).
Ifx,y∈Gℝn, then
(2.10)α̃G(x,y)=infγ∫α¯G(γ(t);γ′(t))|γ′(t)|dt,
where the infimum is taken over all paths connecting x and y in G that are piecewise continuously differentiable (with the understanding that α¯G(z;0)=0 for all z∈G, even though α¯G(z;0) is not defined).


## 2.1. Notation

The notation used conforms largely to that in [23, 24], as was mentioned in Section 1.We denote by{e1,e2,…,en} the standard basis of ℝn and by n the dimension of the Euclidean space under consideration and assume that n≥2. For x∈ℝn we denote by xi its ith coordinate. The following notation is used for Euclidean balls and spheres: (2.1)Bn(x,r):={y∈ℝn:|x-y|<r},Sn-1(x,r):={y∈ℝn:|x-y|=r},Bn:=Bn(0,1),Sn-1:=Sn-1(0,1).
For x,y,z∈ℝn we denote by xyẑ the smallest angle between the vectors x-y and z-y at 0.We use the notationℝn¯:=ℝn∪{∞} for the one-point compactification of ℝn, equipped with the chordal metric. Thus an open ball of ℝn¯ is an open Euclidean ball, an open half-space, or the complement of a closed Euclidean ball. We denote by ∂G, Gc, and G¯ the boundary, complement, and closure of G, respectively, all with respect to ℝn¯.We also need some notation for quantities depending on the underlying Euclidean metric. Forx∈Gℝn we write (2.2)δ(x):=d(x,∂G):=min{|x-z|:z∈∂G}.
For a path γ in ℝn we denote by ℓ(γ) 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.Forx,y∈Gℝn¯ we define (2.3)qx:=supa∈∂G|a-y||a-x|,qy:=supb∈∂G|b-x||b-y|.
The numbers qx and qy are called the Apollonian parameters of x and y (with respect to G) and by the definition (2.4)αG(x,y)=log(qxqy).
The balls (in ℝn¯!), (2.5)Bx:={z∈ℝn¯:|z-x||z-y|<1qx},By:={z∈ℝn¯:|z-y||z-x|<1qy},
are called the Apollonian balls about x and y, respectively. We collect some immediate results regarding these balls; similar results obviously hold with x and y interchanged.(1)
x∈Bx⊂G and Bx¯∩∂G≠∅.(2)
Ifix and iy denote the inversions in the spheres ∂Bx and ∂By, then
(2.6)y=ix(x)=iy(x).(3)
If∞∉G, we have qx≥1. If, moreover, ∞∉G¯, then qx>1.


## 2.3. Uniformity

Uniform domains were introduced by Martio and Sarvas in [28, 2.12], but the following definition is an equivalent form from [26, equation (1.1)]. 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 G to the whole space if G is uniform; see [30].Definition 2.1.
A domainGℝn is said to be uniform with constant K if for every x,y∈G there exists a path γ, parameterized by arc-length, connecting x and y in G, such that  (1)
ℓ(γ)≤K|x-y|,(2)
Kδ(γ(t))≥min{t,ℓ(γ)-t}.The relevance of uniformity to our investigation comes from [26, Corollary 1] which states that a domain is uniform if and only if kG≈jG. This condition is also equivalent to α̃G≲jG; see [13, Theorem 1.2]. 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)α¯G(x;r)=limt→01tαG(x,x+tr|r|),
where r∈ℝn∖{0}. If α¯G(x;r) is independent of r in every point of G, then the Apollonian metric is isotropic and we may denote α¯G(x):=α¯G(x;e1) and call this function the density of αG at x. 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.
LetGℝn¯, x∈G and θ∈Sn-1.  (i)
IfBn(x+sθ,s)⊂G for every s>0 and ∞∉G, then let r+=∞.(ii)
IfBn(x+sθ,s)⊂G for every s>0 and ∞∈G, then let r+ to be the largest negative real number such that G⊂Bn(x+r+θ,|r+|).(iii)
Otherwise letr+>0 to be the largest real number such that Bn(x+r+θ,r+)⊂G.Definer- in the same way but using the vector -θ instead of θ. We define the Apollonian spheres throughxin directionθ by (2.8)S+:=Sn-1(x+r+θ,r+),S-:=Sn-1(x-r-θ,r-)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 5.8]).
LetGℝn¯ be open, x∈G∖{∞} and θ∈Sn-1. Let r± be the radii of the Apollonian spheres S± at x in the direction θ. Then
(2.9)α¯G(x;θ)=12r++12r-,
where one understands 1/∞=0.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 1.4]).
Ifx,y∈Gℝn, then
(2.10)α̃G(x,y)=infγ∫α¯G(γ(t);γ′(t))|γ′(t)|dt,
where the infimum is taken over all paths connecting x and y in G that are piecewise continuously differentiable (with the understanding that α¯G(z;0)=0 for all z∈G, even though α¯G(z;0) is not defined).


## 3. The Proof of the Main Theorem

Proof.
In this proof we denote byδ the distance to the boundary of D, not of G=D∖{p}. It is enough to prove the inequality αG≳α̃G, because other way inequality always holds. Let x,y∈G and denote B:=Bn(p,δ(p)/2). Let γxy be a path connecting x and y such that αD(γxy)=α̃D(x,y); note that such a length-minimizing path exists by [10, Theorem 1.5].Case 1.
(a)x,y∈D∖B and γxy∩B=∅.
Letz∈∂D be such that δ(p)=|p-z|. Let V be the collection of n+1 boundary points of D where they form the vertices of an n-simplex. Denote by Bt:=Bn(c,t) the largest ball with radius t and centered at c such that Bt is inside the n-simplex [V]; see Figure 1. Define l=t/2. Denote by Bl:=Bn(c,l) the ball with radius l and centered at c. Define Ω=ℝn∖V. Let BT⊂Ω be a ball tangent to Bl with maximal radius, denoted by T.
ChooseL=5max{|p-c|,T}. Consider the ball Bn(c,L) centred at c with radius L and denote it by BL. Then we see that V∪{∞}⊂ℝn¯∖D. Since
(3.1)∂Ω=∂(ℝn∖V)=V∪{∞}⊂ℝn¯∖D,
we see that
(3.2)α¯D(w;r)≥α¯Ω(w;r)
for r∈Sn-1.
We now estimate the density of the Apollonian spheres (see Definition2.2) in Ω passing through w∈γxy and in the direction r∈Sn-1. In order to compare the density α¯Ω(w;r) with the densities α¯G(w;r) and α¯D(w;r), we consider two possibilities of the choice of w∈γxy w.r.t. BL.
We first assume thatw∈ℝn∖BL. Denote by F the ray from w along r. Consider a sphere S1 with radius R1 and centered at x∈F such that S1 is tangent to Bl. Denote θ:=xwĉ. Construction of BT gives that, for |θ|<π/2, the Apollonian spheres passing through w and in the direction r are smaller in size than the sphere S1.
This gives(3.3)α¯Ω(w;r)≥12R1=l+(d+l)cosθ(d+l)2-l2,
where d:=d(w,Bl) and R1 is obtained using the cosine formula in the triangle ∆xwc.
Now the sphere with radiusR2 and centre at q passing through w and p gives
(3.4)|w-p|2=R2cos(θ-ψ),
where ψ=cwp̂ and q∈F. If the Apollonian spheres (passing through w and in the direction r) are affected by the boundary point p, then by Lemma 2.3 we have
(3.5)α¯G(w;r)=1R2+1r+≤1R2+α¯D(w;r)=2cos(θ-ψ)|w-p|+α¯D(w;r),
where r+ denotes the radius of the smaller Apollonian sphere which touches ∂D. Denote ϕ:=wpĉ. Since p∈BL, using the sine formula in the triangle ∆wpc we get
(3.6)sinψ=|p-c||w-p|sinϕ≤|p-c||w-p|.
Then we see that
(3.7)cos(θ-ψ)≤cosθ+sinψ≤cosθ+|p-c||w-p|.
Thus, from (3.5) we get
(3.8)α¯G(w;r)≤2(cosθ+|p-c|/|w-p|)|w-p|+α¯D(w;r)=2cosθ|w-p|+2|p-c||w-p|2+α¯D(w;r).
Since w∉BL, we notice that the Euclidean triangle inequalities of the triangle ∆wpc give |w-p|≈d. We then obtain
(3.9)α¯G(w;r)≲cosθd+ld2+α¯D(w;r)≈l+(d+l)cosθ(d+l)2-l2+α¯D(w;r).
We next assume thatw∈B¯L. It is clear that if α¯Ω(w;r)=0 then ∂Ω is contained in a hyperplane, which contradicts our assumption. Thus if w∈BL¯, then α¯Ω(w;r)>0, and since the density function is continuous it has a greatest lower bound; namely, there exists a constant k>0 such that for r∈S1 we have
(3.10)α¯Ω(w;r)≥k.
Therefore, (3.3) and (3.10) together give
(3.11)α¯Ω(w;r)≥min{l+(d+l)cosθ(d+l)2-l2,k}.
Sinceγxy∩B=∅, we note that |w-p|≥δ(p)/2 for all w∈γxy. Thus, if the Apollonian spheres passing through w and in the direction r∈Sn-1 are affected by the boundary point p, then by Lemma 2.3(3.12)α¯G(w;r)≤1|w-p|+12r+≤1|w-p|+α¯D(w;r)≤2δ(p)+α¯D(w;r)≈k+α¯D(w;r)
hold, where r+ denotes the radius of the smaller Apollonian sphere which touches ∂D. Then (3.2), (3.9), (3.11), and (3.12) together give
(3.13)α¯G(w;r)≲min{l+(d+l)cosθ(d+l)2-l2,k}+α¯D(w;r)≲α¯D(w;r).
Thus, by the definition of the inner metric and Lemma 2.4, we get the relation
(3.14)αG(γxy)≤KαD(γxy)=Kα̃D(x,y)
for some constant K. This gives
(3.15)α̃G(x,y)≲α̃D(x,y)≈αD(x,y)≤αG(x,y),
where the second inequality holds by assumption and the third holds trivially, as G is a subdomain of D.
(b)x,y∈D∖B and γxy intersects B.
Letγ be an intersecting part of γxy from x1 to x2 (if there are more intersecting parts, we proceed similarly). Let γ′ be the shortest circular arc on ∂B from x1 to x2, as shown in Figure 2.
Using the density bounds (3.2) and (3.10), we get k≤α¯D(u;r)≤2/δ(u) for every u∈γ′. Then we see that the inequalities
(3.16)αD(γ)≥ℓ(γ)k,αD(γ′)≤4ℓ(γ′)δ(p)
hold. But since ℓ(γ)≥|x1-x2| and ℓ(γ')≤(π/2)|x1-x2|, we have ℓ(γ′)≲ℓ(γ). This shows that αD(γxy′)≲αD(γxy), where the path γxy′ is obtained from γxy by modifying γ with the circular arc γ′ joining x1 to x2. Since γxy′⊂G∖B, (3.13) implies that αG(γxy′)≲αD(γxy′). So we get
(3.17)αG(x,y)≥αD(x,y)≈α̃D(x,y)=αD(γxy)≳αD(γxy′)≳αG(γxy′)≥α̃G(x,y).
Thus we have shown that αG(x,y)≳α̃G(x,y) holds for all x,y∈D∖B.Case 2 (x,y∈Bn(p,(3/4)δ(p))).
Without loss of generality we assume that|y-p|≤|x-p|. Since ∂G=∂D∪{p}, it is clear by the definition and the monotonicity property of the Apollonian metric that
(3.18)α¯G(x,y)≥max{log|x-p||y-p|,αD(x,y)}.
Let γ≔γ1∪γ2, where γ1 is the path which is circular about the point p from y to (|y-p|(x-p)/|x-p|)+p and γ2 is the radial part from (|y-p|(x-p)/|x-p|)+p to x, as shown in Figure 3.
Since the Apollonian spheres are not affected by the boundary pointp in the circular part, we have
(3.19)α¯G(γ1(t);γ1′(t))≤α¯Bn(p,δ(p))(γ1(t);γ1′(t))=1σ(p)-|y-p|+1δ(p)+|y-p|=2σ(p)σ(p)2-|y-p|2,
where the first equality holds since the Apollonian metric equals the hyperbolic metric in a ball. For γ2(t), by monotonicity in the domain of definition, we see that
(3.20)α¯G(γ2(t);γ2′(t))≤α¯Bn(p,δ(p))∖{p}(γ2(t);γ2′(t))=1|p-γ2(t)|+1δ(p)-|p-γ2(t)|.
Hence, by Lemma 2.4 we have
(3.21)α̃G(x,y)≤αG(γ)≤∫γ12δ(p)δ(p)2-|y-p|2|dy|+∫|y-p||x-p|(1t+1δ(p)-t)dt=2δ(p)ℓ(γ1)δ(p)2-|y-p|2+log(|x-p||y-p|δ(p)-|y-p|δ(p)-|x-p|)≤327ℓ(γ1)δ(p)+log(|x-p||y-p|δ(p)-|y-p|δ(p)-|x-p|).
Since u↦u3(δ(p)-u) is increasing for 0<u<3δ(p)/4 and we have
(3.22)|y-p|≤|x-p|≤3δ(p)4,
for the choice u=|x-p|, the inequality
(3.23)|x-p|3(δ(p)-|x-p|)≥|y-p|3(δ(p)-|y-p|)
holds. This inequality is equivalent to
(3.24)log(|x-p||y-p|δ(p)-|y-p|δ(p)-|x-p|)≤4log|x-p||y-p|.
Using αD≈α̃D, we easily get αD(x,y)≳ℓ(γ1)/δ(p). We have thus shown that
(3.25)α̃G(x,y)≤KαD(x,y)+4log|x-p||y-p|≤(K+4)αG(x,y),
for some constant K.Case 3.
x∉Bn(p,3δ(p)/4) and y∈B.
Letw∈Sn-1(p,(3/4)δ(p)) be such that
(3.26)|y-w|=d(y,Sn-1(p,34δ(p))).
Let γ:=γ1∪γ2, where γ1=[y,w] and γ2 is a path connecting w and x such that
(3.27)αG(γ2)=α̃G(w,x).
As we discussed in the previous case, we have
(3.28)αG(γ1)≤4log3δ(p)4|y-p|≤4log|x-p||y-p|≤4αG(x,y).
Since x,w∉B, it follows by Case 1 that
(3.29)αG(γ2)=α̃G(w,x)≲α̃D(w,x)≈αD(w,x)≤2jD(w,x).
It is now sufficient to see that jD(w,x)≲αG(x,y).
Ifδ(w)≤δ(x), then the triangle inequality |w-x|≤|w-p|+|x-p| and the fact δ(w)≥δ(p)/4 together give
(3.30)jD(w,x)≤log(1+4|w-p|+4|x-p|δ(p))=log(4+4|x-p|δ(p)),
where the equality holds due to the fact that w∈Sn-1(p,(3/4)δ(p)). But for s≥3/2, we have log(4+2s)≤5logs. For the choice s=|x-p|/|y-p|, the inequality (3.30) reduces to
(3.31)jD(w,x)≤log(4+2|x-p||y-p|)≤5log|x-p||y-p|≤5αG(x,y),
where the first inequality holds since |y-p|≤δ(p)/2 and the last holds by the definition of the Apollonian metric.
We next move on to the caseδ(w)≥δ(x). If |x-y|≥3δ(x), we see (by the triangle inequality |b-y|≥|x-y|-|b-x|) that
(3.32)αG(x,y)≥supb∈∂Dlog|b-y||b-x|≥log(|x-y|δ(x)-1)
holds. Using (3.32) and the fact that |x-p|/|y-p|≥3/2, we get
(3.33)αG(x,y)≥{log(|x-y|δ(x)-1)for|x-y|δ(x)≥3,log32otherwise.
Since |x-y|≥δ(p)/4, we get the following upper bound for jD(w,x):
(3.34)jD(w,x)≲log(1+|x-y|+δ(p)δ(x))≤log(1+5|x-y|δ(x)),
where the first inequality follows by the triangle inequality |w-x|≤|x-y|+|y-w| and the fact that |y-w|≤(3/4)δ(p). We see that the function f(s)=(s-1)4-(1+5s) is increasing for s≥3, so f(s)≥f(3)=0. Thus, for the choice s=|x-y|/δ(x)≥3, we get
(3.35)jD(w,x)≲log(1+5|x-y|δ(x))≤4log(|x-y|δ(x)-1)≤4αG(x,y),
where the last inequality holds by (3.32). On the other hand, if |x-y|/δ(x)<3, then jD(w,x) is bounded above by 4log2 and αG(x,y) is bounded below by log(3/2), so the inequality jD(w,x)≲αG(x,y) is clear. Thus for the choice of w,x, and y we obtain
(3.36)αG(γ2)≲jD(w,x)≲αG(x,y),
which concludes that
(3.37)α̃G(x,y)≤αG(γ)≲αG(x,y).
We have now verified the inequality in all of the possible cases, so the proof is complete.Figure 1
The largest ballBT tangent to Bl and contained in Ω=ℝn∖V, where V={v1,v2,v3}.Figure 2
The geodesic pathγxy (w.r.t. the Apollonian inner metric α̃D) connecting x and y intersects B, and its modification γ′ from x1 to x2 along the circular part.Figure 3
A short pathγ=γ1∪γ2 connecting x and y in Bn(p,(3/4)σ(p)).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.
LetDℝn be a domain whose boundary does not lie in a hyperplane. Suppose that (pi)i=1k is a finite nonempty sequence of points in D and define G:=D∖{p1,p2,…,pk}. Assume that αD≈α̃D and jD≈kD. Then Inequality (9) in Table 1, αG≈α̃G≪jG≈kG, holds.



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*Source: 191858-2010-06-24.xml*