††footnotetext:  File: main.tex, printed: 2024-8-28, 2.37

# Neargeodesics in Gromov hyperbolic John domains in infinite dimensional
Banach spaces

Vasudevarao Allu  Vasudevarao Allu, Discipline of Mathematics, School of Basic
Sciences, Indian Institute of Technology Bhubaneswar, Argul, Bhubaneswar,
PIN-752050, Odisha (State), India<EMAIL_ADDRESS>and  Abhishek Pandey
Abhishek Pandey, Discipline of Mathematics, School of Basic Sciences, Indian
Institute of Technology Bhubaneswar, Argul, Bhubaneswar, PIN-752050, Odisha
(State), India<EMAIL_ADDRESS>
###### Abstract.

Rasila, Talponen and Zhang in [Proc. Amer. Math. Soc. 146 (2018), 3863–3873]
have proposed a general question that which Banach space properties can be
characterized by the corresponding quasihyperbolic metric? A domain $D$ in a
Banach space $E$ is said to be Gromov hyperbolic if it is $\delta\geq
0$-hyperbolic with respect to quasihyperbolic metric. In this paper we show
that every neargeodesic in a Gromov hyperbolic John domain in infinite
dimensional Banach space is a cone arc. Thus, our result gives an answer to
the above problem in the sense that it gives a necessary condition for a John
domain to be a Gromov hyperbolic space in infinite dimensional Banach space.
Moreover, our main result gives an improvement of a result of Li [Theorem 1,
Int. J. Math. 25 (2014)].

###### Key words and phrases:

Quasihyperbolic metric, Gromov hyperbolic spaces, John domain, neargeodesics,
quasiconformal mappings, quasihyperbolic geodesic, cone arc

###### 2010 Mathematics Subject Classification:

Primary 30C65, 30L10, 30F45. Secondary 30C20

##  1\. Introduction

Let $\Omega$ be an open ball or half space in $\mathbb{R}^{n}$, we denote
$h_{\Omega}$ be the hyperbolic metric in $\Omega$ with constant curvature
$-1$. If $\Omega$ is an upper half plane then $h_{\Omega}$ is defined in
$\Omega$ by the means of the density

$\rho(z)=\frac{1}{{\rm Im\,z}}=\frac{1}{d(z,\partial\Omega)}.$

This density can be used to introduce an analogue of the hyperbolic metric in
an arbitrary subdomain $D$ in $\mathbb{R}^{n}$. The quasihyperbolic metric in
$\mathbb{R}^{n}$ is a generalization of hyperbolic metric, which was
introduced by Gehring and Palka [3] and they have studied the characterization
of domains $D$ in the one point compactification of $\mathbb{R}^{n}$ which are
quasiconformally equivalent to ball in $\mathbb{R}^{n}$. Infact such domains
are homogeneous via quasiconformal mappings. Gehring and Palka [3] have proved
that the maximal dilatation of such quasiconformal mappings can be estimated
in terms of quasihyperbolic metric and using these estimates they obtained
useful results related to the homogeneity of domains with respect to
quasiconformal family. Thus, the importance of the quasihyperbolic metric is
quite clear as it is well-behaved with the quasiconformal mappings. The
quasihyperbolic metric can be naturally defined on a wide range of metric
spaces, including infinite dimensional Banach spaces. The study of this
concept is known as free quasiconformality which has been introduced and
developed by Väisälä (see [12, 13]).
In this paper, we mainly focus on the geometry of neargeodesics in John
domains in infinite dimensional Banach spaces.

##  2\. Preliminaries

In this section we mention some basic as well as advanced concepts and related
results in the literature, which are useful to state and prove our main
result.

###  2.1. Metric Geometry

Let $(X,d)$ be a metric space. A curve is a continuous function
$\gamma:[a,b]\rightarrow X$. If a curve $\gamma:[a,b]\rightarrow X$ is an
embedding of $[a,b]$, then it is called an arc. Let $\mathcal{P}$ denote set
of all partitions $a=t_{0}<t_{1}<t_{2}<\cdots<t_{n}=b$ of the interval
$[a,b]$. The length of the curve $\gamma$ in the metric space $(X,d)$ is

$l_{d}(\gamma)=\sup_{\mathcal{P}}\sum_{k=0}^{n-1}d(\gamma(t_{k}),\gamma(t_{k+1})).$

A curve is said to be rectifiable if $l_{d}(\gamma)<\infty$. A metric space
$X$ is said to be rectifiably connected if every pair of points $x,y\in X$ can
be joined by a rectifiable curve. For a rectifiable curve $\gamma$ we define
arc length $s:[a,b]\rightarrow\left[0,l_{d}(\gamma)\right]$ by
$s(t)=l_{d}(\gamma|_{[a,t]})$. The arc length function is of bounded
variation. For any rectifiable curve $\gamma:[a,b]\to X$, there is a unique
map $\gamma_{s}:[0,l_{d}(\gamma)]\to X$ such that $\gamma=\gamma_{s}\circ s$,
and such a curve $\gamma_{s}$ is called the arclength parametrization of
$\gamma$.

For $x,y\in X$, the inner length metric $\lambda_{X}(x,y)$ is defined by

$\lambda_{X}(x,y)=\inf\\{l_{d}(\gamma):\gamma\mbox{ is a rectifiable arc
joining }x\mbox{ and }y\\}.$

Let $\rho:X\rightarrow[0,\infty]$ be a Borel function. The $\rho$-length of a
rectifiable curve $\gamma$ is

$\int_{\gamma}\rho\,ds=\int_{a}^{b}\rho(\gamma(t))\,ds(t)=\int_{0}^{l_{d}(\gamma)}\rho\circ\gamma_{s}(t)\,dt.$

If $X$ is rectifiably connected then $\rho$ induces a distance function which
is defined by

$d_{\rho}(x,y)=\inf\int_{\gamma}\rho\,ds,$

where infimum is taken over all rectifiable curves joining $x$ and $y$ in $X$.
We note that, in general, $d_{\rho}$ need not to be a metric however
$d_{\rho}$ is a metric if $\rho$ is positive and continuous. If $d_{\rho}$ is
a metric, we say $(X,d_{\rho})$ is a conformal deformation of $(X,d)$ by the
conformal factor $\rho$.

A curve $\gamma:[a,b]\to X$ is said to be geodesic if for all
$t,t^{\prime}\in[a,b]$,

$d(\gamma(t),\gamma(t^{\prime}))=|t-t^{\prime}|.$

A metric space $X$ is said to be a geodesic metric space if every pair of
points $x,y\in X$ can be joined by a geodesic, and is said to be proper if
every closed ball is compact. A metric space $(X,d)$ is said to be intrinsic
or path metric space/ or length metric space if for all $x,y\in X$, we have

$d(x,y)=\lambda_{X}(x,y).$

######  Definition 2.1.

A metric space $(X,d)$ is said to be minimally nice if it is locally compact,
rectifiably connected and non-complete.

###  2.2. Gromov hyperbolicity

Gromov hyperbolicity was introduced by Gromov in the 1980s. It is often
assumed that the under lying space is geodesic metric space and usually it is
a proper metric space. As we will see later that infinite dimensional Banach
space with quasihyperbolic metric need not be geodesic space, therefore we
adapt the definition of Gromov hyperbolicity which works well with the non
geodesic spaces. For this, we refer to the work of Väisälä [17] where the
concept of Gromov product has been used.

######  Definition 2.2.

Let $(X,d)$ be a metric space and let $p\in X$. The Gromov Product $(x|y)_{p}$
of $x,y\in X$ with respect to $p$ is defined by

$(x|y)_{p}=\frac{1}{2}\Big{(}d(x,p)+d(y,p)-d(x,y)\Big{)}.$

Therefore, $(x|y)_{p}$ measures to what extent the triangle inequality for the
triangle $\Delta xyp$ is far from being an equality.

######  Definition 2.3.

Let $\delta\geq 0$. A metric space $(X,d)$ is said to be $\delta$-hyperbolic
if for every $x,y,z,p\in X$, the following inequality holds

$(x|z)_{p}\geq min\Bigl{\\{}(x|y)_{p},(y|z)_{p}\Bigr{\\}}-\delta.$

A metric space $(X,d)$ is said to be Gromov hyperbolic if it is
$\delta$-hyperbolic for some $\delta\geq 0$.

###  2.3. Examples

Clearly, $\mathbb{R}$ is a 0-hyperbolic. Any tree $T$ is $0$-hyperbolic. The
unit disk with hyperbolic metric is Gromov hyperbolic with $\delta=log\,3$.
The complex plane is not Gromov hyperbolic.

We shall mention an important result here due to Väisälä [16, 3.7], which is
known as stability theorem. Infact, stability theorem says that in an
intrinsic hyperbolic space any two quasi-isometric paths with same initial and
terminal points $x$, $y$ runs close to each other even if $|x-y|$ is large. To
mention this theorem we first need to mention the concept of quasi-isometry
and Hausdorff distance.

######  Definition 2.4.

Let $X$ and $Y$ be two non-empty subsets of a metric space $(M,d)$. We define
their Hausdorff distance $d_{\mathcal{H}}(X,Y)$ by

$d_{\mathcal{H}}(X,Y)=\max\left\\{\sup_{x\in X}d(x,Y),\,\sup_{y\in
Y}d(X,y)\right\\},$

where $d(a,B)=\inf_{b\in B}d(a,b)$ denotes the distance from a point $a\in M$
to a subset $B\subset M$.

######  Definition 2.5.

Let $\lambda\geq 1$ and $\mu\geq 0$ and $(X,d)$ and $(Y,d^{\prime})$ be metric
spaces. We say that a map $f:X\to Y$ is a $(\lambda,\mu)$-quasi-isometry if

$\lambda^{-1}d(x,y)-\mu\leq d^{\prime}(f(x),f(y))\leq\lambda d(x,y)+\mu$

for all $x,y\in X$. In the case where $f:I\to Y$ is a map of an real interval
$I$, we say that such a map $(\lambda,\mu)$-quasi-isometric path.

######  Remark 2.1.

An arc $\gamma$ joining $x$ and $y$ in a metric space $X$ is said to be
$\lambda$-quasigeodesic, $\lambda\geq 1$, if

$l(\gamma[u,v])\leq\lambda d(u,v)$

for all $u,v\in\gamma$. In such a case the arc length parametrization
$\gamma_{s}:[0,l(\gamma)]\to\gamma$ satisfies the inequality

$\lambda^{-1}|t-t^{\prime}|\leq
d(\gamma_{s}(t),\gamma_{s}(t^{\prime}))\leq\lambda|t-t^{\prime}|.$

Thus $\gamma$ is $(\lambda,0)$-quasi-isometry.

The following theorem (see [16, 3.7]) is vital to prove our main result.

Theorem A. Suppose that $X$ is an intrinsic $\delta$-hyperbolic space and that
$\gamma$ and $\alpha$ are $(\lambda,\mu)$-quasi-isometric path with same
initial and terminal points. Then there exists a constant $M$ such that

$d_{\mathcal{H}}(\gamma,\alpha)\leq M,$

where $M$ depends only on $\delta,\lambda,\mu$ and $d_{\mathcal{H}}$ denote
the Hausdorff distance between $\gamma$ and $\alpha$.

Next we mention an important result due to Väisälä (see [17, Theorem 3.18])
which says that quasi-isometries preserves hyperbolicity.

######  Theorem 2.1.

Let $X$ and $Y$ be intrinsic spaces and let $f:X\to Y$ be a
$(\lambda,\mu)$-quasi-isometry. If $Y$ is $\delta$-hyperbolic, then $X$ is
$\delta^{\prime}$-hyperbolic with
$\delta^{\prime}=\delta^{\prime}(\delta,\lambda,\mu)$.

###  2.4. Quasihyperbolic metric

In view of our frame work, we consider the quasihyperbolic metric in the
setting of Banach spaces.

######  Definition 2.6.

Let $E$ be a real Banach space of dimension at least $2$ and let $D\subsetneq
E$ be a domain (open, connected nonempty set). Let the metric induced by norm
on D is $|.|$. Define $k:D\to(0,\infty)$ by

$k(z)=\frac{1}{\Delta(z)}=\frac{1}{d(z,\partial D)}.$

Define the quasihyperbolic length of a rectifiable curve $\gamma$ in $D$ by

$l_{k}(\gamma)=\int_{\gamma}\rho(z)\,ds=\int_{\gamma}\frac{ds}{d(z,\partial
D)}.$

Let $k_{D}$ denote the quasihyperbolic metric in $D$ which is defined by

(2.1) $k_{D}(x,y)=\inf_{\gamma}l_{k}(\gamma)$

where the infimum is taken over all rectifiable curves $\gamma$ in $D$ joining
$x$ to $y$.

Observe that $(D,k_{D})$ is always an intrinsic space. For further geometric
properties of hyperbolic metric, we refer to [4] and [22]. Quasihyperbolic
metric satisfies the following properties.

  * (i)

$k_{D}=h_{D}$ when $D$ is a half space in $\mathbb{R}^{n}$.

  * (ii)

$k_{D}\leq h_{D}\leq 2k_{D}$ when $D$ is a ball in $\mathbb{R}^{n}$.

######  Remark 2.2.

[2, Proposition 2.8] If $(X,d)$ is a locally compact, incomplete and
rectifiably connected metric space then $(X,k_{X})$ is a complete, proper and
geodesic metric space.

Finally, we recall some estimates for the quasihyperbolic metric which have
been first introduced by Gehring and Palka [3, 2.1] in $\mathbb{R}^{n}$.
Later, Väisälä [12, Lemma 2.2] have proved these inequalities in the case of
Banach spaces. Let $D\subsetneq E$ be a domain and $x,y\in D$ and let $\gamma$
be a rectifiable curve joining $x$ and $y$. Then we have the following:

(2.2) $k(x,y)\geq
log\left(1+\frac{|x-y|}{\min\\{\Delta(x),\Delta(y)\\}}\right)\geq
log\frac{\Delta(y)}{\Delta(x)}$

and

(2.3) $l_{k}(\gamma)\geq
log\left(1+\frac{l(\gamma)}{\min\\{\Delta(x),\Delta(y)\\}}\right).$

We are going to use frequently (2.2) and (2.3) to prove our main results.

###  2.5. Quasi-hyperbolic geodesic

A rectifiable curve $\gamma$ from $a$ to $b$ in $D$ is said to be
quasihyperbolic geodesic if $k_{D}(a,b)=l_{k}(\gamma)$. Obviously each subarc
of a quasi-hyperbolic geodesic is again a geodesic. Observe that quasi-
hyperbolic geodesic is curve $\gamma$ for which the infimum in (2.1) is
attained.

At this juncture, the natural question is does quasi-hyperbolic geodesic
always exists? A result of Gehring and Osgood [4] shows that for a proper
subdomain $D$ in $\mathbb{R}^{n}$, the answer to this question is affirmative.
For important results on quasihyperbolic geodesics in domain in
$\mathbb{R}^{n}$, we refer to G. Martin [9]. Moreover, Martin [9] has proved
that quasihyperbolic geodesics in $\mathbb{R}^{n}$ are $C^{1}$ smooth. It is
well known that a quasihyperbolic geodesic between any two points exists if
dim ($E$) is finite (see [4, Lemma 1]). Note that the quasihyperbolic geodesic
may not exists in an infinite-dimensional Banach spaces (see [12, Example 2.9]
and [14, 3.5]). However, Väisälä [16, Theorem 2.1] has proved the existence of
quasihyperbolic geodesic when $E$ is reflexive Banach space and $D$ is a
convex domain. Furthermore, in 2011, Martio and Väisälä [10, Theorem 2.11]
proved the existence as well as uniqueness of quasihyperbolic geodesic when
$E$ is uniformly convex Banach space and $D$ is a convex domain.

###  2.6. Uniform Domains

######  Definition 2.7.

Let $c\geq 1$. A domain $D$ in $E$ is said to be $c$-uniform in the norm
metric if for each pair of points $x,y\in D$, there exists a rectifiable arc
$\gamma$ in $D$ joining $x$ to $y$ such that

  * (i)

$\min\\{l(\gamma[x,z]),l(\gamma[z,y])\\}\leq c\,d(z,\partial D)$, for all
$z\in\gamma$, and

  * (ii)

$l(\gamma)\leq c|x-y|$,

where $l(\gamma)$ denotes the arc length of $\gamma$ in $(D,|.\,|)$, where
$|.\,|$ is metric induced by norm, and $\gamma[x,z]$ denotes the part of
$\gamma$ between $x$ and $z$. Also we say that curve $\gamma$ is a double
$c$-cone arc.

###  2.7. Inner uniform domains

######  Definition 2.8.

Let $c\geq 1$. A domain $D$ is said to be $c$-inner uniform in the norm metric
if for each pair of points $x,y\in D$, there exists a rectifiable arc $\gamma$
in $D$ joining $x$ to $y$ such that

  * (i)

$\min\\{l(\gamma[x,z]),l(\gamma[z,y])\\}\leq c\,d(z,\partial D)$, for all
$z\in\gamma$,

  * (ii)

$l(\gamma)\leq c\lambda_{D}(x,y)$.

###  2.8. John Domain

######  Definition 2.9.

Let $c\geq 1$. A domain $D$ is said to be $c$-John in the norm metric if for
each pair of points $x,y\in D$ there exists a rectifiable curve $\gamma$ in
$D$ joining $x$ to $y$ such that for all $z\in\gamma$

(2.4) $\min\\{l(\gamma[x,z]),l(\gamma[z,y])\\}\leq c\,d(z,\partial D)$

and we say that such a curve $\gamma$ is a $c$-cone arc.

######  Remark 2.3.

We mention some of the remarks with regards to the connection between the
aforesaid domains.

  * (i)

$c$-uniform implies inner $c$-uniform implies $c$-John

  * (ii)

$\mathbb{R}^{2}\setminus\\{(x,0):x\geq 0\\}$ is inner uniform but not unifrom.

  * (iii)

$\mathbb{R}^{2}\setminus\\{(n,0):n\in\mathbb{N}\\}$ is a John domain but not
inner uniform.

###  2.9. Gromov hyperbolicity of domains

######  Definition 2.10.

Let $\delta\geq 0$. A domain $D$ in a Banach space $E$ is said to be
$\delta$-hyperbolic if $(D,k_{D})$ is $\delta$-hyperbolic.

######  Definition 2.11.

A domain $D$ in a Banach space $E$ is said to be Gromov hyperbolic if
$(D,k_{D})$ is $\delta$-hyperbolic for some $\delta\geq 0$.

All $c$\- uniform and inner $c$\- uniform domains are Gromov
$\delta=\delta(c)$ hyperbolic.
$\mathbb{R}^{2}\setminus\\{(n,0):n\in\mathbb{N}\\}$ is a John domain which is
not hyperbolic. The broken tube (see [12, 2.12] and for a detailed treatment
we refer to [15]) in an infinite dimensional seperable Hilbert space is a
classical example of a domain which is hyperbolic but not John and hence not
uniform.

###  2.10. Quasihyperbolic geodesic in Uniform Domains in $\mathbb{R}^{n}$

Gehring and Osgood [4] have proved that each quasihyperbolic geodesic in a
$c$-uniform domain $D\in\mathbb{R}^{n}$ is a double $b$-cone arc, where the
constant $b$ depends only on $c$.

###  2.11. Quasihyperbolic Geodesic in John Domain in $\mathbb{R}^{n}$

Since John domains can be thought of one sided uniform domains, it is natural
to ask whether the result also holds for John domains or not? In fact, this
problem has been proposed by Gehring et. al. [5] in the following form.

######  Problem 2.5.

Suppose $D\subset\mathbb{R}^{n}$ is a $c-$ John domain and that $\gamma$ is a
quasihyperbolic geodesic in $D$. Is $\gamma$ a $b$-cone arc for some $b=b(c)$?

In 1989, Gehring et. al. [5] proved the following result.

Theorem B. [5, Theorem 4.1] If $D\subset\mathbb{R}^{2}$ is a simply connected
$c$-John domain then every quasihyperbolic or hyperbolic geodesic in $D$ is a
$b$-cone arc, where $b$ only depends on $c$.

Infact, Gehring et. al. [5] have constructed several examples which shows that
a quasihyperbolic geodesic in a $c$\- John domain need not to be a $b$-cone
arc with $b=b(c)$ unless $n=2$ and $D$ is simply connected. Thus, this suggest
us that we need some extra restriction on $c$-John domain so that the Problem
2.5 holds good. This raises the following natural question.

######  Problem 2.6.

Determine necessary and/or sufficient conditions for quasihyperbolic geodesics
in a John space to be cone arcs.

In $1989$, Heinonen [6, Question 2] posed the following problem.

######  Problem 2.7.

Suppose $D\subset\mathbb{R}^{2}$ is a $c$-John domain which is
quasiconformally equivalent to the unit ball $\mathbb{B}^{n}$ in
$\mathbb{R}^{n}$ and $\gamma$ is a quasihyperbolic geodesic in $D$. Is
$\gamma$ a $b=b(c)$-cone arc for some constant $b$?

In $2001$, Bonk, Heinonen and Koskela [2, Proposition 7.12] gave an
affirmative answer to Probelm 2.7. In particular, they have proved the
following result.

Theorem C. [2, Proposition 7.12] If $D\subset\mathbb{R}^{n}$ is a bounded
$a$-John domain which is homeomorphic to inner $c$-uniform domain via
$K$-quasiconformal map, then each quasihyperbolic geodesic in $D$ is a
$b$-cone arc with $b=b(a,c,K,n)$ Since every ball is inner uniform, Theorem C
gives an affirmative answer to Problem 2.7.

##  3\. Metric Space setting

In 2022, Zhou, Li and Rasila [21] consider the Problem 2.6 further and proved
the following.

######  Theorem 3.1.

[21, Theorem 1.1] Let $(D,d)$ be a locally compact, rectifiablly connected and
non complete metric space, and $k$ be the quasihyperbolic metric of $(D,d)$.
If $(D,d)$ is $a$-John and if $(D,k)$ is $K$-roughly starlike and Gromov
$\delta$-hyperbolic. Then every quasihyperbolic geodesic in $D$ is a $b$-cone
arc, where $b$ depends only on $a,K$ and $\delta$.

######  Remark 3.1.

Observe that any proper subdomain $D$ in $\mathbb{R}^{n}$ is locally compact,
rectifiablly connected and noncomplete with respect to the usual metric.
Theorefore, Theorem 3.1 gives an improvement of Theorem C.

The next result of Zhou and Ponnusamy [19, 20] shows that the condition of
rough starlikeness is not required in Theorem 3.1. Thus the following results
of Zhou and Ponnusamy [20] give an improvement of Theorem 3.1.

######  Theorem 3.2.

[20, Theorem 1.4] Let $(D,d)$ be a locally compact, rectifiablly connected and
non complete metric space, and $k$ be the quasihyperbolic metric of $(D,d)$.
If $(D,d)$ is $a$-John and if $D$ is Gromov hyperbolic i.e., $(D,k_{D})$ is
$\delta$-hyperbolic. Then every quasihyperbolic geodesic in $D$ is a $b$-cone
arc with $b=b(a,\delta)$.

##  4\. Banach space setting

The main aim of this paper is to study Problem 2.6 by replacing
quasihyperbolic geodesic with neargeodesics in Banach space settings precisely
in infinite dimansional Banach spaces. Note that the quasihyperbolic geodesic
may not exists in the infinite-dimensional Banach spaces (see [12, Example
2.9] and [14, Remark 3.5]). However, these examples of domains are in Hilbert
spaces such that the complement of these domains are uncountable. For example
of a domain that is not geodesic with respect to quasihyperbplic metric such
that the complement of domain is countable we refer to [7, Example 4.1]. In
order to overcome this shortage, Väisälä [13] introduced the concept of
neargeodesic. We recall the definition of neargeodesic.

######  Definition 4.1.

Let $D\subsetneq E$ be a domain and $c\geq 1$. An arc $\gamma$ in $D$ is said
to be $c$-neargeodesic if for all $x,y\in\gamma$, we have

$l_{k}(\gamma[x,y])\leq ck_{D}(x,y).$

Thus an arc $\gamma$ is a quasihyperbolic geodesic if, and only if, it is a
$1$-neargeodesic. In [13] Väisälä proved the following existence theorem for
neargeodesics.

######  Theorem 4.1.

[13, Theorem 3.3] Let $c>1$. Then for every pair of points $z_{1},z_{2}\in D$
there exists a $c$-neargeodesic joining $z_{1}$ and $z_{2}$.

Observe that the constant $b$ in Theorem B depends on $n$, so it is natural to
ask the following question.

######  Problem 4.1.

Does there exists a dimension free anologue of Theorem B. In other words does
it holds in the setting of Banach spaces or not if we replace quasihyperbolic
geodesic with neargeodesics?

Li [8, Theorem 1] gave an affirmative answer to Problem 4.1 and proved the
following

######  Theorem 4.2.

If $D\subset E$ is an $a$-John domain which is homeomorphic to $c$-inner
uniform domain via an $(M,C)-CQH$. Let $z_{1},z_{2}\in D$ and $\gamma$ is a
$c_{0}$-neargeodesic joining $z_{1}$ and $z_{2}$ in $D$. Then $\gamma$ is a
$b$-cone arc with $b=b(a,c,c_{0},M,C)$.

##  5\. Main Result

In this section we state our main result of this paper. We observe that in
Theorem 3.1 and 3.2, authors have considered that metric space $(X,D)$ which
is minimally nice. Once you have such metric space, Remark 2.2 immediately
tells us that $(X,k_{X})$ is a geodesic metric space i.e., for any papir of
points quasihyperbolic geodesic exists.

Note that if we consider a domain $D$ in an infinite dimensional Banach space
then $D$ with the metric induced by norm satisfies all the properties of
minimally nice space except locally compact. This motivates us to ask does
Theorem 3.1 and 3.2 type results still holds if we consider John domain $D$ in
infinite dimensional Banach space? In such a case we know that $(D,k_{D})$
need not be geodesic and hence the natural choice is to consider the
neargeodesics at the place of quasihyperbolic geodesics.

######  Theorem 5.1.

Let $E$ be an infinite dimensional Banach space and $D\subsetneq E$ is a
$c$-John domain and suppose that $D$ is Gromov hyperbolic i.e. $(D,k_{D})$ is
$\delta$-hyperbolic for some $\delta\geq 0$. Then every $c_{0}$-neargeodesic
in $D$ is a $b$-cone arc, where $b$ depends only on $c$, $c_{0}$ and $\delta$.

######  Remark 5.1.

  * (1)

Observe that Theorem 5.1 is different from Theorem 3.1 and Theorem 3.2. Note
that in Theorem 5.1 $D$ is a domain in infinite dimansional Banach spce
therefore $D$ is not locally compact with metric induced by norm. Hence Remark
2.2 is also no more valid and we can’t use the results of Gromov hyperbolicity
for geodesic metric space. Further, in this paper, we are considering the
neargeodesics rather than quasihyperbolic geodesic.

  * (2)

To prove our main result we need the results on Gromov hyperbolicity for
general metric spaces that are need not to be geodesic. Thanks to the work of
Väisälä [17] where he has established the theory of Gromov hyperbolicity for
domains in Banach spaces which obviously need not to be geodesic (in
quasihyperbolic metric) when the Banach space is infinite dimensional.

  * (2)

In view of Theorem 2.1 and the following two facts:

    * (i)

all inner $c$-uniform domains are Gromov hyperbolic with respect to
quasihyperbolic metric (see [16, Remark 2.16])

    * (ii)

every $(M,C)-CQH$ map is $(M,C)$-quasi-isometry in quasihyperbolic metric

we conclude that Theorem 5.1 is an improvement of Theorem 4.2 of Li [8,
Theorem 1].

##  6\. Proof of Main Result

We shall devide the proof of Theorem 5.1 into two theorems. Theorem 6.1 below
gives a sufficient condition for a neargeodesic to be a cone arc and Theorem
6.2 below says that in Gromov hyperbolic John domains, every neargeodesic
satisfies this sufficient condition. Therefore, combining these two theorem we
obtain the proof of our main result.

######  Theorem 6.1.

Let $\gamma$ be a $c_{0}$\- neargeodesic joining $x,y$ such that the following
holds: If $z_{1},z_{2}\in\gamma$ such that each $w\in\gamma[z_{1},z_{2}]$
satisfies $\Delta(w)\leq 2\min\\{\Delta(z_{1}),\Delta(z_{2})\\}$, then there
exists a constant $A\geq 1$ such that

$k_{D}(z_{1},z_{2})\leq A.$

Then $\gamma$ is a $C$-cone arc, where $C$ depends only on $A$ and $c_{o}$.

Proof. Suppose $\gamma$ is a $c_{0}$\- neargeodesic joining $x,y$ and if
$z_{1}$ and $z_{2}$ are points on $\gamma$ satisfying for each
$w\in\gamma[z_{1},z_{2}]$, $\Delta(w)\leq
2\min\\{\Delta(z_{1}),\Delta(z_{2})\\}$, then there exists a constant $A\geq
1$ such that

$k_{D}(z_{1},z_{2})\leq A.$

Since $\Delta$ is continous and $\gamma$ is compact, choose a point
$z_{0}\in\gamma$ with

$\Delta(z_{0})=\max_{z\in\gamma}\Delta(z)$

Then there exists unique non-negative integer $n$ such that

$2^{n}\Delta(x)\leq\Delta(z_{0})\leq 2^{n+1}\Delta(x).$

Let $x=x_{0}$. For each $i=1,2,\ldots,n$, let $x_{i}$ be the first point on
$\gamma[x,z_{0}]$ such that

$\Delta(x_{i})=2^{i}\Delta(x_{0}).$

Similarly, there exists $m\geq 0$ such that

$2^{m}\Delta(y)\leq\Delta(z_{0})\leq 2^{m+1}\Delta(y).$

Let $y=y_{0}$. For each $j=1,2,\ldots,m$, let $y_{j}$ be the first point on
$\gamma[z_{0},y]$ such that

$\Delta(y_{j})=2^{j}\Delta(y_{0}).$

In this manner we have devided the arc $\gamma$ into $(n+m+1)$ subarcs

$\gamma[x_{0},x_{1}],\ldots,\gamma[x_{n-1},x_{n}],\gamma[x_{n},y_{m}],\gamma[y_{m},y_{m+1}],\ldots,\gamma[y_{1},y_{0}].$

It is easy to observe that all these subarcs are also $c_{0}$-neargeodesic
between their respective end points. Let $u$ and $v$ be any two adjacent
points along $\gamma$. That is $u$ and $v$ are $x_{i},x_{i+1}$ or
$x_{n},y_{m}$ or $y_{j+1},y_{j}$, $0\leq i\leq n-1$ and $0\leq j\leq m-1$. By
the construction, it is clear that for each $z\in\gamma[u,v]$, we have

$\Delta(z)=2\min\\{\Delta(u),\Delta(v)\\}.$

By assumption, there exists a constant $A\geq 1$ such that

$k_{D}(u,v)\leq A.$

To show that $\gamma$ is a $C$-cone arc for some $C$, we need to show that

(6.1) $\min\\{l(\gamma[x,w],l(\gamma[w,y]))\\}\leq C\Delta(w)\,\,\,\,\,\mbox{
for all }w\in\gamma.$

In order to prove (6.1), it is sufficient to prove either $l(\gamma[x,w]\leq
C\Delta(w)$ or $l(\gamma[w,y])\leq C\Delta(w)$. For this we need to consider
three cases depending upon the location of $w$ on $\gamma$.

  * Case 1.

Suppose $w\in\gamma[x_{i},x_{i+1}]$ for some $0\leq i\leq n-1$. Then in view
of (2.2), we have

$log\frac{\Delta(x_{i})}{\Delta(w)}\leq k_{D}(w,x_{i})=k_{D}(x_{i},w)\leq
k_{D}(x_{i},x_{i+1})\leq A.$

Therefore, we have

(6.2) $\Delta(x_{i})\leq e^{A}\Delta(w).$

Moreover, we have

(6.3) $l(\gamma[x,w])\leq\sum_{t=0}^{i}l(\gamma[x_{s},x_{s+1}])$

In view of (2.3), we have

$\displaystyle
log\left(1+\frac{l(\gamma[x_{s},x_{s+1}])}{\min\\{\Delta(x_{s}),\Delta(x_{s+1})\\}}\right)$
$\displaystyle\leq$ $\displaystyle l_{k}(\gamma[x_{s},x_{s+1}])$
$\displaystyle\leq$ $\displaystyle c_{0}\,k_{D}(x_{s},x_{s+1})$
$\displaystyle\leq$ $\displaystyle c_{0}\,A$

Therefore, this gives us that

$log\frac{l(\gamma[x_{s},x_{s+1}])}{\Delta(x_{s})}\leq
log\left(1+\frac{l(\gamma[x_{s},x_{s+1}])}{\Delta(x_{s})}\right)\leq c_{0}\,A$

and hence

(6.4) $l(\gamma[x_{s},x_{s+1}])\leq e^{c_{0}\,A}\Delta(x_{s})$

Using (6.4) in (6.3), we obtain

$\displaystyle l(\gamma[x,w])$ $\displaystyle\leq$
$\displaystyle\sum_{t=0}^{i}e^{c_{0}\,A}\Delta(x_{t})$ $\displaystyle\leq$
$\displaystyle 2\,e^{c_{0}\,A}\,\Delta(x_{i})$ $\displaystyle\leq$
$\displaystyle 2\,e^{A(c_{0}+1)}\,\Delta(w)$

and hence we have

$l(\gamma[x,w])\leq C\,\Delta(w),\,\,\,\,\,\mbox{ where }C=2e^{A(c_{0}+1)}.$

Theorefore, we have

(6.6) $\min\\{l(\gamma[x,w],l(\gamma[w,y]))\\}\leq C\Delta(w)\,\,\,\,\,\mbox{
for all }w\in\gamma[x_{i},x_{i+1}].$

  * Case 2.

Suppose $w\in\gamma[x_{n},y_{m}]$. Then it is easy to see that

$log\frac{\Delta(x_{n})}{\Delta(w)}\leq k_{D}(w,x_{n})=k_{D}(x_{n},w)\leq
k_{D}(x_{n},y_{m})\leq A,$

which implies

$\Delta(x_{n})\leq e^{A}\,\Delta(w).$

An easy computation shows that

$\displaystyle l(\gamma[x,w])$ $\displaystyle\leq$
$\displaystyle\sum_{i=0}^{n-1}l(\gamma[x_{i},x_{i+1}])+l(\gamma[x_{n},y_{m}])$
$\displaystyle\leq$
$\displaystyle\sum_{i=0}^{m-1}e^{c_{0}\,A}\Delta(x_{i})+e^{c_{0}\,A}\Delta(x_{n})$
$\displaystyle\leq$ $\displaystyle 3e^{c_{0}\,A}\Delta(x_{n})$
$\displaystyle\leq$ $\displaystyle 3\,e^{A(c_{0}+1)}\,\Delta(w).$

That is,

$l(\gamma[x,w])\leq C\,\Delta(w),\,\,\,\,\,\mbox{ where }C=3\,e^{A(c_{0}+1)}$

and hence we have

(6.7) $\min\\{l(\gamma[x,w],l(\gamma[w,y]))\\}\leq C\Delta(w)\,\,\,\,\,\mbox{
for all }w\in\gamma[x_{n},y_{m}].$

  * Case 3.

Suppose $w\in\gamma[y_{j},y_{j+1}]$ for some $0\leq j\leq m-1$. Using the same
argument as in earlier, we obtain

$l(\gamma[w,y])\leq C\,\Delta(w),\,\,\,\,\,\mbox{ where }C=2\,e^{A(c_{0}+1)}$

and hence

(6.8) $\min\\{l(\gamma[x,w],l(\gamma[w,y]))\\}\leq C\Delta(w)\,\,\,\,\,\mbox{
for all }w\in\gamma[y_{j},y_{j+1}].$

Thus, combining (6.6), (6.7) and (6.8), we conclude that $\gamma$ is a $C$\-
cone arc, where $C=3\,e^{A(c_{0}+1)}$. ∎

######  Theorem 6.2.

Let $D$ be a $c$\- John domain in an infinite dimensional Banach space $E$
such that $(D,k_{D})$ is Gromov hyperbolic space. Fix $x,y\in D$ and let
$\gamma$ be a $c_{0}$ neargeodesic joining $x,y\in D$. Then there exists a
constant $A\geq 1$ depending only $c$, $c_{0}$ and $\delta$ such that the
following holds: If there are $u,v\in\gamma$ such that each
$w\in\gamma{[u,v]}$ satisfies $\Delta(w)\leq 2\min\\{\Delta(u),\Delta(v)\\}$,
then

$k_{D}(u,v)\leq A.$

Proof. Let $D$ be a $c$-John domain in a Banach space $E$ such that
$(D,k_{D})$ is Gromov $\delta$-hyperbolic. By definition $(D,k_{D})$ is an
intrinsic space. Fix $x,y\in D$ and $c_{0}>1$. In view of Theorem 4.1, let
$\gamma$ be a $c_{0}$-neargeodesic joining $x$ and $y$. In view of Remark 2.1,
it is clear that a $c$-neargeodesic $\gamma$ is $(c,0)$\- quasi-isometry.
Since $D$ is a $c$-John, there is a $c$-cone arc $\alpha$ joining $x$ and $y$.
To use Theorem A, we first need to show that every $c$-cone arc $\alpha$ is
$(\lambda,\mu)$-quasi-isometry for some $\lambda$ and $\mu$ depending only on
$c$. Which we will prove in the following lemma

######  Lemma 6.9.

Let $c\geq 1$ and $\alpha$ be a $c$-cone arc. Then its arc length
parametrization $\alpha_{s}$ satisfies the following inequality

$\frac{1}{3c}|t-t^{\prime}|-3c\,log(3c)\leq
k_{D}(\alpha_{s}(t),\alpha_{s}(t^{\prime}))\leq 3c|t-t^{\prime}|+3c\,log(3c)$

and hence $\alpha$ is $(3c,3c\,log(3c))$-quasi-isometry.

Proof. Let $\alpha$ be a $c$-cone arc joining $x$ and $y$ in $D$ and
$z_{1},z_{2}\in\alpha$. Without loss of generality we can assume that
$z_{1}\in\alpha[x,z_{2}]$. Since $\alpha$ is a $c$-cone arc, for each
$w\in\alpha[z_{1},z_{2}]$, we have

$l(\alpha[w,z_{2}])\leq\min\\{l(\alpha[x,w]),l(\alpha[w,y])\\}\leq c\Delta(w)$

and hence

(6.10) $l(\alpha[w,z_{2}])\leq c\Delta(w).$

Next, our aim is to show that $\Delta(z_{2})\leq 2c\Delta(w)$. To prove this,
we consider two cases:

  * Case (i)

If $d(w,z_{2})<\Delta(y)/2$, then

$\Delta(w)\geq\Delta(z_{2})-d(w,z_{2})\geq\Delta(z_{2})/2$

and hence, $\Delta(z_{2})\leq 2\Delta(w)$.

  * Case (ii)

Since $\alpha$ is a $c$-cone arc we obtain

$c\Delta(w)\geq l(\alpha[z_{2},w])\geq d(z_{2},w)\geq\Delta(z_{2})/2$

and hence

(6.11) $\Delta(z_{2})\leq 2c\Delta(w)$

By adding (6.10) and (6.11), we obtain

$l(\alpha[w,z_{2}])+\Delta(z_{2})\leq 3c\Delta(w)$

which gives,

(6.12) $\frac{1}{\Delta(w)}\leq\frac{3c}{l(\alpha[w,z_{2}])+\Delta(z_{2})}$

A simple computation shows that

$\displaystyle k_{D}(z_{1},z_{2})$ $\displaystyle\leq$ $\displaystyle
l_{k}(\alpha[z_{1},z_{2}])=\int_{\alpha[z_{1},z_{2}]}\frac{ds}{\Delta(w)}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{l(\alpha[z_{1},z_{2}])}\frac{3c}{t+\Delta(z_{2})}\,dt.$
$\displaystyle l_{k}(\alpha[z_{1},z_{2}])$ $\displaystyle\leq$ $\displaystyle
3c(log(l(\alpha[z_{1},z_{2}])+\Delta(z_{2}))-log(\Delta(z_{2}))$
$\displaystyle\leq$ $\displaystyle
3c\,log\left(1+\frac{l(\alpha[z_{1},z_{2}])}{\Delta(z_{2})}\right)$
$\displaystyle\leq$ $\displaystyle
3c\,log\left(1+\frac{c\Delta(z_{1})}{\Delta(z_{2})}\right)$
$\displaystyle\leq$ $\displaystyle
3c\left(log\left(\frac{\Delta(z_{1})}{\Delta(z_{2})}\right)+log3c\right).$
(6.16) $\displaystyle l_{k}(\alpha[z_{1},z_{2}])$ $\displaystyle\leq$
$\displaystyle 3ck_{D}(z_{1},z_{2})+3c\,log\,3c$

Theorefore, from (6.16), it is clear that the arc length parametrization of
$\alpha$ satisfies the inequality

$\frac{1}{3c}|t-t^{\prime}|-3c\,log(3c)\leq
k_{D}(\alpha_{s}(t),\alpha_{s}(t^{\prime}))\leq 3c|t-t^{\prime}|+3c\,log(3c)$

and hence $\alpha$ is $(3c,3c\,log(3c))$-quasi-isometry. This completes the
proof of Lemma 6.9. ∎

Since $\gamma$ is $(c_{0},0)$-quasi-isometry and $\alpha$ is
$(3c,3c\,log(3c))$-quasi-isometry in view of Theorem A, we have that
quasihyperbolic Hausdorff distance satisfies the following inequality

$k_{D\mathcal{H}}(\gamma,\alpha)\leq R=R(c,c_{0},\delta).$

Let $u,v\in\gamma$ such that each $w\in\gamma{[u,v]}$ satisfies $\Delta(w)\leq
2\min\\{\Delta(u),\Delta(v)\\}$. Our aim is to show that there exists a
constant $A\geq 1$ such that $k_{D}(u,v)\leq A$. For this we need to estimate
$k_{D}(u,v)$. Since $k_{D\mathcal{H}}(\gamma,\alpha)\leq R$ i.e.,

$\max\left\\{\sup_{x\in\gamma}k_{D}(x,\alpha),\sup_{y\in\alpha}k_{D}(\gamma,y)\right\\}\leq
R$

Theorefore,

$\sup_{x\in\gamma}k_{D}(x,\alpha)\leq R$

In particular,

$M_{1}=k_{D}(u,\alpha)=\inf_{b\in\alpha}k_{D}(u,b)\leq R$

and

$M_{2}=k_{D}(v,\alpha)=\inf_{b\in\alpha}k_{D}(v,b)\leq R$

There exists a sequence $b_{i}$ and $b_{j}$ such that $k_{D}(u,b_{i})$
converges to $M_{1}$ and $k_{D}(u,b_{j})$ converges to $M_{2}$. Since $\alpha$
is compact, $b_{i}$ and $b_{j}$ have convergent subsequence which converges to
say $Z_{u}$ and $Z_{v}$ respectively such that $k_{D}(u,Z_{u})\leq R$ and
$k_{D}(v,Z_{v})\leq R$. Therefore,

$log\frac{\Delta(u)}{\Delta(Z_{u})}\leq k_{D}(u,Z_{u})\leq R$

implies

(6.17) $\Delta(Z_{u})\leq e^{R}\Delta(u)$

and

$log\frac{\Delta(v)}{\Delta(Z_{v})}\leq k_{D}(v,Z_{v})\leq R$

implies

(6.18) $e^{-R}\Delta(v)\leq\Delta(Z_{v}).$

Since each $w\in\gamma{[u,v]}$ satisfies

(6.19) $\Delta(w)\leq 2\min\\{\Delta(u),\Delta(v)\\}$

by taking $w=v$ in (6.19), we obtain

(6.20) $\Delta(v)=2\Delta(u)$

and by taking $w=u$ in (6.19), we obtain

(6.21) $\Delta(u)\leq 2\Delta(v).$

Using (6.21) in (6.17), we obtain

(6.22) $\Delta(Z_{u})\leq e^{R}\Delta(u)\leq 2e^{R}\Delta(v).$

Using (6.18) in (6.22), we obtain

$\Delta(Z_{u})\leq 2e^{2R}\Delta(Z_{v})$

which implies that

(6.23) $\frac{\Delta(Z_{u})}{\Delta(Z_{v})}\leq 2e^{2R}.$

In view of (6), we have

(6.24) $k_{D}(Z_{u},Z_{v})\leq
3c\,log\left(1+\frac{c\Delta(Z_{u})}{\Delta(Z_{v})}\right).$

Using (6.23) in (6.24), we obtain

(6.25) $k_{D}(Z_{u},Z_{v})\leq 3c\,log\left(1+2ce^{2R}\right).$

Therefore, we have

$\displaystyle k_{D}(u,v)$ $\displaystyle\leq$ $\displaystyle
k_{D}(u,Z_{u})+k_{D}(Z_{u},Z_{v})+k_{D}(Z_{v},v)$ $\displaystyle\leq$
$\displaystyle 2R+k_{D}(Z_{u},Z_{v})$ $\displaystyle\leq$ $\displaystyle
2R+3c\,log(1+2ce^{2R})\,\,\,\,\,\,\,(\mbox{by }\eqref{last-eq}$

This completes the proof of Theorem 6.2 by taking $A=2R+3c\,log(1+2ce^{2R})$.
∎

Proof of Theorem 5.1. Let $D$ be a $c$\- John domain such that $(D,k_{D})$ is
$\delta$-hyperbolic for some $\delta$ and let $\gamma$ be a $c_{0}$\-
neargeodesic in $D$. Then from Theorem 6.2 we have that if there are
$u,v\in\gamma$ such that each $w\in\gamma{[u,v]}$ satisfies $\Delta(w)\leq
2\min\\{\Delta(u),\Delta(v)\\}$, then $k_{D}(u,v)\leq A.$ Therefore, by
Theorem 6.1, we conclude that $\gamma$ is a $b$-cone arc, where $b$ depends
only on $c,\delta$ and $c_{0}$. This completes the proof of Theorem 5.1.

Acknowledgement: The first named author thanks SERB-CRG, and the second named
author thanks PMRF-MHRD (Id: 1200297), Govt. of India for their support.

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