Examples of hyperbolic spaces without the properties of quasi-ball or bounded eccentricity

Qizheng You and Jiawen Zhang School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, 200433, China. [email protected] School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, 200433, China. [email protected]

Abstract.

In this note, we present examples of non-quasi-geodesic metric spaces which are hyperbolic (i.e., satisfying the Gromov’s $4$ -point condition) while the intersection of any two metric balls therein does not either “look like” a ball or has uniformly bounded eccentricity. This answers an open question posed in [2].

Keywords: Gromov’s hyperbolic spaces, Non-(quasi)-geodesic, Quasi-ball property, Bounded eccentricity property

1. Introduction

In the seminal work [4], Gromov introduced a notion of hyperbolicity for metric spaces which encodes the information of metric curvatures for the underlying spaces, with prototypes from classic hyperbolic geometry. Gromov’s hyperbolic spaces have attractd a lot of interest since they are discovered and have fruitful applications in various aspects of mathematics (see, e.g., [1, 3]).

Recall that a geodesic metric space $(X,d)$ is called hyperbolic (in the sense of Gromov [4]) if there exists $\delta>0$ such that for any geodesic triangle in $(X,d)$ , the union of the $\delta$ -neighbourhoods of any two sides of the triangle contains the third. Gromov also provided a characterisation for his hyperbolicity using the so-called Gromov product:

(x|y)_{p}=\frac{1}{2}(d(x,p)+d(y,p)-d(x,y))\quad\text{for}\quad x,y,p\in(X,d).

He proved in [4] that a geodesic metric space $(X,d)$ is hyperbolic if and only if the following condition holds:

Gromov’s $4$ -point condition: There exists $\delta>0$ such that

(x|y)_{p}\geq\min\{(x|z)_{p},(y|z)_{p}\}-\delta\quad\text{for all}\quad x,y,z,p\in(X,d).

Note that the statement of the Gromov’s $4$ -point condition does not require that the underlying space $(X,d)$ is geodesic. Hence in the general context, we say that a (not necessarily geodesic) metric space is hyperbolic if the Gromov’s $4$ -point condition holds.

Later in [2], Chatterji and Niblo discovered new characterisations of Gromov’s hyperbolicity for geodesic metric spaces using the geometry of intersections of balls. More precisely, for a geodesic metric space, they showed that it is hyperbolic in the sense of Gromov if and only if the following holds:

Quasi-ball property: The intersection of any two metric balls is at uniformly bounded Hausdorff distance from a ball.

They also considered the eccentricity of the intersection of balls. Recall that for a metric space $(X,d)$ and $\delta>0$ , we say that the eccentricity of a subset $S$ of $X$ is less than $\delta$ if there exist $c,c^{\prime}\in X$ and $R\geq 0$ such that

(1.1)

B(c,R)\subseteq S\subseteq B(c^{\prime},R+\delta).

Here we use $B(x,r):=\{y\in X:d(x,y)\leq r\}$ to denote the metric ball. The eccentricity of $S$ is the infimum of $\delta$ satisfying (1.1). By convention, the eccentricity of the empty set is $0$ . Chatterji and Niblo proved in [2] that a geodesic metric space is hyperbolic if and only if the following holds:

Bounded eccentricity property: The intersection of any two metric balls has uniformly bounded eccentricity.

In [2, Section 4], Chatterji and Niblo also discussed the situation of non-geodesic metric spaces. They recorded an example due to Viktor Schroeder (see [2, Example 18]) that there exists a non-geodesic metric space with the quasi-ball property but not hyperbolic (i.e., does not satisfy the Gromov’s $4$ -point condition). However, the other direction is unclear and hence, they asked the following:

Question 1.1.

Does there exist a non-geodesic hyperbolic metric space which does not satisfy the quasi-ball property or the bounded eccentricity property?

In this short note, we provide an affirmative answer to Question 1.1 by constructing concrete examples. The main result is the following:

Theorem 1.2.

There exists a non-quasi-geodesic hyperbolic (i.e., satisfying the Gromov’s $4$ -point condition) space which does not satisfy either the quasi-ball property or the bounded eccentricity property.

Our construction is motivated by Gromov’s observation in [4, Section 1.2] (also suggested in [2, Section 4]) that for a metric space $(X,d)$ , we can endow another metric $d^{\prime}$ on $X$ defined by

(1.2)

d^{\prime}(x,y)=\ln(1+d(x,y))\quad\text{for}\quad x,y\in X

such that $(X,d^{\prime})$ satisfies the Gromov’s $4$ -point condition. We show that if $(X,d)$ is geodesic and unbounded, then $(X,d^{\prime})$ cannot be quasi-geodesic (see Corollary 3.3). Then we study the relation of the quasi-ball property and the bounded eccentricity property between $(X,d)$ and $(X,d^{\prime})$ (see Lemma 3.5, Lemma 3.7 and Lemma 3.9). Finally, we show in Example 3.10 that for the Euclidean space $X=\mathbb{R}^{2}$ with the Euclidean metric $d$ , the construction $(X,d^{\prime})$ in (1.2) provides an example to conclude Theorem 1.2.

2. Preliminaries

Here we collect some necessary notions and notation for this note.

Let $(X,d)$ be a metric space. For $x\in X$ and $R\geq 0$ , denote the (metric) ball by $B(x,R)=\{y\in X:d(x,y)\leq R\}$ . We say that $(X,d)$ is bounded if there exist $x\in X$ and $R\geq 0$ such that $X=B(x,R)$ , and unbounded if it is not bounded. For a subset $A\subset X$ and $R\geq 0$ , denote $\mathcal{N}_{R}(A)=\{x\in X:d(x,A)\leq R\}$ the $R$ -neighbourhood of $A$ in $X$ . For subsets $A,B\subset X$ , the Hausdorff distance between $A$ and $B$ is

d_{H}(A,B)=\inf\{R\geq 0:A\subseteq\mathcal{N}_{R}(B)\text{ and }B\subseteq\mathcal{N}_{R}(A)\}.

Recall that a path in a metric space $(X,d)$ is a continuous map $\gamma:[a,b]\to X$ . A path $\gamma$ is called rectifiable if its length

\ell(\gamma):=\sup\left\{\sum_{i=1}^{n}d(\gamma(t_{i-1}),\gamma(t_{i})):a=t_{0}<t_{1}<\cdots<t_{n}=b,n\in\mathbb{N}\right\}

is finite. Usually it is convenient to change the parameter $t\in[a,b]$ to the standard arc parameter $s\in[0,\ell(\gamma)]$ as follows. Define a map $\varphi:[a,b]\rightarrow[0,\ell(\gamma)]$ by $t\mapsto s=\ell(\gamma|_{[a,t]})$ , and set $\tilde{\gamma}:[0,\ell(\gamma)]\to X$ by $\tilde{\gamma}:=\gamma\circ\varphi^{-1}$ . Then we have $\ell(\tilde{\gamma}|_{[s_{1},s_{2}]})=|s_{1}-s_{2}|$ for any $0\leq s_{1}\leq s_{2}\leq\ell(\gamma)$ .

Now we recall the notion of (quasi-)geodesics.

Definition 2.1.

Let $(X,d)$ be a metric space.

(1)

Given $x,y\in X$ , a geodesic between $x$ and $y$ is an isometric embedding $\gamma:[0,d(x,y)]\rightarrow X$ with $\gamma(0)=x$ and $\gamma(d(x,y))=y$ . The space $(X,d)$ is called geodesic if for any $x,y$ in $X$ , there exists a geodesic between $x$ and $y$ .
(2)

Given $x,y\in X,L\geq 1$ and $C\geq 0$ , an $(L,C)$ -quasi-geodesic between $x$ and $y$ is a map $\gamma:[0,T]\rightarrow X$ such that $\gamma(0)=x,\gamma(T)=y$ and

$\frac{1}{L}|a-b|-C\leq d(\gamma(a),\gamma(b))\leq L|a-b|+C\quad\text{for all}\quad a,b\in[0,T].$

The space $(X,d)$ is called $(L,C)$ -quasi-geodesic if for any $x,y$ in $X$ , there exists an $(L,C)$ -quasi-geodesic between $x$ and $y$ . We also say that $(X,d)$ is quasi-geodesic if it is $(L,C)$ -quasi-geodesic for some $L$ and $C$ .

We also need the notion of ultrametric space. Recall that a metric space $(X,d)$ is called ultrametric if there exists $\delta>0$ such that for any points $x,y,z\in X$ , we have

d(x,y)\leq\mathrm{max}\{d(x,z),d(y,z)\}+\delta.

The following is due to Gromov:

Lemma 2.2 ([4, Section 1.2]).

An ultrametric space satisfies the Gromov’s $4$ -point condition.

Recall from Section 1 that for a metric space $(X,d)$ , Gromov considered another metric $d^{\prime}$ on $X$ defined in (1.2) and noticed that

$\displaystyle d^{\prime}(x,y)$	$\displaystyle\leq$	$\displaystyle\ln(1+d(x,z)+d(z,y))$
	$\displaystyle\leq$	$\displaystyle\ln(2+2\mathrm{max}\{d(x,z),d(y,z)\})$
	$\displaystyle=$	$\displaystyle\mathrm{max}\{d^{\prime}(x,z),d^{\prime}(y,z)\}+\ln 2.$

Combining with Lemma 2.2, we obtain the following:

Lemma 2.3 ([4, Section 1.2]).

For a metric space $(X,d)$ , the new metric $d^{\prime}$ on $X$ defined in (1.2) satisfies the Gromov’s $4$ -point condition.

3. Proof of Theorem 1.2

This whole section is devoted to the proof of Theorem 1.2, which is divided into several parts.

Firstly, we would like to study the property of (quasi-)geodesic for the new metric $d^{\prime}$ defined in (1.2). To simplify the notation, for a path $\gamma$ in $X$ , we denote $\ell(\gamma)$ and $\ell^{\prime}(\gamma)$ its length with respect to the metric $d$ and $d^{\prime}$ , respectively.

We need the following lemma:

Lemma 3.1.

Let $(X,d)$ be a metric space and $d^{\prime}$ be the metric on $X$ defined in (1.2). A path $\gamma:[a,b]\rightarrow X$ in $X$ is rectifiable with respect to $d$ if and only if it is rectifiable with respect to $d^{\prime}$ . In this case, we have $\ell(\gamma)=\ell^{\prime}(\gamma)$ .

Proof.

Firstly, we assume that $\gamma$ is rectifiable with respect to $d$ , i.e., $\ell(\gamma)<\infty$ . Note that $\ln(1+x)\leq x$ holds for all $x\geq 0$ . Hence for any partition $a=t_{0}<t_{1}<\cdots<t_{n}=b$ , we have

\sum_{i=1}^{n}d^{\prime}(\gamma(t_{i-1}),\gamma(t_{i}))\leq\sum_{i=1}^{n}d(\gamma(t_{i-1}),\gamma(t_{i}))\leq\ell(\gamma),

which implies that $\ell^{\prime}(\gamma)\leq\ell(\gamma)$ . In particular, $\gamma$ is rectifiable with respect to $d^{\prime}$ .

Conversely, we assume that $\gamma$ is rectifiable with respect to $d^{\prime}$ , i.e., $\ell^{\prime}(\gamma)<\infty$ . Without loss of generality, we can assume that $\gamma$ is parametrised by the standard arc parameter with $a=0$ and $b=\ell^{\prime}(\gamma)$ . Note that for any $\alpha\in(0,1)$ , there exists $\delta>0$ such that $\alpha x\leq\ln(1+x)$ holds for all $x\in[0,\delta]$ . Given a partition $a=s_{0}<s_{1}<\cdots<s_{m}=b$ , we choose a refinement $a=t_{0}<t_{1}<\cdots<t_{n}=b$ such that $|t_{i-1}-t_{i}|<\ln(1+\delta)$ holds for all $i$ . Hence we have

d^{\prime}(\gamma(t_{i-1}),\gamma(t_{i}))\leq l^{\prime}(\gamma|_{[t_{i-1},t_{i}]})=|t_{i-1}-t_{i}|<\ln(1+\delta),

which implies that $d(\gamma(t_{i-1}),\gamma(t_{i}))<\delta$ due to (1.2). Therefore, we obtain

\sum_{i=1}^{m}d(\gamma(s_{i-1}),\gamma(s_{i}))\leq\sum_{i=1}^{n}d(\gamma(t_{i-1}),\gamma(t_{i}))\leq\frac{1}{\alpha}\cdot\sum_{i=1}^{n}d^{\prime}(\gamma(t_{i-1}),\gamma(t_{i}))\leq\frac{1}{\alpha}\cdot\ell^{\prime}(\gamma)

for all $\alpha\in(0,1)$ . Letting $\alpha\to 1$ and taking the supremum of the left hand side, we obtain that $\ell(\gamma)\leq\ell^{\prime}(\gamma)$ , which concludes the proof. ∎

As a direct corollary, we obtain the following:

Corollary 3.2.

Let $(X,d)$ be a geodesic metric space which contains at least two elements, and $d^{\prime}$ be the metric defined in (1.2). Then the metric space $(X,d^{\prime})$ is not geodesic.

Proof.

By assumption, we take two distinct points $x,y\in X$ . If $(X,d^{\prime})$ is geodesic, we choose a geodesic $\gamma$ between $x$ and $y$ . In particular, $\gamma$ is rectifiable with respect to $d^{\prime}$ . Hence by Lemma 3.1, we know that $\gamma$ is also rectifiable with respect to $d$ and we have

\ell^{\prime}(\gamma)=\ell(\gamma)\geq d(x,y)>d^{\prime}(x,y),

where the last inequality follows from the assumption that $x\neq y$ . This is a contradiction to the assumption that $\gamma$ is a geodesic between $x$ and $y$ with respect to the metric $d^{\prime}$ . Hence we conclude the proof. ∎

Moreover, with an extra hypothesis, the new metric $d^{\prime}$ cannot be even quasi-geodesic.

Corollary 3.3.

Let $(X,d)$ be an unbounded geodesic metric space, and $d^{\prime}$ be the metric defined in (1.2). Then for any $L\geq 1$ and $C\geq 0$ , the metric space $(X,d^{\prime})$ cannot be $(L,C)$ -quasi-geodesic.

To prove Corollary 3.3, we need the following lemma to tame quasi-geodesics. The idea is similar to [1, Lemma III.H.1.11], but the setting is slightly different.

Lemma 3.4.

Let $(X,d)$ be a geodesic metric space and $d^{\prime}$ be the metric defined in (1.2). Given an $(L,C)$ -quasi-geodesic $\gamma:[a,b]\rightarrow X$ with respect to $d^{\prime}$ , there exists a continuous and rectifiable $(L,C^{\prime})$ -quasi-geodesic $\gamma^{\prime}:[a,b]\rightarrow X$ with respect to $d^{\prime}$ satisfying the following:

(1)

$\gamma^{\prime}(a)=\gamma(a)$ and $\gamma^{\prime}(b)=\gamma(b)$ ;
(2)

$C^{\prime}=3(L+C)$ ;
(3)

$\ell^{\prime}(\gamma^{\prime}|_{[t,t^{\prime}]})\leq k_{1}d^{\prime}(\gamma^{\prime}(t),\gamma^{\prime}(t^{\prime}))+k_{2}$ for all $t,t^{\prime}\in[a,b]$ , where $k_{1}=L(L+C)$ and $k_{2}=(LC^{\prime}+4)(L+C)$ ;
(4)

$d^{\prime}_{H}(\mathrm{Im}(\gamma),\mathrm{Im}(\gamma^{\prime}))\leq L+C$ .

Careful readers might already notice that in the situation of [1, Lemma III.H.1.11], we need to assume that the new metric $d^{\prime}$ on $X$ is geodesic instead of the current setting that the original metric $d$ is geodesic. Although the proof is similar, here we also provide one for convenience to readers.

Proof of Lemma 3.4.

Define $\gamma^{\prime}$ to agree with $\gamma$ on $\Sigma:=\{a,b\}\cup(\mathbb{Z}\cap(a,b))$ , then choose geodesic segments with respect to $d$ joining the images of successive points in $\Sigma$ and define $\gamma^{\prime}$ by concatenating linear reparameterisations of these geodesic segments.

Let $[t]$ denote the point of $\Sigma$ closest to $t\in[a,b]$ . Note that the $d^{\prime}$ -distance of the images of successive points in $\Sigma$ is at most $L+C$ , and hence we have

d^{\prime}(\gamma^{\prime}(t),\gamma^{\prime}([t]))=\ln\left(1+d(\gamma^{\prime}(t),\gamma^{\prime}([t]))\right)\leq\ln(1+\exp(L+C)-1)\leq L+C,

which implies that $d^{\prime}_{H}(\mathrm{Im}(\gamma),\mathrm{Im}(\gamma^{\prime}))\leq L+C$ . Since $\gamma$ is an $(L,C)$ -quasi-geodesic with respect to $d^{\prime}$ , and $\gamma([t])=\gamma^{\prime}([t])$ for all $t\in[a,b]$ , we have

d^{\prime}(\gamma^{\prime}(t),\gamma^{\prime}(t^{\prime}))\leq d^{\prime}(\gamma^{\prime}([t]),\gamma^{\prime}([t^{\prime}]))+2(L+C)\leq L|[t]-[t^{\prime}]|+C+2(L+C)\leq L|t-t^{\prime}|+3(L+C),

and similarly, we have

(3.1)

d^{\prime}(\gamma^{\prime}(t),\gamma^{\prime}(t^{\prime}))\geq\frac{1}{L}|t-t^{\prime}|-3(L+C)

for all $t,t^{\prime}\in[a,b]$ . Hence $\gamma^{\prime}$ is an $(L,C^{\prime})$ -quasi-geodesic with respect to $d^{\prime}$ .

For any integers $s,s^{\prime}\in\Sigma$ with $s\leq s^{\prime}$ , Lemma 3.1 tells us that

\ell^{\prime}(\gamma^{\prime}|_{[s,s^{\prime}]})=\ell(\gamma^{\prime}|_{[s,s^{\prime}]})=\sum_{k=s}^{s^{\prime}-1}\ell(\gamma^{\prime}|_{[k,k+1]})\leq|s-s^{\prime}|(L+C).

Similarly for any $s,s^{\prime}\in\Sigma$ , we have $\ell^{\prime}(\gamma^{\prime}|_{[s,s^{\prime}]})\leq(|s-s^{\prime}|+2)(L+C)$ . Hence for any $t,t^{\prime}\in[a,b]$ , we have

\ell^{\prime}(\gamma^{\prime}|_{[t,t^{\prime}]})\leq(|[t]-[t^{\prime}]|+2)(L+C)+(L+C)\leq(|t-t^{\prime}|+4)(L+C).

Combining with Inequality (3.1), we obtain that $\ell(\gamma^{\prime}|_{[t,t^{\prime}]})\leq k_{1}d^{\prime}(\gamma^{\prime}(t),\gamma^{\prime}(t^{\prime}))+k_{2}$ for $k_{1},k_{2}$ defined in (3). Hence we conclude the proof. ∎

Proof of Corollary 3.3.

Assume that $(X,d^{\prime})$ is $(L,C)$ -quasi-geodesic for some $L\geq 1$ and $C\geq 0$ . For any $x,y\in X$ , choose an $(L,C)$ -quasi geodesic $\gamma:[0,T]\rightarrow X$ (with respect to $d^{\prime}$ ) connecting them. Lemma 3.4 implies that there is a rectifiable path $\gamma^{\prime}:[0,T]\rightarrow X$ which is an $(L,3L+3C)$ -quasi-geodesic (with respect to $d^{\prime}$ ) connecting $x$ and $y$ . Moreover, we have $\ell^{\prime}(\gamma^{\prime})\leq k_{1}d^{\prime}(x,y)+k_{2}$ for $k_{1}=L(L+C)$ and $k_{2}=(3L(L+C)+4)(L+C)$ . Hence Lemma 3.1 implies that

k_{1}d^{\prime}(x,y)+k_{2}\geq\ell^{\prime}(\gamma^{\prime})=\ell(\gamma^{\prime})\geq d(x,y)=\exp(d^{\prime}(x,y))-1.

Note that $(X,d)$ is unbounded, which implies that $(X,d^{\prime})$ is also unbounded. Hence taking $d^{\prime}(x,y)\to\infty$ , we conclude a contradiction and finish the proof. ∎

Next, we move to study the relation of the quasi-ball property and the bounded eccentricity property between the metric spaces $(X,d)$ and $(X,d^{\prime})$ . Again to save the notation, for $x\in X$ and $r\geq 0$ , we denote $B(x,r)$ and $B^{\prime}(x,r)$ the closed balls with respect to the metrics $d$ and $d^{\prime}$ , respectively. For $A\subset X$ and $\delta\geq 0$ , we denote $\mathcal{N}_{\delta}(A)$ and $\mathcal{N}^{\prime}_{\delta}(A)$ the $\delta$ -neighbourhood of $A$ with respect to the metrics $d$ and $d^{\prime}$ , respectively.

Lemma 3.5.

Let $(X,d)$ be a metric space and $d^{\prime}$ be the metric on $X$ defined in (1.2). Then $(X,d)$ has the quasi-ball property if and only if $(X,d^{\prime})$ has the quasi-ball property.

Proof.

Firstly, we assume that $(X,d^{\prime})$ has the quasi-ball property, i.e., there exists $\delta\geq 0$ such that the intersection of any two balls in $(X,d^{\prime})$ is $\delta$ -close to another ball (i.e., their Hausdorff distance is bounded by $\delta$ ). Note that there exists $\alpha=\alpha(\delta)>1$ such that $x\leq\alpha\ln(1+x)$ for all $x\in[0,\delta]$ . This implies that for $A,B\subseteq X$ with $A\subseteq\mathcal{N}^{\prime}_{\delta}(B)$ , we have $A\subseteq\mathcal{N}_{\alpha\delta}(B)$ .

Given two balls $B(x,s)$ and $B(y,t)$ in $(X,d)$ , it is clear that $B(x,s)=B^{\prime}(x,\ln(1+s))$ and $B(y,t)=B^{\prime}(y,\ln(1+t))$ . Hence by the assumption, there exists another ball $B^{\prime}(c,r)$ in $(X,d^{\prime})$ such that

d^{\prime}_{H}\left(B^{\prime}(x,\ln(1+s))\cap B^{\prime}(y,\ln(1+t)),B^{\prime}(c,r)\right)\leq\delta.

It follows from the previous paragraph that in this case, we have

d_{H}\left(B(x,s)\cap B(y,t),B(c,\exp(r)-1)\right)=d_{H}\left(B^{\prime}(x,\ln(1+s))\cap B^{\prime}(y,\ln(1+t)),B^{\prime}(c,r)\right)\leq\alpha\delta.

Hence $(X,d)$ has the quasi-ball property.

The converse holds similarly, using the fact that $\ln(1+x)\leq x$ holds for all $x\geq 0$ . ∎

Combining Lemma 2.3, Corollary 3.3 and Lemma 3.5, we obtain the following, which concludes part of Theorem 1.2.

Corollary 3.6.

Let $(X,d)$ be a geodesic metric space which is not hyperbolic (e.g., the Euclidean space $\mathbb{R}^{2}$ with the Euclidean metric) and $d^{\prime}$ be the metric on $X$ defined in (1.2). Then $(X,d^{\prime})$ is a non-quasi-geodesic metric space which satisfies the Gromov’s $4$ -point condition but not the quasi-ball property.

Concerning the bounded eccentricity property, we have the following:

Lemma 3.7.

Let $(X,d)$ be a metric space and $d^{\prime}$ be the metric on $X$ defined in (1.2). If $(X,d)$ satisfies the bounded eccentricity property, so does $(X,d^{\prime})$ .

Proof.

Given two balls $B^{\prime}(x,\ln(1+r))$ and $B^{\prime}(y,\ln(1+s))$ in $(X,d^{\prime})$ , we assume that their intersection $Y$ is non-empty. Note that

Y=B^{\prime}(x,\ln(1+r))\cap B^{\prime}(y,\ln(1+s))=B(x,r)\cap B(y,s)

is again an intersection of balls in $(X,d)$ . Hence by the assumption, there exist $c,c^{\prime}\in X$ and $R\geq 0$ such that

B(c,R)\subseteq Y\subseteq B(c^{\prime},R+\delta_{0}).

Therefore in $(X,d^{\prime})$ , we have

B^{\prime}(c,\ln(1+R))\subseteq Y\subseteq B^{\prime}(c^{\prime},\ln(1+R+\delta_{0})).

Then the eccentricity of $Y$ in $(X,d^{\prime})$ is bounded above by

\ln(1+R+\delta_{0})-\ln(1+R)=\ln(1+\frac{\delta_{0}}{1+R})\leq\ln(1+\delta_{0})\leq\delta_{0},

which concludes the proof. ∎

Remark 3.8.

Readers might wonder whether the converse to Lemma 3.7 holds. Unfortunately, in general this is false. Note that if we have

B^{\prime}(c,\ln(1+R))\subseteq Y\subseteq B^{\prime}(c^{\prime},\ln(1+R)+\delta_{0})

in $(X,d^{\prime})$ , then it implies that

B(c,R)\subseteq Y\subseteq B(c^{\prime},\exp(\delta_{0})\cdot(1+R)-1)

in $(X,d)$ . Hence the eccentricity of $Y$ in $(X,d)$ is bounded by $(\exp(\delta_{0})-1)(1+R)$ . However, if $R$ (i.e., the radius of the ball contained in $Y$ ) does not have a uniform upper bound, then neither does the eccentricity of $Y$ .

Remark 3.8 suggests the following partial converse to Lemma 3.7:

Lemma 3.9.

Let $(X,d)$ be a metric space and $d^{\prime}$ be the metric on $X$ defined in (1.2). Assume that there exists a sequence of subsets $\{Y_{n}\}_{n\in\mathbb{N}}$ of $X$ satisfying the following:

(1)

each $Y_{n}$ is the intersection of two balls in $(X,d)$ ;
(2)

there exists $M>0$ such that for all $n\in\mathbb{N}$ , the radius (with respect to $d$ ) of any ball contained in $Y_{n}$ is bounded above by $M$ ;
(3)

the eccentricity of $Y_{n}$ in $(X,d)$ is not uniformly bounded.

Then $(X,d^{\prime})$ does not satisfy the bounded eccentricity property.

Proof.

Assume the opposite, i.e., there exists $\delta_{0}>0$ such that the eccentricity of the intersection of any two balls in $(X,d^{\prime})$ is uniformly bounded by $\delta_{0}$ . Hence by condition (1) we know that for each $n\in\mathbb{N}$ , there exist $c_{n},c_{n}^{\prime}\in X$ and $r_{n}\geq 0$ such that

B^{\prime}(c_{n},\ln(1+r_{n}))\subseteq Y_{n}\subseteq B^{\prime}(c_{n}^{\prime},\ln(1+r_{n})+\delta_{0}).

Hence in $(X,d)$ , we have

B(c_{n},r_{n})\subseteq Y_{n}\subseteq B(c_{n},\exp(\delta_{0})\cdot(1+r_{n})-1).

By condition (2), we know that $r_{n}\leq M$ for all $n\in\mathbb{N}$ . Therefore, we have

\exp(\delta_{0})\cdot(1+r_{n})-1-r_{n}=(\exp(\delta_{0})-1)(1+r_{n})\leq(\exp(\delta_{0})-1)(1+M),

which is a contradiction to condition (3) in the assumption. ∎

Example 3.10.

Now we show that there exists a non-quasi-geodesic metric space which satisfies the Gromov’s $4$ -point condition but not the bounded eccentricity property. For example, take $(X,d)$ to be the Euclidean space $\mathbb{R}^{2}$ equipped with the Euclidean metric, and let $d^{\prime}$ be the metric on $X$ defined in (1.2). Lemma 2.3 and Corollary 3.3 imply that $(X,d^{\prime})$ is not quasi-geodesic but satisfies the Gromov’s $4$ -point condition.

For each $n\in\mathbb{N}$ , take $x_{n}=(0,0)$ and $y_{n}=(2n,0)$ and set

Y_{n}=B(x_{n},n+1)\cap B(y_{n},n+1).

It is easy to see that in $(X,d)$ , the biggest ball contained in $Y_{n}$ is $B((n,0),1)$ . Moreover, the diameter of $Y_{n}$ is

d\left((n,\sqrt{2n+1}),(n,-\sqrt{2n+1})\right)=2\sqrt{2n+1},

which implies that the eccentricity of $Y_{n}$ cannot be uniformly bounded. Therefore applying Lemma 3.9, we conclude the result.

Proof of Theorem 1.2.

Combining Corollary 3.6 and Example 3.10, we conclude the proof for Theorem 1.2. ∎

Finally, recall that in [5] the following weaker form of the bounded eccentricity property was considered: For a metric space $(X,d)$ , there exist $\lambda>0$ and $\delta>0$ such that for any two balls $B_{1},B_{2}\subseteq X$ with non-empty intersection there exist $z,z^{\prime}\in X$ and $r\geq 0$ such that

B(z,r)\subseteq B_{1}\cap B_{2}\subseteq B(z^{\prime},\lambda r+\delta).

Wenger showed in [5] that this condition also implies Gromov’s hyperbolicity for geodesic metric spaces.

We remark that this weaker form of the bounded eccentricity property does not hold either for the space $(X,d^{\prime})$ constructed in Example 3.10.

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