A New Intrinsic Metric on Metric Spaces

Abstract

A new intrinsic metric called $S_{D}$-metric is introduced for a general metric space (X, d), where D is a non-trivial bounded closed subset of X. The metric $S_{D}$ can be used to define a strongly hyperbolic metric on X. We consider the convergence of metric spaces $\{(X,S_{D_n})\}_{n=1}^{\infty }$ for a sequence of non-trivial bounded closed subsets $\{D_n\}_{n=1}^{\infty }$. The distortion property of the new metric on the unit ball $\mathbb {B}^n$ is also studied under the Möbius transformations of the unit ball.

1 Introduction

Let (X, d) be a metric space. Fix a base point $o\in X$, the Gromov product of $x, x'\in X$ with respect to o is defined as

$$\begin{aligned} (x|x')_o:=\frac{1}{2}\left( d(o,x)+d(o,x')-d(x,x')\right) . \end{aligned}$$

We say that a metric space (X, d) is Gromov $\delta$-hyperbolic for some $\delta \ge 0$, if

$$\begin{aligned} (x|y)_o\ge \min \{(x|z)_o,(z|y)_o\}-\delta \end{aligned}$$

for all $x, y, z,o\in X$.

Gromov hyperbolic space is an important metric space, which plays an important role in many mathematical branches. Although Gromov hyperbolicity yields a very satisfactory theory, for certain analytic purposes, hyperbolicity by itself is not enough, and one needs certain enhancements. In the paper [24], B. Nica and J. Špakula gave the following enhancements of hyperbolicity by introducing the notion of strongly hyperbolic space.

Definition 1.1

[24, Definition 4.1] We say that a metric space (X, d) is strongly hyperbolic with parameter $\epsilon >0$ if

$$\begin{aligned} \exp (-\epsilon (x|y)_o)\le \exp (-\epsilon (x|z)_o)+\exp (-\epsilon (z|y)_o) \end{aligned}$$

for all $x, y, z, o\in X$; equivalently, the four-point condition

$$\begin{aligned} \exp \left( \frac{\epsilon }{2}(d(x,y)+d(z,t))\right) \le \exp \left( \frac{\epsilon }{2}(d(x,z)+d(y,t))\right) +\exp \left( \frac{\epsilon }{2}(d(x,t)+d(z,y))\right) \end{aligned}$$

holds for all $x, y, z, t\in X$.

In the paper [24], the authors obtained the following important theorem for strongly hyperbolic space.

Theorem 1.1

[24, Theorem 4.2] Let X be a strongly hyperbolic space with parameter $\epsilon$. Then X is an $\epsilon$-good, $\log 2/\epsilon$-hyperbolic space. Furthermore, X is strongly bolic provided that X is roughly geodesic.

Theorem 1.1 shows that the strongly hyperbolic space has better properties than general hyperbolic spaces. Thus it is interesting to determine which hyperbolic metric is a strongly hyperbolic metric or to construct a strongly hyperbolic metric on a given metric space. Recently, these problems have been considered in the two papers [22, 33]. In particular, in [33], the second author and Zhang established the following theorem.

Theorem 1.2

[33] Suppose that (X, d) is a Ptolemy space, then the metric space $(X,\log (1+d))$ is a strongly hyperbolic space with parameter 2.

Moreover, they considered the following distance function $s_p$, which was introduced and used by Bonk and Kleiner in the paper [4], where

$$\begin{aligned} s_p(x,y)= \frac{d(x,y)}{[1+d(x,p)][1+d(y,p)]} \end{aligned}$$

for $x,y\in X$ and a given $p\in X$. Using Theorem 1.2, Zhang and Xiao proved the following theorem.

Theorem 1.3

[33] Suppose (X, d) is a Ptolemy space and $p\in X$. The metric space $(X,\log (1+s_p))$ is a strongly hyperbolic space with parameter $\epsilon =2$. Thus $(X,\log (1+s_p))$ is a $\log 2/2$-hyperbolic space.

In this paper, we consider the following generalization of the distance function $s_p$ in any metric space (X, d). Here, we say a bounded closed subset D of a metric space (X, d) is non-trivial if D contains at least two points and introduce the following quantity.

Definition 1.2

Let D be a non-trivial bounded closed subset of a metric space (X, d). For $x,y\in X$, define the following quantity

$$\begin{aligned} s_{D}(x,y)=\frac{d(x,y)}{[1+d_D(x)][1+d_D(y)]}, \end{aligned}$$

where $d_D(x)=\sup \{d(x,y): y\in D\}$.

Usually, $s_D$ is not a metric since it may not satisfy the triangle inequality. We remark that when D is trivial, that is D is a single point p, $s_D$ just happens to be $s_p$ defined by Bonk and Kleiner in [4].

Another important motivation for introducing the quantity $s_{D}$ comes from the geometric function theory. During the past decades, in order to make up for the absence of the hyperbolic metric on general domains of higher-dimensional Euclidean spaces, many hyperbolic intrinsic metrics have been introduced and studied in a series of papers (see [1, 3, 7, 9,10,11, 14, 15, 27, 29, 30, 34] and the references therein), for example, Möbius-invariant Cassinian metric [20], Apollonian metric [3, 17], Seittenranta’s metric [13, 29], scale-invariant Cassinian metric [19, 23], half Apollonian metric [12] and $\widetilde{j}$-metric [10], the triangular ratio metric [21, 26]. Some key features of classical hyperbolic metric are inherited by these metrics but not all. Each metric might be used to discover some intricate features of mappings and domains not detected by other metrics. Here we say a new metric is intrinsic if it is locally comparable with the original metric. Our new quantity $s_D$ differs slightly from other intrinsic metrics, and thus it may potentially be a great help for new discoveries about intrinsic geometry of domains.

The paper is organized as follows: In Section 2, firstly, we show that the quantity $s_{D}$ is a Ptolemaic metric on X, that is $(X, s_{D})$ is a Ptolemy space if the metric space (X, d) is a Ptolemy space. Using Theorem 1.2, it is easy to obtain the metric space $(X, \log (1+ s_{D}))$ is a strongly hyperbolic space. Secondly, for general metric space, we obtained a real metric $S_D$ by a standard technique and proved that the metric $S_D$ is not equivalent to the existing some hyperbolic type metrics by some examples. In Section 3, motivated by the recent works of Herron in [16], we study the relations between the convergence of sets $\{D_n\}_{n=1}^{\infty }$ and the convergence of the associated sequence of metric spaces $\{(X,s_{D_n})\}_{n=1}^{\infty }$ or $\{(X,S_{D_n})\}_{n=1}^{\infty }$. In Section 4, we mostly focus on the special case where (X, d) is the n-dimensional Euclidean space $\mathbb {R}^n$ with the Euclidean distance. We show how the $s_{\partial \mathbb {B}^n}$-metric behaves under the Möbius maps from the unit ball $\mathbb {B}^n$ to itself.

2 $s_{D}$ Metric and Strongly Hyperbolicitiy

In this section, we firstly prove that $s_D$ is truly a metric in the Ptolemy space case.

Definition 2.1

A metric space (X, d) is called Ptolemy space if the following Ptolemy inequality

$$\begin{aligned} d(x_1,x_2)d(x_3,x_4)\le d(x_1,x_4)d(x_2,x_3)+d(x_1,x_3)d(x_2,x_4) \end{aligned}$$

holds for all quadruples $x_1,x_2,x_3,x_4\in X$. If (X, d) is a Ptolemy metric space, we say that d is a Ptolemaic metric on X.

Many important spaces have been proved to be Ptolemy spaces, for examples, the inner-product space, CAT(0) spaces [28] and the boundaries of CAT($-1$) spaces [8]; in particular, it was proved that the snowflake space $(X,d_{\alpha })$ is a Ptolemy space when $0<\alpha \le \frac{1}{2}$, where (X, d) is a metric space and $d_{\alpha }(x,y)=d(x,y)^{\alpha }$ for $x,y\in X$. We refer to Foertsch and Schroeder [8], Ibragimov [18], Xiao and Gu [32] for Ptolemy spaces. In the following, we show that if (X, d) is a Ptolemy space, the quantity $s_D$ is a Ptolemaic metric on X.

Theorem 2.1

Let D be a non-trivial bounded closed subset of a Ptolemy metric space (X, d). Then the function

$$\begin{aligned} s_{D}(x,y)=\frac{d(x,y)}{[1+d_D(x)][1+d_D(y)]} \end{aligned}$$

is a metric, where $d_D(x):=\sup \{d(x,y): y\in D\}$ and $(X, s_{D})$ is a Ptolemy space.

Proof

Obviously, $s_{D}(x,y)\ge 0$, $s_{D}(x,y)= s_{D}(y,x)$ and $s_{D}(x, y)=0$ if and only if $x=y$. So it is enough to show that the triangle inequality holds. Note that for any $x, y, z\in X$, the triangle inequality

$$\begin{aligned} s_{D}(x, y)\le s_{D}(x, z)+ s_{D}(z, y) \end{aligned}$$

is equivalent to

$$\begin{aligned} d(x,y)[1+d_D(z)]\le d(x,z)[1+d_D(y)]+d(y,z)[1+d_D(x)]. \end{aligned}$$

Since (X, d) is a Ptolemy space, for any $z^{*}\in D$, we have

$$\begin{aligned} d(x,y)d(z,z^{*})&\le d(x,z)d(y,z^{*})+d(y,z)d(x,z^{*})\\&\le d(x,z)d_D(y)+d(y,z)d_D(x). \end{aligned}$$

By the arbitrariness of $z^{*}$, we have

$$\begin{aligned} d(x,y)d_D(z)\le d(x,z)d_D(y)+d(y,z)d_D(x). \end{aligned}$$

Thus

$$\begin{aligned} d(x,y)[1+d_D(z)]\le d(x,z)[1+d_D(y)]+d(y,z)[1+d_D(x)], \end{aligned}$$

which implies that the triangle inequality holds and so $s_{D}$ is a metric on X.

In the following, we will show that $(X, s_{D})$ also is a Ptolemy space. For any $x_i\in X$ for $i=1,2,3,4$, put $p_i=1+d_D(x_i)$ and $d_{ij}=d(x_i,x_j)$. Using these notations, we have $s_{D}(x_i,x_j)=d_{ij}/p_ip_j$ for $i,j\in \{1,2,3,4\}$. Since (X, d) is a Ptolemy space, we have

$$\begin{aligned} d_{12}d_{34}\le d_{13}d_{24}+d_{14}d_{23}. \end{aligned}$$

Thus

$$\begin{aligned} \frac{d_{12}d_{34}}{p_1p_2p_3p_4}\le \frac{d_{13}d_{24}}{p_1p_2p_3p_4}+\frac{d_{14}d_{23}}{p_1p_2p_3p_4}. \end{aligned}$$

That is

$$\begin{aligned} s_{D}(x_1,x_2) s_{D}(x_3,x_4)\le s_{D}(x_1,x_3) s_{D}(x_2,x_4)+ s_{D}(x_1,x_4) s_{D}(x_2,x_3), \end{aligned}$$

which shows that $(X, s_{D})$ is a Ptolemy space. $\square$

Since $s_{D}$ is a Ptolemaic metric on X, we can obtain the following result from Theorem 1.2.

Theorem 2.2

Suppose (X, d) is a Ptolemy and D is a non-trivial bounded closed subset of X. Then the metric space $(X, \log (1 + s_{D}(x, y)))$ is a strongly hyperbolic space with parameter $\epsilon =2$. Thus $(X, \log (1 + s_{D}(x, y)))$ is a $\log 2/2$-hyperbolic space.

Usually, $s_D$ is not a metric in general metric space since it may not satisfy the triangle inequality. Fortunately, there is a standard technique which forces the triangle inequality: we define

$$\begin{aligned} S_D(x,y)=\inf \left\{ \sum _{i=1}^ks_D(x_i,x_{i-1}): x=x_0,\ldots ,x_k=y\in X\right\} \end{aligned}$$

for $x,y\in X$. Then we have the following result.

Theorem 2.3

Let (X, d) be a metric space and $D\subset X$ is a non-trivial bounded closed subset of X.

• (1) For $x,y\in X$, we have

  $$\begin{aligned} \frac{1}{4}s_D(x,y)\le & {} S_D(x,y)\le s_D(x,y)\nonumber \\\le & {} \min \left\{ \frac{1}{d_D(x)+1}+\frac{1}{d_D(y)+1},4\frac{d(x,y)}{(2+\mathrm {diam}D)^2}\right\} . \end{aligned}$$

  (2.1)

  In particular, $S_D$ is a distance function on X.

• (2) The identity map $id: (X,d)\rightarrow (X,S_D)$ is a $\vartheta$-quasimöbius homeomorphism, where $\vartheta (t)=16t$.

• (3) $(X, S_D)$ is a bounded metric space and

  $$\begin{aligned} \mathrm {diam}(X,S_D)\le \frac{4}{\mathrm {diam}D+2}. \end{aligned}$$

Here, we recall that an embedding $f:(X,d_X)\rightarrow (Y,d_Y)$ between two metric space is $\vartheta$-quasimöbius if $\vartheta :[0,\infty )\rightarrow [0,\infty )$ is a homeomorphism and for all quadruples x, y, z, w of distinct points in X,

$$\begin{aligned} \frac{d_X(x,y)d_X(z,w)}{d_X(x,z)d_X(y,w)}\le t\Rightarrow \frac{d_Y(f(x),f(y))d_Y(f(z),f(w))}{d_Y(f(x),f(z))d_Y(f(y),f(w))}\le \vartheta (t). \end{aligned}$$

In order to prove Theorem 2.3, we need the following lemma.

Lemma 2.1

Suppose $u,v\in X$, then $|\frac{1}{d_D(u)+1}-\frac{1}{d_D(v)+1}\Big |\le s_D(u,v)$.

Proof

Without loss of generality, we assume that $d_D(u)\ge d_D(v)$. For any $\varepsilon >0$, there is $x_u\in D$ such that $d(x,x_u)\le d_D(x)\le d(x,x_u)+\varepsilon$. Then

$$\begin{aligned} \Big |\frac{1}{d_D(u)+1}-\frac{1}{d_D(v)+1}\Big |&= \Big |\frac{d_D(u)-d_D(v)}{(d_D(u)+1)(d_D(v)+1)}\Big |\\&\le \frac{d_D(u)-d_D(v)}{(d_D(u)+1)(d_D(v)+1)}\\&\le \frac{d(u,x_u)+\varepsilon -d(v,x_u)}{(d_D(u)+1)(d_D(v)+1)}\\&\le \frac{d(u,v)+\varepsilon }{(d_D(u)+1)(d_D(v)+1)}. \end{aligned}$$

Letting $\varepsilon \rightarrow 0$, we have the desired result. $\square$

Now, we are in the position to prove Theorem 2.3.

Proof of Theorem 2.3

(1) Obviously, we have $S_D(x,y)\le s_D(x,y)$. For $x,y\in X,z\in D$, we have $d(x,y)\le d(x,z)+d(y,z)\le d_D(x)+d_D(y)$, thus

$$\begin{aligned} s_D(x,y)&=\frac{d(x,y)}{(1+d_D(x))(1+d_D(y))}\le \frac{d(x,z)+d(y,z)}{(1+d_D(x))(1+d_D(y))}\\&\le \frac{d_D(x)+d_D(y)}{(1+d_D(x))(1+d_D(y))}\le \frac{1}{1+d_D(x)}+\frac{1}{1+d_D(y)}. \end{aligned}$$

Since $D\subseteq B(x,d_D(x))$, we have $2d_D(x)\ge \mathrm {diam}D$. Thus, we have

$$\begin{aligned} s_{D}(x,y)=\frac{d(x,y)}{(1+d_D(x))(1+d_D(y))}\le 4\frac{d(x,y)}{(2+\mathrm {diam}D)^2}. \end{aligned}$$

For $x,y\in X$, we asume $d_D(x)\le d_D(y)$. Let $x_0\cdots ,x_k$ be an arbitrary sequence of points in X with $x_0=x$ and $x_k=y$. We consider two cases. If $\frac{1}{d_D(x_i)+1}\ge \frac{1}{2}\frac{1}{d_D(x)+1}$ for all i, by the triangle inequality respect to d, we have

$$\begin{aligned} \sum _{i=1}^ks_D(x_i,x_{i-1})&=\sum _{i=1}^k\frac{d(x_i,x_{i-1})}{(d_D(x_i)+1)(d_D(x_{i-1})+1)}\\&\ge \frac{1}{4}\frac{1}{(d_D(x)+1)^2}\sum _{i=1}^kd(x_i,x_{i-1})\\&\ge \frac{1}{4}\frac{1}{(d_D(x)+1)(d_D(y)+1)}d(x,y)\\&=\frac{1}{4}s_D(x,y). \end{aligned}$$

Suppose that there exists some $j\in \{0,\ldots ,k\}$ such that $\frac{1}{d_D(x_j)+1}<\frac{1}{2}\frac{1}{d_D(x)+1}$. Since $d_D(x)\le d_D(y), d(x,y)\le 2d_D(y)$, so

$$\begin{aligned} s_D(x,y)=\frac{d(x,y)}{(d_D(x)+1)(d_D(y)+1)}\le 2\frac{1}{d_D(x)+1}. \end{aligned}$$

By Lemma 2.1, we have

$$\begin{aligned} \sum _{i=1}^ks_D(x_i,x_{i-1})&\ge \sum _{i=1}^j\Big |\frac{1}{d_D(x_i)+1}-\frac{1}{d_D(x_{i-1}+1)}\Big |\\&\ge \Big |\frac{1}{d_D(x_j)+1}-\frac{1}{d_D(x)+1}\Big |\\&\ge \frac{1}{2}\frac{1}{d_D(x)+1}\ge \frac{1}{4}s_D(x,y), \end{aligned}$$

which implies that $S_D(x,y)\ge \frac{1}{4}s_D(x,y)$.

(2) From (1), when x, y, z, w is a quadruple of distinct points in X, we have

$$\begin{aligned} \frac{S_D(x,y)S_D(z,w)}{S_D(x,z)S_D(y,w)}\le 16\frac{s_D(x,y)s_D(z,w)}{s_D(x,z)s_D(y,w)}=16\frac{d(x,y)d(z,w)}{d(x,z)d(y,w)}, \end{aligned}$$

which implies that the identity map $id: (X,d)\rightarrow (X,S_D)$ is a $\vartheta (t)$-quasimöbius homeomorphism, where $\vartheta (t)=16t$.

(3) For any $x,y\in X$, we have $2d_D(x), 2d_D(y)\ge \mathrm {diam} D$. From inequality (2.1), we have

$$\begin{aligned} S_D(x,y)\le \frac{1}{d_D(x)+1}+\frac{1}{d_D(y)+1}\le \frac{4}{\mathrm {diam}D+2} \end{aligned}$$

for any $x,y\in X$. Thus $(X, S_D)$ is a bounded metric space. $\square$

In the following, we will compare our metric to other hyperbolic type metrics to demonstrate that the metric we define is not equal to some existing metrics. It carries its own significance. First, we recall the notion of t-metric, which was introduced by Rainio and Vuorinen in [25].

Definition 2.2

[25] Let G be some non-empty, open, proper and connected subset of a metric space X. Choose some metric $\eta _G$ defined on the closure of G and denote $\eta _G(x)= \eta _G(x,\partial G)=\inf \{\eta _G(x, z) | z\in \partial G\}$ for all $x\in G$. The t-metric for a metric $\eta _G$ in a domain G is a function $t_G : G \times G\rightarrow [0, 1]$,

$$\begin{aligned} t_G(x, y)=\frac{\eta _G(x, y)}{\eta _G(x, y)+\eta _G(x) +\eta _G(y)} \end{aligned}$$

for all $x, y\in G$.

In the following, we focus on the special case where $G\varsubsetneq \mathbb {R}^n$ and $\eta _G$ is the Euclidean distance. Distance between the points x, y in $\mathbb {R}^n$ is denoted by $|x-y|$ and let $\delta _G(x)=\inf \{|x-z|:z\in \partial G\}$ for a domain $G\subsetneq \mathbb {R}^n$. We consider the following hyperbolic type metrics for G: The triangular ratio metric: $Tr_G: G\times G\rightarrow [0, 1]$,

$$\begin{aligned} Tr_G(x,y)=\frac{|x-y|}{\inf _{z\in \partial G}\{|x-z|+|y-z|\}}, \end{aligned}$$

the $j^{*}_G$ metric $j^{*}_G: G\times G\rightarrow [0, 1]$,

$$\begin{aligned} j^{*}_G(x,y)=\frac{|x-y|}{|x-y|+2\min \{\delta _G(x),\delta _G(y)\}}, \end{aligned}$$

and the point pair function: $p_G: G\times G\rightarrow [0, 1]$,

$$\begin{aligned} p_G(x,y)=\frac{|x-y|}{\sqrt{|x-y|^2+4\delta _G(x)\delta _G(y)}}. \end{aligned}$$

The relation of the above four kinds of metrics has been obtained in [25].

Theorem 2.4

[25, Theorem 3.8] For all domains $G\varsubsetneq \mathbb {R}^n$ and all points $x, y\in G$, the following inequalities hold:

• (1) $j^{*}_G(x, y)/2\le t_G(x, y)\le j^{*}_G(x, y),$

• (2) $p_G(x, y)/2 \le t_G(x, y)\le p_G(x, y),$

• (3) $Tr_G(x, y)/2\le t_G(x, y)\le Tr_G(x, y)$.

Remark 2.1

For all domains $G\varsubsetneq \mathbb {R}^n$ with a non-trivial boundary $D=\partial G$, we consider the relations between the new metric $s_D$ and the other four metrics $j^{*}_G, p_G,Tr_G, t_G$. From Theorem 2.4, it is enough to compare the relations between $s_D$ and $Tr_G$. By the definitions of $s_{D}$ and the triangular ratio metric $Tr_{G}$, it is easy to obtain that $s_{D}(x,y)\le Tr_G(x,y)$ for all $x,y\in G$. In the following, we suppose $G=\{z\in \mathbb {C}: |z|<1\}$ and $D=\partial G=\{z\in \mathbb {C}: |z|=1\}$. We have $d_{D}(x)=\sup \{|x-z|:z\in \partial G\}=1+|x|$ and

$$\begin{aligned} s_{D}(x,y)=\frac{|x-y|}{(2+|x|)(2+|y|)} \le \frac{|x-y|}{|x-y|+1-|x|+1-|y|}=t_{G}(x,y) \end{aligned}$$

for $x,y\in G$.

For $n\in \mathbb {N}$, let $x_n=1-\frac{1}{n}$ and $y_n=(1-\frac{1}{n})e^{i/n}$. We obtain

$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }\frac{t_{G}(x_n,y_n)}{s_{ D}(x_n,y_n)}&=\lim _{n\rightarrow \infty }\frac{(2+|x_n|)(2+|y_n|)}{|x_n-y_n|+1-|x_n|+1-|y_n|}\\&=\lim _{n\rightarrow \infty }\frac{(3-1/n)^2}{(1-1/n)|e^{i/n}-1|+2-2(1-1/n)}=+\infty , \end{aligned} \end{aligned}$$

which shows that there does not exist a fixed constant K such that

$$\begin{aligned} t_{G}(x,y)\le K s_{D}(x,y) \end{aligned}$$

for all $x,y\in G$. Thus there does not exist a fixed constant K such that

$$\begin{aligned} Tr_{G}(x,y)\le K s_{D}(x,y). \end{aligned}$$

This example shows that the metric $s_{D}$ on G defined is not equivalent to the other four metrics $j^{*}_G, p_G, Tr_G, t_G$ when $G=\{z\in \mathbb {C}: |z|<1\}$.

3 Convergence Results on $s_{D}$ Under Hausdorff Metric

In this section, we study the convergence of sets with an eye toward making sure the convergence of the associated sequence of metric spaces defined in this paper. Our main result was motivated by the following theorem obtained by Herron [16].

Theorem 3.1

[16] Suppose a sequence $(A_i )$ of closed subsets of $\widehat{\mathbb {R}}^n$ converges, with respect to chordal Hausdorff distance, to a closed set $A\ne \{\infty \}$. Then, with respect to pointed Gromov–Hausdorff distance, the quasihyperbolizations of $(\mathbb {R}^n{\setminus } A_i )$ converge to the quasihyperbolization of $\mathbb {R}^n{\setminus } A$.

Let (X, d) be an arbitrary metric space. The distance from a point z to a set A is denoted by $\mathrm {dist}(z,A)=\inf \{d(z,y):y\in A\}$. The Hausdorff distance between subsets $A,B\subset X$ is

$$\begin{aligned} \begin{aligned} d_H(A,B)&=\max \{\sup _{b\in B}(d(b,A)),\sup _{a\in A}d(a,B)\}\\&=\inf \{\varepsilon : A \subset B(\varepsilon ),B\subset A(\varepsilon )\}, \end{aligned} \end{aligned}$$

where $E(\varepsilon )=\{y\in X:d(y, E)<\varepsilon \}$.

Lemma 3.1

Suppose $D_1,D_2$ are two non-trivial bounded closed subsets of a metric space (X, d). Then

$$\begin{aligned} |d_{D_2}(x)-d_{D_1}(x)|\le d_H(D_1,D_2) \end{aligned}$$

for all $x\in X$.

Proof

For any $\varepsilon >0$, there is $x_1\in D_1$ such that $d(x,x_1)\le d_{D_1}(x)\le d(x,x_1)+\varepsilon$. Thus for any $x_2\in D_2$, we have

$$\begin{aligned} \begin{aligned}d_{D_1}(x)-d_{D_2}(x)&\le d(x,x_1)-d_{D_2}(x)+\varepsilon \\&\le d(x,x_1)-d(x,x_2)+\varepsilon \\&\le d(x_1,x_2)+\varepsilon . \end{aligned} \end{aligned}$$

By the arbitrariness of $x_2$, we have

$$\begin{aligned} d_{D_1}(x)-d_{D_2}(x)\le d(x_1, D_2)+\varepsilon \le d_H(D_1,D_2)+\varepsilon . \end{aligned}$$

By the arbitrariness of $\varepsilon$, we have

$$\begin{aligned} d_{D_1}(x)-d_{D_2}(x)\le d_H(D_1,D_2). \end{aligned}$$

By the same argument, we have

$$\begin{aligned} d_{D_2}(x)-d_{D_1}(x)\le d_H(D_1,D_2). \end{aligned}$$

In conclusion, we have

$$\begin{aligned} |d_{D_2}(x)-d_{D_1}(x)|\le d_H(D_1,D_2). \end{aligned}$$

$\square$

From Lemma 3.1, we have the following result.

Lemma 3.2

Suppose $\{D_n\}_{n=1}^{\infty }$ is a sequence of non-trivial bounded closed subsets of a metric space (X, d). If $d_H(D_n, D)\rightarrow 0$ as $n\rightarrow \infty$, then

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{x\in X}|d_{D_n}(x)-d_D(x)|=0. \end{aligned}$$

Proof

From Lemma 3.1, we have $|d_{D}(x)-d_{D_n}(x)|\le d_H(D_n,D)$ for any $x\in X$. Thus

$$\begin{aligned} \sup _{x\in X}|d_{D_n}(x)-d_{D}(x)|\le d_H(D_n,D)\rightarrow 0 \end{aligned}$$

as $n\rightarrow \infty$. $\square$

Theorem 3.2

Suppose $\{D_n\}_{n=1}^{\infty }$ is a sequence of non-trivial bounded closed subsets of a metric space (X, d). If $d_H(D_n, D)\rightarrow 0$ as $n\rightarrow \infty$. Then for all $x,y\in X$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }S_{D_n}(x,y)=S_{D}(x,y). \end{aligned}$$

Proof

We set $\varepsilon _n=d_H(D_n, D)$ for $n\in \mathbb {N}$, then $\varepsilon _n\rightarrow 0$ as $n\rightarrow \infty$. From Lemma 3.1, we have

$$\begin{aligned} d_D(x)-\varepsilon _n\le d_{D_n}(x)\le d_D(x)+\varepsilon _n \end{aligned}$$

for all $x\in X$.

For any $x,y\in X$, we have

$$\begin{aligned} \begin{aligned} s_{D_n}(x,y)&=\frac{d(x,y)}{(1+d_{D_n}(x))(1+d_{D_n}(y))}\\&=\frac{d(x,y)}{(1+d_{D}(x))(1+d_{D}(y))}\frac{(1+d_{D}(x))(1+d_{D}(y))}{(1+d_{D_n}(x))(1+d_{D_n}(y))}\\&= s_{D}(x,y)\frac{(1+d_{D}(x))(1+d_{D}(y))}{(1+d_{D_n}(x))(1+d_{D_n}(y))}\\&\le s_{D}(x,y)\frac{(1+d_{D}(x))(1+d_{D}(y))}{(1+d_{D}(x)-\varepsilon _n)(1+d_{D}(y)-\varepsilon _n)} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} s_{D_n}(x,y)&=\frac{d(x,y)}{(1+d_{D_n}(x))(1+d_{D_n}(y))}\\&=\frac{d(x,y)}{(1+d_{D}(x))(1+d_{D}(y))}\frac{(1+d_{D}(x))(1+d_{D}(y))}{(1+d_{D_n}(x))(1+d_{D_n}(y))}\\&= s_{D}(x,y)\frac{(1+d_{D}(x))(1+d_{D}(y))}{(1+d_{D_n}(x))(1+d_{D_n}(y))}\\&\ge s_{D}(x,y)\frac{(1+d_{D}(x))(1+d_{D}(y))}{(1+d_{D}(x)+\varepsilon _n)(1+d_{D}(y)+\varepsilon _n)}. \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }s_{D_n}(x,y)&\le \lim _{n\rightarrow \infty } s_{D}(x,y)\frac{(1+d_{D}(x))(1+d_{D}(y))}{(1+d_{D}(x)-\varepsilon _n)(1+d_{D}(y)-\varepsilon _n)}\\ {}&= s_{D}(x,y) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }s_{D_n}(x,y)&\ge \lim _{n\rightarrow \infty } s_{D}(x,y)\frac{(1+d_{D}(x))(1+d_{D}(y))}{(1+d_{D}(x)+\varepsilon _n)(1+d_{D}(y)+\varepsilon _n)}\\&= s_{D}(x,y). \end{aligned} \end{aligned}$$

Those show that

$$\begin{aligned} \lim _{n\rightarrow \infty }s_{D_n}(x,y)=s_{D}(x,y) \end{aligned}$$

for all $x,y\in X$.

Fix $x,y\in X$. For any sequence $x_0=x, x_1,\ldots ,x_k=y$, we have

$$\begin{aligned} \begin{aligned} S_{D_n}(x,y)\le \sum _{i=1}^k s_{D_n}(x_i,x_{i-1}). \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned} \limsup _{n\rightarrow \infty }S_{D_n}(x,y)&\le \limsup _{n\rightarrow \infty }\sum _{i=1}^k s_{D_n}(x_i,x_{i-1})\\&=\lim _{n\rightarrow \infty }\sum _{i=1}^ks_{D_n}(x_i,x_{i-1})\\&=\sum _{i=1}^ks_{D}(x_i,x_{i-1}). \end{aligned} \end{aligned}$$

By the arbitrariness of the sequence $x_0=x, x_1,\ldots ,x_k=y$, we have

$$\begin{aligned} \limsup _{n\rightarrow \infty }S_{D_n}(x,y)\le S_D(x,y). \end{aligned}$$

(3.1)

On the other hand, for any a sequence $x_0=x,x_1\cdots ,x_k=y$ in X, we have

$$\begin{aligned} \begin{aligned} S_{D}(x,y)\le \sum _{i=1}^k s_{D}(x_i,x_{i-1}). \end{aligned} \end{aligned}$$

Note that

$$\begin{aligned} \lim _{n\rightarrow \infty }s_{D_n}(x_i,x_{i-1})=s_{D}(x_i,x_{i-1}) \quad \text {for all }1\le i\le k. \end{aligned}$$

We conclude that for any $\varepsilon >0$, there is a $N\in \mathbb {N}$ such that when $n\ge N$,

$$\begin{aligned} |s_D(x_i,x_{i-1})-s_{D_n}(x_i,x_{i-1})|\le \frac{\varepsilon }{k} \end{aligned}$$

for all $1\le i\le k$. Thus, when $n\ge N$, we have

$$\begin{aligned} \begin{aligned} S_{D}(x,y)\le \sum _{i=1}^k s_{D}(x_i,x_{i-1})\le \sum _{i=1}^k s_{D_n}(x_i,x_{i-1})+\varepsilon . \end{aligned} \end{aligned}$$

By the arbitrariness of the sequence $\{x_i\}_{i=0}^k$, we have

$$\begin{aligned} \begin{aligned} S_{D}(x,y)\le S_{D_n}(x,y)+\varepsilon \end{aligned} \end{aligned}$$

when $n\ge N$. Thus

$$\begin{aligned} \begin{aligned} S_{D}(x,y)\le \liminf _{n\rightarrow \infty }S_{D_n}(x,y)+\varepsilon . \end{aligned} \end{aligned}$$

By the arbitrariness of $\varepsilon$, we have

$$\begin{aligned} S_{D}(x,y)\le \liminf _{n\rightarrow \infty }S_{D_n}(x,y). \end{aligned}$$

(3.2)

From (3.1) and (3.2), we have

$$\begin{aligned} \lim _{n\rightarrow \infty }S_{D_n}(x,y)=S_{D}(x,y). \end{aligned}$$

$\square$

If (X, d) is a bounded Ptolemy metric space, we have the following stronger convergence result. We need to recall the notion of Gromov–Hausdorff distance, which appears in [5, 6]. We also refer to [31] for the Gromov–Hausdorff limit of a sequence of metric spaces.

Definition 3.1

(Gromov–Hausdorff distance) A subset S of a metric space X is said to be $\epsilon$-dense if every point of X lies in the $\epsilon$-neighborhood of S. (Under the same circumstances, S is called a $\epsilon$-net in X.) An $\epsilon$-relation between two metric spaces $X_1$ and $X_2$ is a subset $R\subseteq X_1 \times X_2$ such that:

• (1) for $i=1, 2$, the projection of R to $X_i$ is $\epsilon$-dense, and

• (2) if $(x_1,x_2),(x'_1,x'_2)\in R$ then $| d_{X_1}(x_1, x'_1)-d_{X_2}( x_2, x'_2)|<\epsilon$. The relation is said to be surjective if its projection onto each $X_i$ is surjective. If there exists an $\epsilon$-relation between $X_1$ and $X_2$, then we write $X_1\sim _\epsilon X_2$, and if there is a surjective $\epsilon$-relation, then we write $X_1 \simeq _\epsilon X_2$. We define the Gromov–Hausdorff distance between $X_1$ and $X_2$ to be:

  $$\begin{aligned} d_{GH}(X_1, X_2)=\frac{1}{2}\inf \{\epsilon | X_1 \simeq _\epsilon X_2\}. \end{aligned}$$

Definition 3.2

(Gromov–Hausdorff limit) We say that a sequence of metric spaces $(X_n,d_n)$ converges to (X, d) in the Gromov–Hausdorff metric if and only if $d_{GH}(X_n, X) \rightarrow 0$ as $n\rightarrow \infty$. Write

$$\begin{aligned} \mathrm {GH}-\lim _{n\rightarrow \infty }(X_n,d_n)=(X,d). \end{aligned}$$

Theorem 3.3

Suppose $\{D_n\}_{n=1}^{\infty }$ is a sequence of non-trivial bounded closed subsets of a bounded Ptolemy metric space (X, d). If $d_H(D_n, D)\rightarrow 0$ as $n\rightarrow \infty$, then

$$\begin{aligned} \mathrm {GH}-\lim _{n\rightarrow \infty }(X, s_{D_n})=(X, s_{D}). \end{aligned}$$

Proof

Since (X, d) is a Ptolemy space, all $s_{D_n},n\in \mathbb {N}$ and $s_D$ are metrics on X by Theorem 2.1, For $x,y\in X$, by Lemma 3.1, we have

$$\begin{aligned} |d_{D_n}(x)-d_D(x)|<d_H(D_n,D) \end{aligned}$$

and

$$\begin{aligned} |d_{D_n}(y)-d_D(y)|<d_H(D_n,D). \end{aligned}$$

Since (X, d) is bounded, we have

$$\begin{aligned} \begin{aligned} |d_{D_n}(x)d_{D_n}(y)-d_D(x)d_{D}(y)|&\le d_{D_n}(x)|d_{D_n}(y)-d_{D}(y)|+d_D(y)|d_{D_n}(x)-d_D(x)|\\&<2\mathrm {diam}X d_H(D_n,D). \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} |s_{D_n}(x,y)- s_{D}(x,y)|&=\Big |\frac{d(x,y)}{\left( 1+d_{D_n}(x)\right) \left( 1+d_{D_n}(y)\right) }-\frac{d(x,y)}{\left( 1+d_{D}(x)\right) \left( 1+d_{D}(y)\right) }\Big |\\&\le \mathrm {diam}X\Big |d_{D_n}(x)-d_D(x)+d_{D_n}(y)-d_D(y) \\&\quad +\,d_{D_n}(x)d_{D_n}(y)-d_D(x)d_{D}(y)\Big |\\&\le d_{H}(D,D_n)(2+2\mathrm {diam}X)\mathrm {diam}X. \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \sup _{x,y\in X}|s_{D_n}(x,y)- s_{D}(x,y)|\rightarrow 0 \end{aligned}$$

as $n\rightarrow \infty$.

For any $\varepsilon >0$, there is $N\in \mathbb {N}$ such that for all $n>N$, we have $|s_{D_n}(x,y)- s_{D}(x,y)|<\varepsilon$ for all $x,y\in X$. Take $R_n=\{(x,x):x\in X\}$ for $n>N$. Obviously, $R_n$ is a surjective $\epsilon$-relation between two metric space $(X,s_{D_n})$ and $(X, s_{D})$. Thus $d_{GH}((X,s_{D_n}), (X, s_{D}))<\varepsilon$. That is

$$\begin{aligned} \mathrm {GH}-\lim _{n\rightarrow \infty }(X,s_{D_n})=(X, s_{D}). \end{aligned}$$

$\square$

4 Distortion Property Under Möbius Transformations

The main goal of this section is to study the distortion property of the metric defined in this paper under Möbius transformations. In the field of geometric function theory, distortion means how to use the distance between two points to control the distance between their images under some maps. In [19], Ibragimov studied the distortion properties of the scale-invariant Cassinian metric $\widetilde{\tau }$ of the unit ball under Möbius transformations. Mohapatra and Sahoo also considered the distortion of the $\widetilde{\tau }$-metric under Möbius transformations of a punctured ball onto another punctured ball in [23].

In order to state our result, we need to recall some notions and basic properties of Möbius transformations in the n-dimensional Euclidean space $\mathbb {R}^n$ for $n\ge 2$. Given $x\in \mathbb {R}^n$ and $r>0$, we set $B^{n}(x,r):=\{y\in \mathbb {R}^n: |x-y|< r\}$ and $S^{n-1}(a,r):=\{x\in \mathbb {R}^n: |x-a|=r\}$. The reflection on $S^{n-1}(a,r)$ is the function $\psi$ defined by

$$\begin{aligned} \psi (x)=a+\left( \frac{r}{|x-a|}\right) ^{2}(x-a),\quad x\in \mathbb {R}^n{\setminus }\{a\}, \end{aligned}$$

and $\psi (a)=\infty$, $\psi (\infty )=a$. It is convenient to denote reflection on $\mathbb {S}^{n-1}:=S^{n-1}(0,1)$ by $x\mapsto x^{*}$, i.e., $x^{*}=x/|x|^{2}$ if $x\ne 0,\infty$. For any reflection $\psi$ on $S^{n-1}(a,r)$, the following formula has been obtain in [2, (3.1.5)]):

$$\begin{aligned} \ |\psi (x)-\psi (y)|=\frac{r^{2}|x-y|}{|x-a||y-a|},\quad x,y\in \mathbb {R}^n{\setminus }\{a\}. \end{aligned}$$

(4.1)

In the following, we consider the distortion of the metric $s_{\partial \mathbb {B}^n}$ on the unit ball $\mathbb {B}^n:=B^n(0,1)$ under Möbius transformations of the unit ball in $\mathbb {R}^n$. According to Theorem 3.5.1 of [2], if $\psi$ is a Möbius transformation with $\psi (\mathbb {B}^n)=\mathbb {B}^n$, then $\psi (z)=A(\sigma (x))$, where $\sigma$ is a reflection in some sphere orthogonal to $\partial \mathbb {B}^n$ and A is an orthogonal matrix. Thus it is enough to consider the distortion of the reflection in some sphere orthogonal to $\partial \mathbb {B}^n$.

Theorem 4.1

Let $\phi$ be the reflection in the Euclidean sphere S(a, r) with $\phi (\mathbb {B}^n)=\mathbb {B}^n$. Then for $x,y\in \mathbb {B}^n$, we have

$$\begin{aligned} s_{\partial \mathbb {B}^n}(x,y)\frac{|a|-1}{|a|+1}\le s_{\partial \mathbb {B}^n}(\phi (x),\phi (y))\le s_{\partial \mathbb {B}^n}(x,y)\frac{|a|+1}{|a|-1}. \end{aligned}$$

Proof

Since $\phi (\mathbb {B}^n)=\mathbb {B}^n$, we have S(a, r) and $\partial \mathbb {B}^n$ are orthogonal and

$$\begin{aligned} \phi (0)=a^{*}=\frac{a}{|a|^2}, \end{aligned}$$

which implies that $|a|^2=r^2+1$. Note that

$$\begin{aligned} s_{\partial \mathbb {B}^n}(\phi (x),\phi (y))=\frac{|\phi (x)-\phi (y)|}{(|\phi (x)|+2)|(|\phi (y)|+2)} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \frac{s_{\partial \mathbb {B}^n}(\phi (x),\phi (y))}{s_{\partial \mathbb {B}^n}(x,y)}&=\frac{|\phi (x)-\phi (y)|}{(|\phi (x)|+2)|(|\phi (y)|+2)} \frac{(|x|+2)(|y|+2)|}{|x-y|}\\&=\frac{|\phi (x)-\phi (y)|}{|x-y|} \frac{(|x|+2)(|y|+2)|}{(|\phi (x)|+2)|(|\phi (y)|+2)}\\&=r^2\frac{|x|+2}{(|\phi (x)|+2)|x-a|}\frac{|y|+2}{(|\phi (y)|+2)|y-a|}. \end{aligned} \end{aligned}$$

For $x\in \mathbb {B}^n$, set $\Delta (x)=\frac{|x|+2}{(|\phi (x)|+2)|x-a|}$, we claim that

$$\begin{aligned} \Delta (x)\ge \frac{1}{|a|+1}. \end{aligned}$$

Since $|\phi (x)|=|\phi (x)-\phi (a^{*})|=\frac{|a||x-a^{*}|}{|x-a|}$, we obtain

$$\begin{aligned} \begin{aligned} \Delta (x)&=\frac{|x|+2}{(\frac{|a||x-a^{*}|}{|x-a|}+2)|x-a|}\\&=\frac{|x|+2}{|a||x-a^{*}|+2|x-a|}\\&=\frac{|a|(|x|+2)}{\big ||a|^2x-a\big |+2|a||x-a|}. \end{aligned} \end{aligned}$$

Note that $x\in \mathbb {B}^n$, we have

$$\begin{aligned} \begin{aligned} \big ||a|^2x-a\big |+2|a||x-a|&\le |a|^2|x|+|a|+2|a||x|+2|a|^2\\&=(|x|+2)|a|^2+|a|(1+2|x|)\\&\le (|x|+2)|a|^2+|a|(2+|x|)\\&=(|x|+2)(|a|^2+|a|). \end{aligned} \end{aligned}$$

Thus $\Delta (x)\ge \frac{|a|(|x|+2)}{(|x|+2)(|a|^2+|a|)}=\frac{1}{|a|+1}$, which implies that

$$\begin{aligned} \frac{s_{\partial \mathbb {B}^n}(\phi (x),\phi (y))}{s_{\partial \mathbb {B}^n}(x,y)}\ge r^2\left( \frac{1}{|a|+1}\right) ^2=\frac{|a|-1}{|a|+1}. \end{aligned}$$

Note that $\phi ^2=1$, we have

$$\begin{aligned} \frac{s_{\partial \mathbb {B}^n}(x,y)}{s_{\partial \mathbb {B}^n}(\phi (x),\phi (y))}=\frac{s_{\partial \mathbb {B}^n}(\phi ^2(x),\phi ^2(y))}{s_{\partial \mathbb {B}^n}(\phi (x),\phi (y))}\ge \frac{|a|-1}{|a|+1}. \end{aligned}$$

So

$$\begin{aligned} \frac{s_{\partial \mathbb {B}^n}(\phi (x),\phi (y))}{s_{\partial \mathbb {B}^n}(x,y)}\le \frac{|a|+1}{|a|-1}. \end{aligned}$$

In conclusion, we complete the proof of Theorem. $\square$