# Breaking of multiplicity scaling observed in $K^{*0}/K$ ratio in baryon-rich
QCD matter

Aswini Kumar Sahoo<EMAIL_ADDRESS>Md. Nasim<EMAIL_ADDRESS>Department of Physical Sciences, Indian Institute of Science Education and
Research, Berhampur 760010, India.  Subhash Singha<EMAIL_ADDRESS>Institute of Modern Physics Chinese Academy of Sciences, Lanzhou, 73000,
China.

###### Abstract

In this study, we investigated the influence of collision energy and system
size on hadronic rescattering by analyzing the production of $K^{*0}$ mesons
using the Ultra Relativistic Quantum Molecular Dynamics (UrQMD) model.
Analysis are done in Au+Au collisions at various center-of-mass energies
($\sqrt{s_{NN}}$ = 7.7, 11.5, 19.6, 27, and 200 GeV), as well as in Cu+Cu and
isobaric (Ru+Ru and Zr+Zr) collisions at $\sqrt{s_{NN}}$ = 200 GeV at mid-
rapidity. Our findings reveal that the ratio of $K^{*0}$ mesons to charged
kaons ($K^{*0}/K$) decreases as the collision multiplicity increases.
Moreover, at top RHIC energies, this ratio exhibits a smooth multiplicity
scaling behavior. However, this scaling can be violated due to the formation
of baryon-rich matter at lower beam energies, specifically $\sqrt{s_{NN}}$ =
11.5 GeV or below. These results highlight the importance of considering the
interplay between collision energy, system size, and the chemical composition
of the produced matter when studying the hadronic rescattering effects in
heavy-ion collisions.

###### keywords:

Resonance , Re-scattering, Heavy-ion Collision

## Introduction

Ultra-relativistic nucleus-nucleus collisions generate extremely high energy
densities across a large volume, providing a unique opportunity to explore and
investigate the predictions of Quantum Chromodynamics (QCD) in a laboratory
setting [1]. Several experimental observations have indicated the possible
creation of a strongly coupled medium known as the quark-gluon plasma (QGP) at
such high temperatures and densities [2]. The Relativistic Heavy-ion Collider
(RHIC) and Large Hadron Collider (LHC) have conducted extensive heavy-ion
collisions involving different collision systems such as Au+Au, Pb+Pb, Cu+Cu,
U+U, Zr+Zr, Ru+Ru, Cu+Au, d+Au. These experiments aim to systematically
investigate the properties of the Quark-Gluon Plasma (QGP). Additionally, the
Beam Energy Scan (BES) program at RHIC covers a broad range of collision
energies, allowing for the creation of a QCD medium with varying the net
baryon density in the mid-rapidity region.

Short-lived resonances serve as an excellent tool to study the properties of
the QGP medium formed in heavy-ion collisions [3, 4, 5, 6, 7]. As the QGP
medium evolves, it reaches a critical temperature known as the transition
temperature. At this point, the medium undergoes a transition from a partonic
phase to a hadronic phase. As the medium continues to cool, the inelastic
collisions between particles come to a halt, a stage referred to as chemical
freeze-out (CFO). Following chemical freeze-out, the elastic collisions
between particles cease, which is known as kinetic freeze-out (KFO) [8, 9,
10]. Resonances like $K^{*0}$ are of particular interest as their lifetime is
smaller than the duration of the medium created in heavy-ion collision [11],
and they can decay, rescatter, and regenerated in between CFO and KFO, or the
so-called hadronic phase. The $K^{*0}$ resonance undergoes hadronic decays,
specifically $K^{*0}(\overline{K^{*0}})\rightarrow K^{\pm}\pi^{\mp}$, with a
branching ratio of 2/3 [12]. Following the decay, the daughter kaons and pions
have the potential to undergo rescattering interactions with other hadrons in
the medium, making it challenging to reconstruct the parent resonance from its
decay products. At the same time, the abundant population of kaons and pions
within the medium can lead to the creation of the $K^{*0}$ resonance through
pseudo-elastic scattering. This phenomenon is referred to as regeneration [13,
14, 15, 16].. The interplay between re-scattering and regeneration can be
observed through the $K^{*0}/K$ ratio. The behavior of this ratio,
specifically its suppression or enhancement, with increasing medium
multiplicity $dN_{ch}/dy$, provides insights into the relative significance of
hadronic rescattering compared to regeneration processes. A suppression of the
$K^{*0}/K$ ratio with increasing medium multiplicity would suggest that
hadronic rescattering dominates over regeneration, whereas an enhancement
would indicate the prevalence of regeneration over rescattering. Previous
experimental measurements on $K^{*0}/K$ ratio [17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32] and phenomenological studies [33, 34, 35, 36,
37] have consistently indicated the prevalence of rescattering over
regeneration.

Figure 1 (reference [31]) illustrates the behavior of the $K^{*0}/K$ ratio as
a function of $(dN_{ch}/dy)^{1/3}$, which serves as a proxy for the system
size, at various center-of-mass energies ranging from 62.4 GeV to 5.02 TeV.
Notably, the $K^{*0}/K$ ratio exhibits a decreasing trend with increasing
system size, demonstrating a smooth dependence on $(dN_{ch}/dy)^{1/3}$. These
findings strongly suggest the dominance of the rescattering effect within the
hadronic phase, and further indicate that the impact of rescattering is
primarily dependent on the system’s multiplicity or size.

Figure 1: (Color online) Experimental data on $K^{*0}/K$ ratio as a function
of $<dNdy>^{1/3}$ [17, 18, 19, 20, 25, 26, 27, 28, 29, 30, 31]. The bars and
boxes denote the statistical and systematic uncertainty respectively.

In addition to the dependence on system size, the rescattering within the
hadronic medium is also influenced by the chemical composition of the medium
itself. At high energies achieved at top RHIC and LHC, the created medium is
predominantly mesonic. In this scenario, the decay daughters ($\pi$ and $K$)
of the $K^{*0}$ resonance primarily experience re-scattering with pions,
resulting in meson-meson elastic collisions dominating the rescattering
process. However, at lower collision energies within the RHIC BES energy
regime, the medium created becomes baryon-dominated at mid-rapidity. In this
case, meson-baryon interactions could surpass meson-meson interactions in
terms of significance due to the formation of baryon-rich matter. Furthermore,
the elastic interaction cross-section between meson and baryon increases
significantly at lower energies. Consequently, these meson-meson interactions
at high energies and meson-baryon interactions at lower energies may lead to
distinct rescattering effects on the observed $K^{*0}$ resonance.

In this study, we present a comprehensive investigation of the $K^{*0}/K$
ratio across different collision systems and energies available at RHIC,
utilizing the UrQMD model. Section 2 provides a concise overview of the
physics underlying the UrQMD model, along with the methodology employed for
reconstructing the $K^{*0}$ resonance. The model results are presented and
analyzed in Section 3. Finally, a summary of the findings is provided in
Section 4.

Figure 2: (Color online) Reconstructed $K^{*0}$ resonance from UrQMD model
after background subtraction.

## Model Description and the resonance reconstruction

The UrQMD (Ultra-relativistic Quantum Molecular Dynamics) model, developed
based on a microscopic transport theory, encompasses various aspects of
particle production mechanisms, including resonance decays, string excitation,
and fragmentation. The model focuses on the phase space description of
reactions, considering the stochastic collisions of hadrons. In the UrQMD
model, the projectile and target nuclei are initialized according to a Woods-
Saxon profile in coordinate space. The stochastic collisions between hadrons
are performed similarly to the original cascade models, until the collision
criteria are satisfied. When the relative distance $d_{trans}$ between two
particles in three-dimensional configuration space becomes smaller than a
critical distance, a collision occurs. The critical distance $d_{0}$ is
determined by the corresponding total cross section $\sigma_{tot}$, which
depends on the center-of-mass energy $\sqrt{s}$ and the species and quantum
numbers of the incoming particles. The UrQMD model incorporates more than 50
baryon species, including nucleons, delta and hyperon resonances with masses
up to 2.25 GeV/$c^{2}$, and 45 meson species, including strange meson
resonances. The model also includes the corresponding antiparticles and
isospin-projected states. The interactions in the model encompass baryon-
baryon, meson-baryon, and meson-meson interactions, allowing for a
comprehensive description of the dynamics and interactions within the hadronic
system [38, 39].

Figure 3: (Color online) The $p_{T}$ integrated yield of $K^{*0}$ and charged
kaons as a function of system-size for various collision systems and beam
energies from the UrQMD calculation. .

Figure 4: (Color online) The $K^{*0}/K$ ratio calculated at mid-rapidity as a
function of system-size for various collision systems and beam energies from
the UrQMD calculation.

In our study, we utilized the publicly available version of the UrQMD model.
In this model, the masses of resonances are distributed according to the
Breit-Wigner function, which allows for the simulation of their decay
processes. We performed the hadronic simulation or hadron propagation for a
duration of $\tau$, specifically choosing values of 5 and 20 fm/$c$, in order
to balance computational efficiency. During the simulation, the decay of
unstable particles was enabled to account for their natural decay processes.
The information about the final freeze-out particles was collected from the
file ”test.f14” generated by the model. For the reconstruction of short-lived
particles such as $K^{*0}$, we implemented a method similar to that used in
experimental data analysis. This involved accumulating invariant mass
distributions for different combinations of daughter particles and applying
proper treatments to eliminate background contributions. The yield of the
$K^{*0}$ resonance was then estimated by integrating the resonance mass peak
and correcting for the corresponding branching ratio. The reconstructed
$K^{*0}$ resonance from the UrQMD model after performing combinatorial
background subtraction is shown in Fig. 2. The signal was fitted with Breit-
Wigner function plus first order polynomial. The $K^{*0}$ and
$\overline{K^{*0}}$ is combined and denoted as $K^{*0}$, unless specified.
Similarly the charged kaons are combined and denoted as K.

## Results and Discussions

Figure 3 illustrates the integrated yield (dN/dy) of $K^{*0}$ mesons and
charged kaons as a function of $(dN_{ch}/dy)^{1/3}$. The model calculations
are performed within the rapidity range $|y|<0.5$ for four different collision
systems: Au+Au, Zr+Zr, Ru+Ru, and Cu+Cu, at a center-of-mass energy of
$\sqrt{s_{NN}}$ = 200 GeV. Additionally, calculations are shown for Au+Au
collisions at varying center-of-mass energies: $\sqrt{s_{NN}}$ = 7.7, 11.5,
19.6, and 27 GeV. The results indicate that the yield of charged kaons
exhibits a scaling behavior with multiplicity for all collision systems and
beam energies. In contrast, the yield of $K^{*0}$ mesons follows the scaling
trend at the top RHIC energy, regardless of the collision system. However, for
lower beam energies, the scaling behavior deviates from the observed trend.
These findings of $K^{*0}$ mesons deviates from the scaling trend at lower
beam energies indicates the influence of additional factors on their measured
yield e.g. loss due to re-scattering.

Figure 4 presents the system size dependence of the $K^{*0}/K$ ratio as a
function of $(dN_{ch}/dy)^{1/3}$ in the UrQMD model for two different medium
evolution times: $\tau$ = 5 fm/c and $\tau$ = 20 fm/c. The ratio demonstrates
a decreasing trend with increasing multiplicity, indicating a prevalence of
hadronic rescattering over regeneration with increasing collision centrality.
At a center-of-mass energy of $\sqrt{s_{NN}}$ = 200 GeV, the ratios exhibit
multiplicity scaling irrespective of the collision species. However, as the
collision energy decreases below 11.5 GeV, the multiplicity scaling breaks.
This effect becomes more prominent for a longer medium evolution time of
$\tau$ = 20 fm/c. This breaking of scaling suggests that even at the same
multiplicity or system size, the effect of re-scattering is not the same for
$\sqrt{s_{NN}}$ = 200 GeV and lower energies e.g. 7.7 GeV. This is possible if
the chemical composition of the medium differs at lower energy compared to top
RHIC energy.

The UrQMD model serves as a comprehensive framework for examining the complete
evolution of hadronic interactions concerning propagation time or medium
evolution time. Figure 5 illustrates the rates of meson-meson and meson-baryon
interactions as a function of medium evolution time. These data are derived
from the collision history file ($test.f15$) generated by the UrQMD model. The
presented distributions pertain to two collision energies, specifically
$\sqrt{s_{NN}}$ = 7.7 GeV (Au+Au) and 200 GeV(Au+Au and Cu+Cu). In this case,
only the distributions of meson-meson and meson-baryon interactions are shown
since the $K^{*0}$ resonance decays into $\pi$ and $K$ particles. At
$\sqrt{s_{NN}}$ = 200 GeV, meson-meson interactions dominates over baryon-
meson interactions in both Au+Au and Cu+Cu (also for Ru+Ru and Zr+Zr, not
shown here) and we observed that $K^{*0}/K$ ratio shows smooth multiplicity
scaling in all such collisions system. However, at lower beam energies,
depending on the system’s composition, baryon-meson interactions tend to
dominate over meson-meson interactions. It should be noted that the
interaction cross-section between two mesons, such as between two pions,
differs from the interaction between a meson and a baryon, such as between a
pion and a proton. Consequently, the effect of rescattering can vary, even for
systems of the same size. This observation suggests that the observed
deviation from multiplicity scaling in the $K^{*0}/K$ ratio arises from the
predominance of baryon-meson interactions in baryon-rich matter created at low
center-of-mass energies. This intriguing finding should be corroborated by
experimental data.

Figure 6 recent measurements from the STAR experiment at RHIC BES (Beam Energy
Scan) Phase-I energies ranging from $\sqrt{s_{NN}}$ = 7.7 to 39 GeV [32], as
well as measurements at top RHIC and LHC energies. The measurement at BES
Phase-I indicates a potential deviation from the predicted multiplicity
scaling in the $K^{*0}/K$ ratio. However, due to limited statistics in the
STAR data, it is not possible to draw strong conclusions. The STAR experiment
has conducted the Phase-II of the BES program, collecting a large amount of
data in Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7 to 19.6 GeV [40]. The aim of
this phase is to improve the statistics significantly. Therefore, it is
suggested that it would be interesting to revisit the measurement of the
$K^{*0}/K$ ratio with the high statistics BES-II data to obtain a more
conclusive understanding of the potential breaking of multiplicity scaling.

Figure 5: (Color online) The time-differential distribution of hadronic
interaction for $\sqrt{s_{NN}}$= 7.7 and 200 GeV (Au+Au and Cu+Cu) from UrQMD
model. The blue line refers to meson-meson (M+M) collision and the red line
refers to meson-bayon (M+B) collision. Figure 6: (Color online) Experimental
data on $K^{*0}/K$ ratio as a function of $<dNdy>^{1/3}$ [17, 18, 19, 20, 25,
26, 27, 28, 29, 30, 31, 32]. The bars and boxes denote the statistical and
systematic uncertainty respectively.

## Summary

In summary, we investigated the dependence of $K^{*0}$ production on system
size and collision energy in various collision systems and beam energies
available at RHIC, employing the UrQMD model. Our analysis revealed a
suppression in the ratio of resonance to non-resonance ($K^{*0}/K$) particles,
indicating the prevalence of hadronic rescattering over regeneration in
central A+A collisions. At the top RHIC energy, the ratio exhibited a scaling
behavior with collision multiplicity. However, as the collision energy
decreased, this scaling was found to break, suggesting that meson-baryon
interactions play a more prominent role in a baryon-rich environment at lower
energy regimes, contrasting with the dominance of meson-meson interactions at
higher energies. To gain further insights into rescattering in the BES energy
range, it would be valuable to compare these predicted results with
measurements from high-statistics BES-II data. This comparison can provide a
better understanding of the dynamics of the medium and the interplay between
collision energy, system size, and the chemical composition when studying the
hadronic rescattering .

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# Path Connectivity of Spheres in the Gromov–Hausdorff Class

A. Ivanov, R. Tsvetnikov, and A. Tuzhilin

###### Abstract

The paper is devoted to geometrical investigation of Gromov–Hausdorff distance
on the classes of all metric spaces and of all bounded metric spaces. The main
attention is paid to path connectivity questions. The path connected
components of the Gromov–Hausdorff class of all metric spaces are described,
and the path connectivity of spheres is proved in several particular cases.

Keywords: metric geometry, Gromov–Hausdorff distance, metric spaces, bounded
metric spaces, compact metric spaces, path connectivity

## Introduction

Comparison of metric spaces is an important problem that is interesting as in
pure research, so as in numerous applications. A natural approach is to define
some distance function between metric spaces: The less similar the spaces are,
the more the distance between them should be. Currently, the Gromov–Hausdorff
distance and some its modifications are the most commonly used for these
purposes.

The history of this distance goes back to works of F. Hausdorff [1]. In 1914
he defined a non-negative symmetrical function on pairs of subsets of a metric
space $X$ that is equal to the infimum of the reals $r$ such that the first
subset is contained in the $r$-neighborhood of the second one, and vice-versa.
It turns out that this function satisfies the triangle inequality and further,
it is a metric on the family of all closed bounded subsets of $X$. Later on,
D. Edwards [2] and, independently, M. Gromov [3] generalized the Hausdorff’s
construction to the case of all metric spaces using their isometric embeddings
into all possible ambient metric spaces, see formal definition below. The
resulting function is now referred as the _Gromov–Hausdorff distance_. Notice
that this distance is also symmetric, non-negative and satisfies the triangle
inequality. It always equals zero for a pair of isometric metric spaces,
therefore in this context such spaces are usually identified. But generally
speaking, the Gromov–Hausdorff distance can be infinite and also can vanish on
a pair of non-isometric spaces. Nevertheless, if one restricts himself to the
family $\mathcal{M}$ of isometry classes of all compact metric spaces, then
the Gromov–Hausdorff distance satisfies all metric axioms. The set
$\mathcal{M}$ endowed with the Gromov–Hausdorff distance is called the
_Gromov–Hausdorff space_. Geometry of this metric space turns out to be rather
non-trivial and is studied actively. It is well-known that $\mathcal{M}$ is
path connected, Polish (i.e., complete and separable), geodesic [4], and also
the space $\mathcal{M}$ is not proper, and has no non-trivial symmetries [5].
The Gromov–Hausdorff distance is effectively used in computer graphics and
computational geometry for comparison and transformation of shapes, see for
example [6]. It also has applications in other natural sciences, for example,
C. Sormani used this distance to prove stability of Friedmann cosmology model
[7]. A detailed introduction to geometry of the Gromov–Hausdorff distance can
be found in [8, Ch. 7] or in [9].

The case of arbitrary metric spaces is also very interesting. For such spaces
many authors use some modifications of the Gromov–Hausdorff distance. For
example, pointed spaces, i.e., the spaces with marked points, are considered,
see [10] and [11]. In the present paper we continue the studying of the
classical Gromov–Hausdorff distance on the classes
$\operatorname{\mathcal{G\\!H}}$ and ${\cal B}$ consisting of the
representatives of isometry classes of all metric spaces and of bounded metric
spaces, respectively, that we started in [13]. Here the term “class” is
understood in the sense of von Neumann–Bernays–Gödel set theory (NBG) that
permits to define correctly the distance on the proper class
$\operatorname{\mathcal{G\\!H}}$, and to construct on it an analogue of metric
topology basing on so-called filtration by sets with respect to cardinality,
see below. In paper [13], continuous curves in
$\operatorname{\mathcal{G\\!H}}$ are defined, and it is proved that the
Gromov–Hausdorff distance is an interior generalized semi-metric as on
$\operatorname{\mathcal{G\\!H}}$, so as on ${\cal B}$, i.e., the distance
between points is equal to the infimum of the lengths of curves connecting
these points.

In the present paper the questions of path connectivity in
$\operatorname{\mathcal{G\\!H}}$ are studied, in particular, the path
connected components of $\operatorname{\mathcal{G\\!H}}$ are described. One of
these components is the class ${\cal B}$ of bounded metric spaces. Notice that
the geometry of those components is rather tricky, see [14]. Also in the
present paper the path connectivity of spheres is proved for several
particular cases. Namely, the following results are obtained.

  * •

It is shown that all spheres centered at the single-point metric space in
$\operatorname{\mathcal{G\\!H}}$, in ${\cal B}$, and in $\mathcal{M}$ are path
connected.

  * •

For each bounded metric space $X$, there exists a real $R_{X}$ such that all
the spheres centered at $X$ with radius $r\geq R_{X}$ are path connected in
${\cal B}$. And if $X$ is a compact space, then such spheres are path
connected in $\mathcal{M}$.

  * •

For each generic metric space $M$ (see the definition below), there exists a
real $r_{M}>0$ such that all the spheres centered at $M$ with radius $r\leq
r_{M}$ are path connected in $\operatorname{\mathcal{G\\!H}}$. And if $M$ is a
bounded (respectively, compact) metric space, then such spheres are path
connected in ${\cal B}$ (in $\mathcal{M}$, respectively).

#### Acknowledgements

This work was done at Lomonosov Moscow State University, Interdisciplinary
Science Educational School “Mathematical Methods of Complex Systems Analysis”.
The authors use this opportunity to express their deep respect to Professor
Yuri M. Smirnov, whose books, poetry and lectures about classical opera had
great influence on us. The work is supported by Russian Science Foundation,
Project 21–11–00355.

##  1 Preliminaries

Let $X$ be an arbitrary set. By $\\#X$ we denote the cardinality of $X$, and
let $\mathcal{P}_{0}(X)$ be the set of all its non-empty subsets. A symmetric
mapping $d\colon X\times X\to[0,\infty]$ vanishing on the pairs of the same
elements is called a _distance function on $X$_. If $d$ satisfies the triangle
inequality, then $d$ is referred as a _generalized semi-metric_. If in
addition, $d(x,y)>0$ for all $x\neq y$, then $d$ is called a _generalized
metric_. At last, if $d(x,y)<\infty$ for all $x,y\in X$, then such distance
function is called a _metric_ , and sometimes a _finite metric_ to emphasize
the difference with a generalized metric. A set $X$ endowed with a
(generalized) (semi-)metric is called a _( generalized) (semi-)metric space_.

We need the following simple properties of metrics.

######  Proposition 1.1.

The following statements are valid.

  1. (1)

A non-trivial non-negative linear combination of two metrics given on an
arbitrary set is also a metric.

  2. (2)

A positive linear combination of a metric and a semi-metric given on an
arbitrary set is a metric.

If $X$ is a set endowed with a distance function, then the distance between
its points $x$ and $y$ we usually denote by $|xy|$. To emphasize that the
distance between $x$ and $y$ is calculated in the space $X$, we write
$|xy|_{X}$. Further, if $\gamma\colon[a,b]\to X$ is a continuous curve in $X$,
then its _length $|\gamma|$_ is defined as the supremum of the “lengths of
inscribed polygonal lines”, i.e., of the values
$\sum_{i}\big{|}\gamma(t_{i})\gamma(t_{i+1})\big{|}$, where the supremum is
taken over all possible partitions $a=t_{1}<\cdots<t_{k}=b$ of the segment
$[a,b]$. A distance function on $X$ is called _interior_ if the distance
between any two its points $x$ and $y$ equals to the infimum of lengths of
curves connecting these points. A curve $\gamma$ whose length differs from
$|xy|$ at most by $\varepsilon$ is called _$\varepsilon$ -shortest_. If for
any pair of points $x$ and $y$ of the space $X$ there exists a curve whose
length equals to the infimum of lengths of curves connecting these points and
equals $|xy|$, then the distance is called _strictly interior_ , and the space
$X$ is referred as a _geodesic space_.

Let $X$ be a metric space. For each $A,\,B\in\mathcal{P}_{0}(X)$ and each
$x\in X$ we put

$\displaystyle|xA|=|Ax|=\inf\bigl{\\{}|xa|:a\in
A\bigr{\\}},\qquad|AB|=\inf\bigl{\\{}|ab|:a\in A,\,b\in B\bigr{\\}},$
$\displaystyle d_{H}(A,B)=\max\\{\sup_{a\in A}|aB|,\,\sup_{b\in
B}|Ab|\\}=\max\bigl{\\{}\sup_{a\in A}\inf_{b\in B}|ab|,\,\sup_{b\in
B}\inf_{a\in A}|ba|\bigr{\\}}.$

The function
$d_{H}\colon\mathcal{P}_{0}(X)\times\mathcal{P}_{0}(X)\to[0,\infty]$ is called
the _Hausdorff distance_. It is well-known, see for example [8] or [9], that
$d_{H}$ is a metric on the family ${\cal H}(X)\subset\mathcal{P}_{0}(X)$ of
all non-empty closed bounded subsets of $X$.

Let $X$ and $Y$ be metric spaces. A triple $(X^{\prime},Y^{\prime},Z)$
consisting of a metric space $Z$ and two its subsets $X^{\prime}$ and
$Y^{\prime}$ isometric to $X$ and $Y$, respectively, is called a _realization
of the pair $(X,Y)$_. The _Gromov–Hausdorff distance $d_{GH}(X,Y)$ between $X$
and $Y$_ is defined as the infimum of reals $r$ such that there exists a
realization $(X^{\prime},Y^{\prime},Z)$ of the pair $(X,Y)$ with
$d_{H}(X^{\prime},Y^{\prime})\leq r$.

Notice that the Gromov–Hausdorff distance could take as finite, so as infinite
values, and always satisfies the triangle inequality, see [8] or [9]. Besides,
this distance always vanishes at each pair of isometric spaces, therefore, due
to triangle inequality, the Gromov–Hausdorff distance does not depend on the
choice of representatives of isometry classes. There are examples of non-
isometric metric spaces with zero Gromov–Hausdorff distance between them, see
[12].

Since each set can be endowed with a metric (for example, one can put all the
distances between different points to be equal to $1$), then representatives
of isometry classes form a proper class. This class endowed with the
Gromov–Hausdorff distance is denoted by $\operatorname{\mathcal{G\\!H}}$. Here
we use the concept _class_ in the sense of von Neumann–Bernays–Gödel set
theory (NBG).

Recall that in NBG all objects (analogues of ordinary sets) are called
_classes_. There are two types of classes: _sets_ (the classes that are
elements of other classes), and _proper classes_ (all the remaining classes).
The class of all sets is an example of a proper class. Many standard
operations are well-defined for classes. Among them are intersection,
complementation, direct product, mapping, etc.

Such concepts as a _distance function,(generalized) semi-metric_ and _(
generalized) metric_ are defined in the standard way for any class, as for a
set, so as for a proper class, because the direct products and mappings are
defined. But direct transfer of some other structures, such as topology, leads
to contradictions. For example, if we defined a topology for a proper class,
then this class has to be an element of the topology, that is impossible due
to the definition of proper classes. In paper [13] the following construction
is suggested.

For each class ${\cal C}$ consider a “filtration” by subclasses ${\cal
C}_{n}$, each of which consists of all the elements of ${\cal C}$ of
cardinality at most $n$, where $n$ is a cardinal number. Recall that elements
of a class are sets, therefore cardinality is defined for them. A class ${\cal
C}$ such that all its subclasses ${\cal C}_{n}$ are sets is said to be
_filtered by sets_. Evidently, if a class ${\cal C}$ is a set, then it is
filtered by sets.

Thus, let ${\cal C}$ be a class filtered by sets. When we say that the class
${\cal C}$ satisfies some property, we mean the following: Each set ${\cal
C}_{n}$ satisfies this property. Let us give several examples.

  * •

Let a distance function on ${\cal C}$ be given. It induces an “ordinary”
distance function on each set ${\cal C}_{n}$. Thus, for each ${\cal C}_{n}$
the standard concepts of metric geometry such as open balls and spheres are
defined and are sets. The latter permits to construct standard metric topology
$\tau_{n}$ on ${\cal C}_{n}$ taking the open balls as a base of the topology.
It is clear that if $n\leq m$, then ${\cal C}_{n}\subset{\cal C}_{m}$, and the
topology $\tau_{n}$ on ${\cal C}_{n}$ is induced from $\tau_{m}$.

  * •

More general, a _topology_ on the class ${\cal C}$ is defined as a family of
topologies $\tau_{n}$ on the sets ${\cal C}_{n}$ satisfying the following
_consistency condition_ : If $n\leq m$, then $\tau_{n}$ is the topology on
${\cal C}_{n}$ induced from $\tau_{m}$. A class endowed with a topology is
referred as a _topological class_.

  * •

The presence of a topology permits to define continuous mappings from a
topological space $Z$ to a topological class ${\cal C}$. Notice that according
to NBG axioms, for any mapping $f\colon Z\to{\cal C}$ from the set $Z$ to the
class ${\cal C}$, the image $f(Z)$ is a set, all elements of $f(Z)$ are also
some sets, and hence, the union $\cup f(Z)$ is a set of some cardinality $n$.
Therefore, each element of $f(Z)$ is of cardinality at most $n$, and so,
$f(Z)\subset{\cal C}_{n}$. The mapping $f$ is called _continuous_ , if $f$ is
a continuous mapping from $Z$ to ${\cal C}_{n}$. The consistency condition
implies that for any $m\geq n$, the mapping $f$ is a continuous mapping from
$Z$ to ${\cal C}_{m}$, and also for any $k\leq n$ such that $f(Z)\subset{\cal
C}_{k}$, the mapping $f$ considered as a mapping from $Z$ to ${\cal C}_{k}$ is
continuous.

  * •

The above arguments allows to define _continuous curves in a topological class
${\cal C}$_.

  * •

Let a class ${\cal C}$ be endowed with a distance function and the
corresponding topology. We say that the distance function is _interior_ if it
satisfies the triangle inequality, and for any two elements from ${\cal C}$
such that the distance between them is finite, this distance equals the
infimum of the lengths of the curves connecting these elements.

  * •

Let a sequence $\\{X_{i}\\}$ of elements from a topological class ${\cal C}$
be given. Since the family $\\{X_{i}\\}_{i=1}^{\infty}$ is the image of the
mapping $\mathbb{N}\to{\cal C}$, $i\mapsto X_{i}$, and $\mathbb{N}$ is a set,
then, due to the above arguments, all the family $\\{X_{i}\\}$ lies in some
${\cal C}_{m}$. Thus, the concept of _convergence_ of a sequence in a
topological class is defined, namely, the sequence converges if it converges
with respect to some topology $\tau_{m}$ such that $\\{X_{i}\\}\subset{\cal
C}_{m}$, and hence, with respect to any such topology.

Our main examples of topological classes are the classes
$\operatorname{\mathcal{G\\!H}}$ and ${\cal B}$ defined above. Recall that the
class $\operatorname{\mathcal{G\\!H}}$ consists of representatives of isometry
classes of all metric spaces, and the class ${\cal B}$ consists of
representatives of isometry classes of all bounded metric spaces. Notice that
$\operatorname{\mathcal{G\\!H}}_{n}$ and ${\cal B}_{n}$ are sets for any
cardinal number $n$.

The most studied subset of $\operatorname{\mathcal{G\\!H}}$ is the set of all
compact metric spaces. It is called a _Gromov–Hausdorff space_ and often
denoted by $\mathcal{M}$. It is well-known, see [8, 9, 4], that the
Gromov–Hausdorff distance is an interior metric on $\mathcal{M}$, and the
metric space $\mathcal{M}$ is Polish and geodesic. In paper [13] it is shown
that the Gromov–Hausdorff distance is interior as on the class
$\operatorname{\mathcal{G\\!H}}$, so as on the class ${\cal B}$.

As a rule, to calculate the Gromov–Hausdorff distance between a pair of given
metric spaces is rather difficult, and for today the distance is known for a
few pairs of spaces, see for example [15]. The most effective approach for
this calculations is based on the next equivalent definition of the
Gromov–Hausdorff distance, see details in [8] or [9].

Recall that a _relation_ between sets $X$ and $Y$ is defined as an arbitrary
subset of their direct product $X\times Y$. Thus, $\mathcal{P}_{0}(X\times Y)$
is the set of all non-empty relations between $X$ and $Y$.

######  Definition 1.2.

For any $X,Y\in\operatorname{\mathcal{G\\!H}}$ and any
$\sigma\in\mathcal{P}_{0}(X\times Y)$, the _distortion
${\operatorname{dis}}\,\sigma$ of the relation $\sigma$_ is defined as the
following value:

${\operatorname{dis}}\,\sigma=\sup\Bigl{\\{}\bigl{|}|xx^{\prime}|-|yy^{\prime}|\bigr{|}:(x,y),\,(x^{\prime},y^{\prime})\in\sigma\Bigr{\\}}.$

A relation $R\subset X\times Y$ between sets $X$ and $Y$ is called a
_correspondence_ , if the restrictions of the canonical projections
$\pi_{X}\colon(x,y)\mapsto x$ and $\pi_{Y}\colon(x,y)\mapsto y$ onto $R$ are
surjective. Notice that relations can be considered as partially defined
multivalued mapping. Form this point of view, the correspondences are
multivalued surjective mappings. For a correspondence $R\subset X\times Y$ and
$x\in X$, we put $R(x)=\big{\\{}y\in Y:(x,y)\in R\big{\\}}$ and call $R(x)$
the _image of the element $x$ under the relation $R$_. By $\mathcal{R}(X,Y)$
we denote the set of all correspondences between $X$ and $Y$. The following
result is well-known.

######  Assertion 1.3.

For any $X,Y\in\operatorname{\mathcal{G\\!H}}$, it holds

$d_{GH}(X,Y)=\frac{1}{2}\inf\bigl{\\{}{\operatorname{dis}}\,R:R\in\mathcal{R}(X,Y)\bigr{\\}}.$

We need the following estimates that can be easily proved by means of
Assertion 1.3. By $\Delta_{1}$ we denote the single-point metric space.

######  Assertion 1.4.

For any $X,Y\in\operatorname{\mathcal{G\\!H}}$, the following relations are
valid:

  * •

$2d_{GH}(\Delta_{1},X)={\operatorname{diam}}\,X$;

  * •

$2d_{GH}(X,Y)\leq\max\\{{\operatorname{diam}}\,X,{\operatorname{diam}}\,Y\\}$;

  * •

If at least one of $X$ and $Y$ is bounded, then
$\bigl{|}{\operatorname{diam}}\,X-{\operatorname{diam}}\,Y\bigr{|}\leq
2d_{GH}(X,Y)$.

For topological spaces $X$ and $Y$, their direct product $X\times Y$ is
considered as the topological space endowed with the standard topology of the
direct product. Therefore, it makes sense to speak about _closed relations_
and _closed correspondences_.

A correspondence $R\in\mathcal{R}(X,Y)$ is called _optimal_ if
$2d_{GH}(X,Y)={\operatorname{dis}}\,R$. The set of all optimal correspondences
between $X$ and $Y$ is denoted by $\mathcal{R}_{\operatorname{opt}}(X,Y)$.

######  Assertion 1.5 ([16, 17]).

For any $X,\,Y\in\mathcal{M}$, there exists as a closed optimal
correspondence, so as a realization $(X^{\prime},Y^{\prime},Z)$ of the pair
$(X,Y)$ which the Gromov–Hausdorff distance between $X$ and $Y$ is attained
at.

######  Assertion 1.6 ([16, 17]).

For any $X,\,Y\in\mathcal{M}$ and each closed optimal correspondence
$R\in\mathcal{R}(X,Y)$, the family $R_{t}$, $t\in[0,1]$, of compact metric
spaces, where $R_{0}=X$, $R_{1}=Y$, and for $t\in(0,1)$ the space $R_{t}$ is
the set $R$ endowed with the metric

$\bigl{|}(x,y),(x^{\prime},y^{\prime})\bigr{|}_{t}=(1-t)|xx^{\prime}|+t\,|yy^{\prime}|,$

is a shortest curve in $\mathcal{M}$ connecting $X$ and $Y$, and the length of
this curve equals $d_{GH}(X,Y)$.

Let $X$ be a metric space and $\lambda>0$ a real number. By $\lambda\,X$ we
denote the metric space obtained from $X$ by multiplication of all the
distances by $\lambda$, i.e., $|xy|_{\lambda X}=\lambda|xy|_{X}$ for any
$x,\,y\in X$. If the space $X$ is bounded, then for $\lambda=0$ we put
$\lambda X=\Delta_{1}$.

######  Assertion 1.7.

For any $X,Y\in\operatorname{\mathcal{G\\!H}}$ and any $\lambda>0$, the
equality $d_{GH}(\lambda X,\lambda Y)=\lambda\,d_{GH}(X,Y)$ holds. If
$X,Y\in{\cal B}$, then, in addition, the same equality is valid for
$\lambda=0$.

######  Assertion 1.8.

Let $X\in{\cal B}$ and $\lambda_{1},\lambda_{2}\geq 0$. Then
$2d_{GH}(\lambda_{1}X,\lambda_{2}X)=|\lambda_{1}-\lambda_{2}|\,{\operatorname{diam}}\,X$.

######  Remark 1.9.

If ${\operatorname{diam}}\,X=\infty$, then Assertion 1.8 is not true in
general. For example, let $X=\mathbb{R}$. The space $\lambda\,\mathbb{R}$ is
isometric to $\mathbb{R}$ for any $\lambda>0$, therefore
$d_{GH}(\lambda_{1}\mathbb{R},\lambda_{2}\mathbb{R})=0$ for any
$\lambda_{1},\lambda_{2}>0$.

##  2 Path Connectivity.

In the present section, the path connected components of $\mathcal{M}$, ${\cal
B}$, and $\operatorname{\mathcal{G\\!H}}$ are described, and the path
connectivity of spheres in these classes is investigated.

###  2.1 Path Connectivity Components.

Let us start with several auxiliary results.

######  Lemma 2.1.

If $\gamma\colon[a,b]\to{\cal B}$ is a continuous curve, then
${\operatorname{diam}}\,\gamma(t)$ is a continuous function.

###### Proof.

According to Assertion 1.4, we have:
${\operatorname{diam}}\,\gamma(t)=2d_{GH}\bigl{(}\gamma(t),\Delta_{1}\bigr{)}$.
It remains to use the fact that the distance function is continuous. ∎

######  Lemma 2.2.

If $\gamma\colon[a,b]\to\operatorname{\mathcal{G\\!H}}$ is a continuous curve,
then the distance between its ends is finite. In particular, if the distance
between $X$ and $Y$ is infinite, then such $X$ and $Y$ can not be connected by
a continuous curve.

###### Proof.

Indeed, the image of the segment $[a,b]$ under the continuous mapping $\gamma$
is a compact subset. Therefore, it can be covered by a finite number of balls
of a fixed radius. The distance between any two points from such ball is
finite. Since the Gromov–Hausdorff distance is interior, then such points can
be connected by a curve of finite length. The latter implies that the
endpoints of the curve $\gamma$ can be connected by a “polygonal line” of a
finite length. ∎

######  Lemma 2.3.

If $\gamma\colon[a,b]\to\operatorname{\mathcal{G\\!H}}$ is a continuous curve,
then either ${\operatorname{diam}}\,\gamma(t)=\infty$ for all $t$, or
${\operatorname{diam}}\,\gamma(t)<\infty$ for all $t$, i.e., in the latter
case, the $\gamma$ is a continuous curve in ${\cal B}$.

###### Proof.

It is clear that if $X,\,Y\in\operatorname{\mathcal{G\\!H}}$,
${\operatorname{diam}}\,X<\infty$ and ${\operatorname{diam}}\,Y=\infty$, then
$d_{GH}(X,Y)=\infty$, but the distance between any two points of a continuous
curve is finite according to Lemma 2.2, that implies the lemma’s statement. ∎

######  Corollary 2.4.

Two spaces $X,\,Y\in\operatorname{\mathcal{G\\!H}}$ can be connected by a
continuous curve in $\operatorname{\mathcal{G\\!H}}$, if and only if
$d_{GH}(X,Y)<\infty$. In particular, $\operatorname{\mathcal{G\\!H}}$ is not
path connected, and its path connected components are the classes of spaces
with pairwise finite mutual distances. One of such components coincides with
the class ${\cal B}$.

###### Proof.

Indeed, if the distance between a pair of points in
$\operatorname{\mathcal{G\\!H}}$ is infinite, then, due to Lemma 2.2, there is
no a continuous curve connecting them, and if the distance is finite, then
such continuous curve exists because the distance is interior. Finally, the
distance between any two spaces from ${\cal B}$ is finite according to
Assertion 1.4. ∎

######  Lemma 2.5.

If $\lambda(t)$, $t\in[a,b]$, is a continuous non-negative function and
$\gamma\colon[a,b]\to{\cal B}$ a continuous curve, then $\lambda(t)\gamma(t)$
is a continuous curve in ${\cal B}$.

###### Proof.

Consider the mapping $F\colon[0,\infty)\times{\cal B}\to{\cal B}$ defined as
$F\colon(\lambda,X)\mapsto\lambda\,X$. We will show that $F$ is continuous.
Notice that

$d_{GH}(\lambda X,\mu Y)\leq d_{GH}(\lambda X,\mu X)+d_{GH}(\mu X,\mu
Y)=\frac{1}{2}|\lambda-\mu|\,{\operatorname{diam}}\,X+\mu\,d_{GH}(X,Y).$

Further, for any $\varepsilon>0$, there exists $\delta>0$ such that for any
pair $(\mu,Y)$ with $d_{GH}(X,Y)<\delta$ and $|\lambda-\mu|<\delta$, the
inequalities
$\frac{1}{2}|\lambda-\mu|\,{\operatorname{diam}}\,X<\varepsilon/2$ and
$\mu\,d_{GH}(X,Y)<\varepsilon/2$ are valid, and hence, $d_{GH}(\lambda X,\mu
Y)<\varepsilon$, and the latter means continuity of $F$. It remains to notice
that $\lambda(t)\gamma(t)=F\bigl{(}\lambda(t),\gamma(t)\bigr{)}$. ∎

###  2.2 Path Connectivity of Spheres Centered at Single-Point Metric Space

For a metric space $X\in\operatorname{\mathcal{G\\!H}}$ and a real $r\geq 0$,
we define the _sphere $S_{r}(X)$_ and the _ball $B_{r}(X)$ in
$\operatorname{\mathcal{G\\!H}}$_ in the standard way:

$S_{r}(X)=\\{Y:d_{GH}(X,Y)=r\\}\ \ \text{and}\ \ B_{r}(X)=\\{Y:d_{GH}(X,Y)\leq
r\\}.$

As usual, $X$ and $r$ are called the _center_ and the _radius_ of the sphere
$S_{r}(X)$ and of the ball $B_{r}(X)$, respectively. Notice that for $r>0$,
the spheres and the balls are proper subclasses in
$\operatorname{\mathcal{G\\!H}}$. Also notice that for $X\in{\cal B}$, the
spheres and the balls in the class ${\cal B}$ (in the space $\mathcal{M}$ for
$X\in\mathcal{M}$) can be defined as the intersections of $S_{r}(X)$ and
$B_{r}(X)$ with ${\cal B}$ (with $\mathcal{M}$, respectively).

######  Lemma 2.6.

Let $A$ and $B$ be arbitrary spaces in ${\cal B}$ with nonzero diameters. Then
for any positive $\varepsilon$ such that
$2\varepsilon<{\operatorname{min}}\,\\{{\operatorname{diam}}\,A,{\operatorname{diam}}\,B\\}$,
any $\varepsilon$-shortest curve connecting $A$ and $B$ does not pass through
$\Delta_{1}$.

###### Proof.

Assume the contrary, and let $\gamma$ be an $\varepsilon$-shortest curve
passing through $\Delta_{1}$. Denote by $\gamma_{1}$ and $\gamma_{2}$ the
segments of $\gamma$ between $A$ and $\Delta_{1}$ and between $\Delta_{1}$ and
$B$, respectively. Then $|\gamma|=|\gamma_{1}|+|\gamma_{2}|\geq
d_{GH}(A,\Delta_{1})+d_{GH}(\Delta_{1},B)=\frac{1}{2}({\operatorname{diam}}\,A+{\operatorname{diam}}\,B)$.
On the other hand, $|\gamma|\leq
d_{GH}(A,B)+\varepsilon\leq\frac{1}{2}\max\\{{\operatorname{diam}}\,A,{\operatorname{diam}}\,B\\}+\varepsilon$,
and hence,
${\operatorname{diam}}\,A+{\operatorname{diam}}\,B\leq\max\\{{\operatorname{diam}}\,A,{\operatorname{diam}}\,B\\}+2\varepsilon$.
The latter inequality is not valid for
$2\varepsilon<{\operatorname{min}}\,\\{{\operatorname{diam}}\,A,{\operatorname{diam}}\,B\\}$,
a contradiction. ∎

######  Theorem 1.

Each sphere centered at the single-point metric space $\Delta_{1}$ is path
connected.

###### Proof.

Let $S=S_{r}(\Delta_{1})$ be the sphere of radius $r>0$ centered at the
single-point space $\Delta_{1}$. Then $S\subset{\cal B}$. Consider arbitrary
$A,\,B\in S$ and fix an $\varepsilon$,
$0<2\varepsilon<{\operatorname{min}}\,\\{{\operatorname{diam}}\,A,{\operatorname{diam}}\,B\\}$.
Consider an $\varepsilon$-shortest curve $\gamma(t)$ connecting $A$ and $B$.
According to Lemmas 2.3 and 2.6, the diameter of each space $\gamma(t)$ is
finite and does not equal zero. Define a mapping
$\delta\colon[a,b]\to\mathcal{{\cal B}}$ as follows: $\delta\colon
t\mapsto\bigl{(}2r/{\operatorname{diam}}\,\gamma(t)\bigr{)}\gamma(t)$. Due to
Lemma 2.1, the function $2r/{\operatorname{diam}}\,\gamma(t)$ is continuous,
and so, due to Lemma 2.5, we conclude that $\delta(t)$ is a continuous curve.
And ${\operatorname{diam}}\,\delta(t)=2r$, therefore the curve $\delta$ lies
in the sphere $S$. Theorem is proved. ∎

If $A,\,B\in S_{r}(\Delta_{1})$ are compact spaces, then one can change the
$\varepsilon$-shortest curve in the proof of Theorem 1 by a shortest curve in
$\mathcal{M}$. This implies the following result.

######  Corollary 2.7.

Any sphere $S_{r}(\Delta_{1})$ in $\mathcal{M}$ centered at the single-point
compact space $\Delta_{1}$ is linear connected.

###  2.3 Path Connectivity of Large Spheres Centered at an Arbitrary Bounded
Metric Space

We start with several technical results. The next lemma is a special kind of
Implicit Function Theorem.

######  Lemma 2.8.

Let $F\colon[T_{0},T_{1}]\times[S_{0},S_{1}]\to\mathbb{R}$ be a continuous
function that satisfies the following conditions:

  1. (1)

For each $s\in[S_{0},S_{1}]$, the function $f_{s}(t)=F(t,s)$ is strictly
monotonic;

  2. (2)

There exists an $r$ such that for all $s\in[S_{0},S_{1}]$ the inequality
$F[T_{0},s]\leq r\leq F[T_{1},s]$ holds. Then the set
$\big{\\{}(t,s):F(t,s)=r\big{\\}}$ is an image of embedded continuous curve of
the form $\gamma(s)=\big{(}t(s),s\big{)}$.

###### Proof.

Since the function $f_{s}(t)$ is strictly monotonic and continuous, and
$f_{s}(T_{0})\leq r\leq f_{s}(T_{1})$, then for each $s$ there exists unique
$t=t(s)$ such that $f_{s}\big{(}t(s)\big{)}=r$. Put
$\gamma(s)=\big{(}t(s),s\big{)}$. Notice that $F\big{(}\gamma(s)\big{)}=r$ by
definition of $\gamma$. It remains to show that function $t(s)$ is continuous.

Assume to the contrary that $t(s)$ is discontinuous at some point $s_{0}$, and
let $t_{0}=t(s_{0})$. Then there exists an $\varepsilon>0$ and a sequence
$s_{i}\to s_{0}$ as $i\to\infty$ such that
$\bigl{|}t_{0}-t(s_{i})\bigr{|}\geq\varepsilon$. Due to compactness arguments,
the sequence $t_{i}=t(s_{i})$ contains a convergent subsequence. Assume
without loss of generality that the entire sequence $t_{i}$ converges to some
$t^{\prime}$. Since $|t_{0}-t_{i}|\geq\varepsilon$ for all $i$, then
$t^{\prime}\neq t_{0}$. Since $F$ is continuous and $F(s_{i},t_{i})=r$, then
$F(s_{0},t^{\prime})=r$ too. But $F(s_{0},t_{0})$ is also equal to $r$, that
is impossible, because $f_{s_{0}}$ is strictly monotonic, a contradiction. ∎

######  Lemma 2.9.

For a pair $C,G\in{\cal B}$ of bounded metric spaces with
$2d_{GH}(G,C)>{\operatorname{diam}}\,G>0$, the function
$h(\lambda):=d_{GH}(G,\lambda C)$ is strictly monotonically increasing for
$\lambda\geq 1$.

###### Proof.

Due to assumptions, for any correspondence $R\in\mathcal{R}(G,C)$ we have
${\operatorname{dis}}\,R>{\operatorname{diam}}\,G$. Consider a sequence
$\bigl{(}(g_{i},c_{i}),(g^{\prime}_{i},c^{\prime}_{i})\bigr{)}\in R\times R$
such that
$\bigl{|}|c_{i}c^{\prime}_{i}|-|g_{i}g^{\prime}_{i}|\bigr{|}\to{\operatorname{dis}}\,R$.
Since ${\operatorname{dis}}\,R>{\operatorname{diam}}\,G$, then the inequality
$|c_{i}c^{\prime}_{i}|>|g_{i}g^{\prime}_{i}|$ is valid for sufficiently large
$i$. Assume without loss of generality that it holds for all $i$.

By $R_{\lambda}$ we denote the correspondence from $\mathcal{R}(G,\lambda C)$
that coincides with $R$ as a subset of $G\times C$. Then for $\lambda\geq 1$
we have

${\operatorname{dis}}\,R_{\lambda}\geq\lambda|c_{i}c^{\prime}_{i}|-|g_{i}g^{\prime}_{i}|=(\lambda-1)|c_{i}c^{\prime}_{i}|+|c_{i}c^{\prime}_{i}|-|g_{i}g^{\prime}_{i}|,$

and therefore, passing to the limit as $i\to\infty$, we conclude that

${\operatorname{dis}}\,R_{\lambda}\geq(\lambda-1)\liminf_{i\to\infty}|c_{i}c^{\prime}_{i}|+{\operatorname{dis}}\,R.$

Since ${\operatorname{dis}}\,R>{\operatorname{diam}}\,G>0$, then
$|c_{i}c^{\prime}_{i}|>|g_{i}g^{\prime}_{i}|+\frac{1}{2}{\operatorname{diam}}\,G$
for large $i$, therefore,
$\liminf_{i\to\infty}|c_{i}c^{\prime}_{i}|\geq\frac{1}{2}\,{\operatorname{diam}}\,G$,
and hence,
${\operatorname{dis}}\,R_{\lambda}\geq{\operatorname{dis}}\,R+\frac{\lambda-1}{2}\,{\operatorname{diam}}\,G$.
Since the latter inequality is valid for all correspondences $R\in G\times C$,
we conclude that $d_{GH}(G,\lambda C)\geq
d_{GH}(G,C)+\frac{\lambda-1}{4}\,{\operatorname{diam}}\,G$, and so
$d_{GH}(G,\lambda C)>d_{GH}(G,C)$ for $\lambda>1$. In particular, each
$\lambda C$, $\lambda\geq 1$, taking instead of $C$, satisfies the conditions
of the lemma. So, choosing arbitrary $\lambda_{1},\,\lambda_{2}$,
$1\leq\lambda_{1}<\lambda_{2}$, and substituting to the inequality
$\lambda_{1}C$ and $\lambda_{2}/\lambda_{1}$ instead of $C$ and $\lambda$,
respectively, we get $d_{GH}(G,\lambda_{2}C)>d_{GH}(G,\lambda_{1}C)$. ∎

Let us now prove the path connectivity of spheres of a sufficiently large
radius centered at an arbitrary $G\in{\cal B}$.

######  Theorem 2.

For any bounded metric space $G\in{\cal B}$ and any
$r>{\operatorname{diam}}\,G$, the sphere $S_{r}(G)$ is path connected.

###### Proof.

Since ${\operatorname{diam}}\,G<\infty$, then $S_{r}(G)\subset{\cal B}$, see
Assertion 1.4. Since the case ${\operatorname{diam}}\,G=0$ has been considered
in Theorem 1, assume that ${\operatorname{diam}}\,G>0$.

######  Lemma 2.10.

The ball $B_{r}(G)$ lies inside the ball $B_{3r/2}(\Delta_{1})$.

###### Proof.

Indeed, if $X\in B_{r}(G)$, then according to the triangle inequality, we have

$d_{GH}(\Delta_{1},X)\leq
d_{GH}(\Delta_{1},G)+d_{GH}(G,X)\leq\frac{1}{2}\,{\operatorname{diam}}\,G+r<\frac{3r}{2},$

and hence, $X\in B_{3r/2}(\Delta_{1})$. ∎

######  Lemma 2.11.

The sphere $S_{r}(G)$ lies outside the ball $B_{r/2}(\Delta_{1})$.

###### Proof.

Indeed, if $X\in S_{r}(G)$, then, due to the triangle inequality,

$d_{GH}(\Delta_{1},X)\geq d_{GH}(X,G)-d_{GH}(G,\Delta_{1})>r-r/2=r/2,$

and so, $d_{GH}(\Delta_{1},X)>r/2$. ∎

Consider a pair of arbitrary points $A$ and $B$ lying in the sphere
$S_{r}(G)$. Due to Lemma 2.11, ${\operatorname{diam}}\,A>r$ and
${\operatorname{diam}}\,B>r$, therefore, there exists $\lambda_{A}>0$ and
$\lambda_{B}>0$ such that ${\operatorname{diam}}\,\lambda_{A}A=r$ and
${\operatorname{diam}}\,\lambda_{B}B=r$, i.e., the points
$A^{\prime}=\lambda_{A}A$ and $B^{\prime}=\lambda_{B}B$ lie in the sphere
$S_{r/2}(\Delta_{1})$. Due to Theorem 1, there exists a continuous curve
$\gamma\colon[a,b]\to{\cal B}$ lying in $S_{r/2}(\Delta_{1})$ and such that
$A^{\prime}=\gamma(a)$ and $B^{\prime}=\gamma(b)$. But then, according to
Lemma 2.5, the points $A^{\prime\prime}=3A^{\prime}$ and
$B^{\prime\prime}=3B^{\prime}$ can be connected by a continuous curve
$3\gamma(s)$ lying in $S_{3r/2}(\Delta_{1})$.

Since

$2d_{GH}\bigl{(}G,\gamma(s)\bigr{)}\geq{\operatorname{diam}}\,\gamma(s)-{\operatorname{diam}}\,G>r>{\operatorname{diam}}\,G,$

then Lemma 2.9 can be applied to the points $\gamma(s)$, and according to this
Lemma, the function $f_{s}(t)=d_{GH}\bigl{(}G,t\,\gamma(s)\bigr{)}$,
$t\in[1,3]$, is strictly monotonically increasing. Due to Lemmas 2.10 and
2.11, for each $s$ there exists $t$ such that $f_{s}(t)=r$. Let us put
$F(t,s)=d_{GH}\bigl{(}G,t\,\gamma(s)\bigr{)}$. Applying Lemma 2.8, we obtain a
continuous curve lying in $S_{r}(G)$ and connecting $A$ and $B$. Theorem is
proved. ∎

If $G$ is a compact space, then one can consider a sphere $S_{r}(G)$ in the
Gromov–Hausdorff space $\mathcal{M}$. It is easy to see that in this case all
the curves constructed in the proof of Theorem 2 lie in $\mathcal{M}$.

######  Corollary 2.12.

For each compact metric space $G\in\mathcal{M}$ and each
$r>{\operatorname{diam}}\,G$, the sphere $S_{r}(G)$ in $\mathcal{M}$ is path
connected.

###  2.4 Generic Metric Spaces

Let $X$ be a metric space, $\\#X\geq 3$. By $S(X)$ we denote the set of all
bijective mappings of the set $X$ onto itself, and let $\operatorname{id}\in
S(X)$ be the identical bijection. Put

$\displaystyle s(X)$ $\displaystyle=\inf\bigl{\\{}|xx^{\prime}|:x\neq
x^{\prime},\ x,x^{\prime}\in X\bigr{\\}},$ $\displaystyle t(X)$
$\displaystyle=\inf\bigl{\\{}|xx^{\prime}|+|x^{\prime}x^{\prime\prime}|-|xx^{\prime\prime}|:x\neq
x^{\prime}\neq x^{\prime\prime}\neq x\bigr{\\}},$ $\displaystyle e(X)$
$\displaystyle=\inf\bigl{\\{}{\operatorname{dis}}\,f:f\in S(X),\
f\neq\operatorname{id}\bigr{\\}}.$

We call a metric space $M\in\operatorname{\mathcal{G\\!H}}$, $\\#M\geq 3$,
_generic_ if all the three its characteristics $s(M)$, $t(M)$, and $e(M)$ are
positive.

The following generalization of results from [18] and [19] was obtained by A.
Filin in his diploma.

######  Assertion 2.13.

Let $M,X\in\operatorname{\mathcal{G\\!H}}$ be metric spaces such that
$\\#M\geq 3$ and $r:=d_{GH}(M,X)<s(M)/2$. Then for any real $d$ such that
$r<d\leq s(M)/2$, the following statements are valid.

  1. (1)

There exists a correspondence $R\in\mathcal{R}(M,X)$ such that
${\operatorname{dis}}\,R<2d\leq s(M)$.

  2. (2)

For each such $R$, the family $D_{R}=\bigl{\\{}X_{i}:=R(i)\bigr{\\}}_{i\in M}$
is a partition of the space $X$.

  3. (3)

For any $i,j\in M$, probably coinciding, and for any $x_{i}\in X_{i}$ and
$x_{j}\in X_{j}$, the estimate $\bigl{|}|x_{i}x_{j}|-|ij|\bigr{|}<2d\leq s(M)$
holds.

  4. (4)

For all $i\in M$, the inequality ${\operatorname{diam}}\,X_{i}<2d\leq s(M)$
holds.

  5. (5)

If $d\leq s(M)/4$, then the partition $D_{R}$ is uniquely defined, i.e., if
$R^{\prime}\in\mathcal{R}(M,X)$ is such that
${\operatorname{dis}}\,R^{\prime}<2d$, then $D_{R^{\prime}}=D_{R}$.

  6. (6)

If $d\leq{\operatorname{min}}\,\bigl{\\{}s(M)/4,\,e(M)/4\bigr{\\}}$, then the
correspondence $R\in\mathcal{R}(M,X)$ such that ${\operatorname{dis}}\,R<2d$
is uniquely defined, and hence, $R$ is an optimal correspondence. In this
case, ${\operatorname{dis}}\,R=2d_{GH}(M,X)$,
${\operatorname{diam}}\,X_{i}\leq{\operatorname{dis}}\,R$, and
$\bigl{|}|x_{i}x^{\prime}_{j}|-|ij|\bigr{|}\leq{\operatorname{dis}}\,R$ for
all $i,j\in M$, $i\neq j$.

###### Proof.

(1) Since

$d_{GH}(M,X)=\frac{1}{2}\inf\bigl{\\{}{\operatorname{dis}}\,R:R\in\mathcal{R}(M,X)\bigr{\\}}<d,$

then there exists a correspondence $R\in\mathcal{R}(M,X)$ such that
${\operatorname{dis}}\,R<2d\leq s(M)$.

(2) If for some $x\in X$ one has $\\#R^{-1}(x)>1$, then
${\operatorname{dis}}\,R\geq{\operatorname{diam}}\,R^{-1}(x)\geq s(M)$, a
contradiction. Therefore, the family $\big{\\{}X_{i}:=R(i)\big{\\}}_{i\in M}$
is a partition of the space $X$.

(3) Since ${\operatorname{dis}}\,R<2d$, then for any $i,j\in M$ and any
$x_{i}\in R(i)$, $x_{j}\in R(j)$ we have:

$\bigl{|}|x_{i}x_{j}|-|ij|\bigr{|}\leq{\operatorname{dis}}\,R<2d\leq s(M).$

(4) Since ${\operatorname{dis}}\,R<2d$, then for any $i\in M$ we have
${\operatorname{diam}}\,X_{i}\leq{\operatorname{dis}}\,R<2d\leq s(M)$.

(5) If $d\leq s(M)/4$, and $R^{\prime}\in\mathcal{R}(M,X)$ is another
correspondence such that ${\operatorname{dis}}\,R^{\prime}<2d$, then
$\big{\\{}X^{\prime}_{i}:=R^{\prime}(i)\big{\\}}_{i\in M}$ is also a partition
of $X$ satisfying the properties already proved above. If $D_{R}\neq
D_{R^{\prime}}$, then either there exists an $X^{\prime}_{i}$ intersecting
simultaneously some different $X_{j}$ and $X_{k}$, or there exists an $X_{i}$
intersecting simultaneously some different $X^{\prime}_{j}$ and
$X^{\prime}_{k}$. Indeed, if there is no such an $X^{\prime}_{i}$, then each
$X^{\prime}_{i}$ is contained in some $X_{j}$, and hence,
$\\{X^{\prime}_{p}\\}$ is a sub-partition of $\\{X_{q}\\}$. Since these
partitions are different, then some element $X_{i}$ contains some different
$X^{\prime}_{j}$ and $X^{\prime}_{k}$.

Thus, assume without loss of generality that $X^{\prime}_{i}$ intersects some
different $X_{j}$ and $X_{k}$ simultaneously. Choose arbitrary $x_{j}\in
X_{j}\cap X^{\prime}_{i}$ and $x_{k}\in X_{k}\cap X^{\prime}_{i}$. Then

$2d>{\operatorname{diam}}\,X^{\prime}_{i}\geq|x_{j}x_{k}|>|jk|-2d\geq
s(M)-2d,$

and hence, $d>s(M)/4$, a contradiction.

(6) Let $R^{\prime}\in\mathcal{R}(M,X)$ satisfy
${\operatorname{dis}}\,R^{\prime}<2d$ as in previous Item. Then
$D_{R}=D_{R^{\prime}}$, as it is already proved above. It remains to verify
that $X_{i}=X^{\prime}_{i}$ for each $i\in M$. Assume to the contrary that
there exists a non-trivial bijection $f\colon M\to M$ such that
$X_{i}=X^{\prime}_{f(i)}$. Due to assumptions, ${\operatorname{dis}}\,f\geq
e(M)\geq 4d$. The latter means that there exist $i,j\in M$ such that

$\Bigl{|}|ij|-\bigl{|}f(i)f(j)\bigr{|}\Bigr{|}\geq 4d.$

Let us put $p=f(i)$, $q=f(j)$, and choose arbitrary $x_{i}\in
X_{i}=X^{\prime}_{p}$ and $x_{j}\in X_{j}=X^{\prime}_{q}$. Then

$4d\leq\bigl{|}|pq|-|ij|\bigr{|}=\bigl{|}|pq|-|x_{i}x_{j}|+|x_{i}x_{j}|-|ij|\bigr{|}\leq\bigl{|}|pq|-|x_{i}x_{j}|\bigr{|}+\bigl{|}|x_{i}x_{j}|-|ij|\bigr{|}<2d+2d,$

a contradiction. Thus, a correspondence $R$ whose distortion is close enough
to $\frac{1}{2}d_{GH}(M,X)$ is uniquely defined. So, $R$ is an optimal
correspondence. ∎

######  Lemma 2.14.

Let $M$ be a metric space, and $s(M)>0$. For each pair $\\{i,j\\}\subset M$,
fix a real number $a_{ij}=a_{ji}$ in such a way that $a_{ii}=0$ and
$|a_{ij}|<s(M)$. Change the distance function on $M$ by adding the value
$a_{ij}$ to each distance $|ij|$ and denote by $\rho$ the resulting function.
Put $a=\sup|a_{ij}|$. If $a\leq t(M)/3$, then the following statements are
valid.

  1. (1)

The distance function $\rho$ is a metric on the set $M$.

  2. (2)

For the resulting metric space $M_{\rho}=(M,\rho)$, the inequality
$d_{GH}(M,M_{\rho})\leq a/2$ holds.

  3. (3)

If in addition
$a/2<{\operatorname{min}}\,\bigl{\\{}s(M)/4,\,e(M)/4\bigr{\\}}$, then
$d_{GH}(M,M_{\rho})=a/2$.

  4. (4)

Under assumptions of Item $(\ref{lem:M_sigma:3})$, if the equality $a_{ij}=\pm
a$ holds for all $i\neq j$, then $M_{\rho}$ lies in the sphere $S_{a/2}(M)$;
moreover, if $M_{\rho^{\prime}}$ is constructed in a similar way by the set
$a^{\prime}_{ij}=\pm a$, $i\neq j$, and $\rho_{t}=(1-t)\rho+t\,\rho^{\prime}$,
$t\in[0,1]$, then the set $M$ with the distance function $\rho_{t}$ is a
metric space, the mapping $t\mapsto M^{t}:=(M,\rho_{t})$ is a continuous curve
in $\operatorname{\mathcal{G\\!H}}$, and, if for some pair $\\{i,j\\}$, $i\neq
j$, the equality $a_{ij}=a^{\prime}_{ij}$ holds, then all the spaces $M^{t}$
lie in the sphere $S_{a/2}(M)$.

###### Proof.

(1) It is sufficient to verify that the distance function $\rho$ satisfies the
triangle inequalities. Choose arbitrary three points $i,j,k\in M$, then

$\rho(i,j)+\rho(j,k)-\rho(i,k)=|ij|+a_{ij}+|jk|+a_{jk}-|ik|-a_{ik}\geq
t(M)-3a\geq 0.$

(2) Let $R\in\mathcal{R}(M,M_{\rho})$ be the identical mapping, then

$2d_{GH}(M,M_{\rho})\leq{\operatorname{dis}}\,R=\sup\Bigl{\\{}\bigl{|}|ij|-|ij|-a_{ij}\bigr{|}:i,j\in
M\Bigr{\\}}=\sup\bigl{\\{}|a_{ij}|:i,j\in M\bigr{\\}}=a.$

(3) The assumptions imply that Assertion 2.13, Item (6), can be applied, and
hence, the identical mapping $R$ is an optimal correspondence. The latter
implies the equality required.

(4) Proposition 1.1 implies that $M^{t}$ is a metric space. Further, let
$R^{t}\in\mathcal{R}(M,M^{t})$ be the identical mapping, then
${\operatorname{dis}}\,R^{t}\leq a$, and so $R^{t}$ is an optimal
correspondence. Let $R^{\prime}\in\mathcal{R}(M^{t},M^{s})$ be the identical
mapping too. Then

$2d_{GH}(M^{t},M^{s})\leq{\operatorname{dis}}\,R^{\prime}=|t-s|\,\sup_{i,j\in
M}|a_{ij}-a^{\prime}_{ij}|\leq 2a|t-s|,$

therefore, $M^{t}$ is a continuous curve in $\operatorname{\mathcal{G\\!H}}$.
It remains to notice that if $a_{ij}=a^{\prime}_{ij}$ for some $i\neq j$, then

$\bigl{|}\rho_{t}(i,j)-|ij|\bigr{|}=\bigl{|}(1-t)a_{ij}+t\,a^{\prime}_{ij}\bigr{|}=a,$

so ${\operatorname{dis}}\,R^{t}=a$ for all $t$, and hence,
$d_{GH}(M,M^{t})=a/2$. ∎

###  2.5 Path Connectivity of Small Spheres Centered at Generic Spaces

We need the following notations. Let $A$ and $B$ be non-empty subsets of a
metric space $X$. We put

$|AB|=\inf\bigl{\\{}|ab|:a\in A,\,b\in
B\bigr{\\}},\qquad|AB|^{\prime}=\sup\bigl{\\{}|ab|:a\in A,\,b\in B\bigr{\\}}.$

If we need to emphasis that these values are calculated with respect to a
metric $\rho$, then we write $|AB|_{\rho}$ and $|AB|^{\prime}_{\rho}$ instead
of $|AB|$ and $|AB|^{\prime}$, respectively. Similarly, we sometimes write
${\operatorname{diam}}\,_{\rho}A$ instead of ${\operatorname{diam}}\,A$ by the
same reason.

######  Theorem 3.

Let $M$ be a generic metric space and

$0<r<{\operatorname{min}}\,\bigl{\\{}s(M)/4,\,e(M)/4,\,t(M)/6\bigr{\\}}.$

Then the sphere $S_{r}(M)$ in $\operatorname{\mathcal{G\\!H}}$ is path
connected.

###### Proof.

Choose an arbitrary $X\in S_{r}(M)$. Since the conditions of Assertion 2.13,
Item (6), are valid, then there exists a unique optimal correspondence
$R\in\mathcal{R}(M,X)$. Moreover, ${\operatorname{dis}}\,R=2r$ and, besides,
the correspondence $R$ defines the partition
$D_{R}=\big{\\{}X_{i}:=R(i)\big{\\}}_{i\in M}$ of the space $X$ into subsets
$X_{i}$, whose diameters are at mots $2r$ and such that for any $x_{i}\in
X_{i}$ and $x_{j}\in X_{j}$, the inequality
$\bigl{|}|x_{i}x_{j}|-|ij|\bigr{|}\leq 2r$ holds.

It is easy to see that the distortion of the correspondence $R$ can be
calculated by the following formula:

${\operatorname{dis}}\,R=\max\Bigl{[}\sup_{i}\\{{\operatorname{diam}}\,X_{i}\\},\,\sup_{i\neq
j}\bigl{\\{}|X_{i}X_{j}|^{\prime}-|ij|\bigr{\\}},\,\sup_{i\neq
j}\bigl{\\{}|ij|-|X_{i}X_{j}|\bigr{\\}}\Bigr{]}.$

We consider three possibilities depending on which of the three items the
maximum attains at.

(1) Let ${\operatorname{dis}}\,R=\sup_{i}\\{{\operatorname{diam}}\,X_{i}\\}$.
Define a distance function $\rho$ on the set $X$ as follows: the distances
between the points in each $X_{i}$ do not change, and for $x_{i}\in X_{i}$ and
$x_{j}\in X_{j}$, $i\neq j$, put $\rho(x_{i},x_{j})=|ij|+2r$. Let us show that
$\rho$ is a metric. To do that, it suffices to verify the triangle
inequalities for the points $x_{i},x^{\prime}_{i}\in X_{i}$ and $x_{j}\in
X_{j}$, $j\neq i$. Since the triangle $x_{i}x^{\prime}_{i}x_{j}$ is
equilateral, it suffices to show that $\rho(x_{i},x^{\prime}_{i})\leq
2\rho(x_{i},x_{j})$. We have

$2\rho(x_{i},x_{j})-\rho(x_{i},x^{\prime}_{i})=2|ij|+4r-|x_{i}x^{\prime}_{i}|=\bigl{(}2|ij|-|x_{i}x^{\prime}_{i}|\bigr{)}+4r>0.$

Notice that $|X_{i}X_{j}|_{\rho}=|X_{i}X_{j}|^{\prime}_{\rho}=|ij|+2r$ for any
$i\neq j$.

Define the functions $\rho_{t}$, $t\in[0,1]$, that coincide with the initial
metric on all $X_{i}$ and equal
$\rho_{t}(x_{i},x_{j})=(1-t)|x_{i}x_{j}|+t\bigl{(}|ij|+2r\bigr{)}$ for all
$x_{i}\in X_{i}$ and $x_{j}\in X_{j}$, $i\neq j$. All the functions $\rho_{t}$
are metrics according to Proposition 1.1. By $X^{t}$ we denote the metric
space $(X,\rho_{t})$. If $R^{\prime}\in\mathcal{R}(X^{t},X^{s})$ is the
identical mapping, then

${\operatorname{dis}}\,R^{\prime}=|t-s|\sup\Bigl{\\{}\bigl{|}|x_{i}x_{j}|-|ij|-2r\bigr{|}:i,j\in
M,\,i\neq j\Bigr{\\}}\leq 4r\,|t-s|,$

and so $X^{t}$ is a continuous curve in $\operatorname{\mathcal{G\\!H}}$.

Further, let us show that all the spaces $X^{t}$ lie in the sphere $S_{r}(M)$.
Since the metric $\rho_{t}$ coincides with the initial one at each $X_{i}$,
then ${\operatorname{diam}}\,_{\rho_{t}}X_{i}={\operatorname{diam}}\,X_{i}$.
Further, for $i\neq j$, the function
$\rho_{t}(x_{i},x_{j})=|x_{i}x_{j}|+t\bigl{(}|ij|+2r-|x_{i}x_{j}|\bigr{)}$
increases monotonically as $t$ increases, and

$|x_{i}x_{j}|\leq\rho_{t}(x_{i},x_{j})\leq|ij|+2r.$

Therefore,

$-2r\leq|X_{i}X_{j}|^{\prime}-|ij|\leq|X_{i}X_{j}|^{\prime}_{\rho_{t}}-|ij|\leq
2r\quad\text{and}\quad
2r\geq|ij|-|X_{i}X_{j}|\geq|ij|-|X_{i}X_{j}|_{\rho_{t}}\geq-2r.$

So, we conclude that

$\sup_{i\neq j}\bigl{\\{}|X_{i}X_{j}|^{\prime}_{\rho_{t}}-|ij|\bigr{\\}}\leq
2r\quad\text{and}\quad\sup_{i\neq
j}\bigl{\\{}|ij|-|X_{i}X_{j}|_{\rho_{t}}\bigr{\\}}\leq 2r.$

But
$\sup_{i}\\{{\operatorname{diam}}\,_{\rho_{t}}X_{i}\\}=\sup_{i}\\{{\operatorname{diam}}\,X_{i}\\}=2r$,
therefore, if we take the correspondence $R^{t}\in\mathcal{R}(M,X^{t})$
coinciding with $R$, then

${\operatorname{dis}}\,R^{t}=\max\Bigl{[}\sup_{i}\\{{\operatorname{diam}}\,_{\rho_{t}}X_{i}\\},\,\sup_{i\neq
j}\bigl{\\{}|X_{i}X_{j}|^{\prime}_{\rho_{t}}-|ij|\bigr{\\}},\,\sup_{i\neq
j}\bigl{\\{}|ij|-|X_{i}X_{j}|_{\rho_{t}}\bigr{\\}}\Bigr{]}=2r.$

So, each space $X^{t}$ satisfies the conditions of Assertion 2.13, Item (6),
and hence, $R^{t}$ is an optimal correspondence. Thus,
$2d_{GH}(M,X^{t})={\operatorname{dis}}\,R^{t}=2r$, i.e., the curve $X^{t}$
lies in $S_{r}(M)$.

Now let us deform the metric on the space $X^{1}$ as follows: for points
$x,\,x^{\prime}$ lying in the same $X_{i}$, we put
$\nu_{t}(x,x^{\prime})=(1-t)|xx^{\prime}|$, $t\in[0,1]$, and for points lying
in different sets $X_{i}$, we do not change the metric. It is easy to see that
all $\nu_{t}$ are metrics, and for the metric space $X^{1}_{t}=(X,\nu_{t})$,
the next relations are valid:

$\displaystyle{\operatorname{diam}}\,_{\nu_{t}}X_{i}=(1-t)\,{\operatorname{diam}}\,X_{i}\leq
2r,$
$\displaystyle|X_{i}X_{j}|^{\prime}_{\nu_{t}}-|ij|=|X_{i}X_{j}|^{\prime}_{\rho}-|ij|=2r,$
$\displaystyle|ij|-|X_{i}X_{j}|_{\nu_{t}}=|ij|-|X_{i}X_{j}|_{\rho}=-2r.$

Consider the correspondence $R$ as a correspondence between $M$ and
$X^{1}_{t}$, and denote it by $R^{t}_{1}\in\mathcal{R}(M,X^{1}_{t})$. Then
${\operatorname{dis}}\,R^{t}_{1}=2r$, therefore, $R^{t}_{1}$ is an optimal
correspondence, and $d_{GH}(M,X^{1}_{t})=r$, i.e., all the spaces $X^{1}_{t}$
lie in the sphere $S_{r}(M)$. The same arguments as above show that the
mapping $t\mapsto X^{1}_{t}$ is a continuous curve in
$\operatorname{\mathcal{G\\!H}}$. The space $X^{1}_{1}$ is obtained from $M$
by adding the value $2r$ to all the distances $|ij|$, $i\neq j$. This space is
denoted by $M^{+}$.

Thus, we have constructed a continuous curve in
$S_{r}(M)\subset\operatorname{\mathcal{G\\!H}}$ connecting the space $X\in
S_{r}(M)$ with the space $M^{+}\in S_{r}(M)$.

(2) Let ${\operatorname{dis}}\,R=\sup_{i\neq
j}\bigl{\\{}|X_{i}X_{j}|^{\prime}-|ij|\bigr{\\}}$. Consider the same family
$\rho_{t}$ of metrics as in Case (1). Since

$|X_{i}X_{j}|^{\prime}-|ij|\leq|X_{i}X_{j}|^{\prime}_{\rho_{t}}-|ij|\leq 2r,$

and $\sup_{i\neq j}\bigl{\\{}|X_{i}X_{j}|^{\prime}-|ij|\bigr{\\}}=2r$, then
$\sup_{i\neq j}\bigl{\\{}|X_{i}X_{j}|^{\prime}_{\rho_{t}}-|ij|\bigr{\\}}=2r$
for all $t$. As it is shown in Case (1),
${\operatorname{diam}}\,_{\rho_{t}}X_{i}={\operatorname{diam}}\,X_{i}\leq 2r$
and $\big{|}|ij|-|X_{i}X_{j}|_{\rho_{t}}\big{|}\leq 2r$, therefore, the family
$X^{t}=(X,\rho_{t})$ lies in the sphere $S_{r}(M)$. The arguments from Case
(1) prove the continuity of the curve $t\mapsto X^{t}$, whose ending point can
be connected with the space $M^{+}\in S_{r}(M)$ by means of the same
construction. Thus, again we have constructed a continuous curve in
$S_{r}(M)\subset\operatorname{\mathcal{G\\!H}}$ connecting the space $X\in
S_{r}(M)$ with the space $M^{+}\in S_{r}(M)$.

(3) Let ${\operatorname{dis}}\,R=\sup\bigl{\\{}|ij|-|X_{i}X_{j}|:i\neq
j\bigr{\\}}$. In this case, we define the metric $\rho$ in a different way,
namely, we put $\rho(x_{i},x_{j})=|ij|-2r$ for $i\neq j$, and
$\rho_{t}(x_{i},x_{j})=(1-t)|x_{i}x_{j}|+t\bigl{(}|ij|-2r\bigr{)}$. Let us
show that $\rho$ is a metric too. The value $|ij|-2r$ is positive, because
$r<s(M)/4$, and hence $|ij|-2r>|ij|-s(M)/2\geq s(M)/2>0$. It remains to verify
the triangle inequalities. If all three points belong to the same $X_{i}$,
then the inequality coincides with the one in $X$, and hence it holds.
Further, if $x_{i},x^{\prime}_{i}\in X_{i}$ and $x_{j}\in X_{j}$, $j\neq i$,
then the triangle $x_{i}x^{\prime}_{i}x_{j}$ is equilateral, and it suffices
to show that $\rho(x_{i},x^{\prime}_{i})\leq 2\rho(x_{i},x_{j})$. Since
$r<s(M)/4$, we have

$2\rho(x_{i},x_{j})-\rho(x_{i},x^{\prime}_{i})=2|ij|-4r-|x_{i}x^{\prime}_{i}|>2|ij|-s(M)-|x_{i}x^{\prime}_{i}|=\big{(}|ij|-s(M)\big{)}+\big{(}|ij|-|x_{i}x^{\prime}_{i}|\big{)}.$

It remains to notice that $|ij|-s(M)\geq 0$ by definition of $s(M)$, and
$|ij|-|x_{i}x^{\prime}_{i}|>2r$ because ${\operatorname{diam}}\,X_{i}\leq 2r$
and $|ij|\geq s(M)>4r$. At last, if $x_{i}\in X_{i}$, $x_{j}\in X_{j}$, and
$x_{k}\in X_{k}$ for pairwise distinct $i,\,j,\,k$, then

$\rho(x_{i},x_{j})+\rho(x_{j},x_{k})-\rho(x_{i},x_{k})=|ij|+|jk|-|ik|-2r\geq
t(M)-2r>6r-2r=4r>0.$

Thus, $\rho$ is a metric. Notice that
$|X_{i}X_{j}|_{\rho}=|X_{i}X_{j}|^{\prime}_{\rho}=|ij|-2r$ for any $i\neq j$.

In this case the function
$\rho_{t}(x_{i},x_{j})=|x_{i}x_{j}|-t\bigl{(}2r+|x_{i}x_{j}|-|ij|\bigr{)}$
decreases monotonically as $t$ increases. Since

$|ij|-|X_{i}X_{j}|\leq|ij|-|X_{i}X_{j}|^{\prime}_{\rho_{t}}\leq 2r,$

and $\sup_{i\neq j}\bigl{\\{}|ij|-|X_{i}X_{j}|\bigr{\\}}=2r$, then
$\sup_{i\neq j}\bigl{\\{}|ij|-|X_{i}X_{j}|_{\rho_{t}}\bigr{\\}}=2r$ for all
$t$. Reasonings similar to Cases (1) and (2) show that
${\operatorname{diam}}\,_{\rho_{t}}X_{i}={\operatorname{diam}}\,X_{i}\leq 2r$
and $\big{|}|X_{i}X_{j}|^{\prime}_{\rho_{t}}-|ij|\big{|}\leq 2r$, therefore,
the family $X^{t}=(X,\rho_{t})$ forms a continuous curve lying in the sphere
$S_{r}(M)$. Further, the arguments similar to the ones given in the previous
cases allow us to construct a continuous curve in
$\operatorname{\mathcal{G\\!H}}$ connecting the space $X\in S_{r}(M)$ with the
space $M^{-}\in S_{r}(M)$ by a curve lying in the sphere $S_{r}(M)$. Here the
space $M^{-}$ is obtained from the space $M$ by subtracting the value $2r$
from all the distances $|ij|$, $i\neq j$.

To conclude the proof, we apply Lemma 2.14, Item (4), and construct continuous
curves in $S_{r}$ connecting the spaces $M^{+}$ and $M^{-}$ with the space
$M^{\pm}$, where some nonzero distances have the form $|ij|+2r$, $i\neq j$,
and the remaining ones have the form $|ij|-2r$, $i\neq j$. That can be done
since $\\#M\geq 3$ according to the assumptions. Theorem is proved. ∎

If the space $M$ is finite, and $X$ is a compact metric space, then it is easy
to see that all the curves constructed in the proof of Theorem 3 lies in
$\mathcal{M}$. Thus, the following result holds.

######  Corollary 2.15.

Let $M$ be a finite generic space, and

$0<r<{\operatorname{min}}\,\bigl{\\{}s(M)/4,\,e(M)/4,\,t(M)/6\bigr{\\}}.$

Then the sphere $S_{r}(M)$ in $\mathcal{M}$ is path connected.

In conclusion, we give several examples of generic spaces.

######  Example.

A finite metric space $M$ with pairwise distinct non-zero distances obviously
has $s(M)>0$ and $e(M)>0$. Adding a positive constant $\varepsilon$ to all
non-zero distances we get $t(M)\geq\varepsilon$ in addition.

The following elegant construction has been suggested to us by Konstantin
Shramov and gives an opportunity to obtain a generic metric space of an
arbitrary cardinality.

######  Example.

Consider an arbitrary infinite set $X$. It is well-known (Zermelo’s theorem)
that any set can be well-ordered, i.e., there exists a total order such that
any subset contains a least element. It is easy to show, see for example [20,
Ch. 2], that an order-preserving bijection of a well-ordered set is identical.

Let us fix some well-order on $X$ and denote it by $\leq$. Construct an
oriented graph $G_{o}$ on the vertex set $X$ connecting by an oriented edge
$(x,y)$ each pair of vertices $x,\,y\in X$ such that $x\lvertneqq y$. It
follows from the above that the automorphism group of this graph is trivial.
Now reconstruct $G_{o}$ to another oriented graph $H_{o}$ changing each its
oriented edge $e=(x,y)$ by three vertices $u_{e},\,v_{e},\,w_{e}$ and four
oriented edges $(x,u_{e})$, $(u_{e},v_{e})$, $(v_{e},y)$, and $(v_{e},w_{e})$.
Finally, denote by $H$ the non-oriented graph corresponding to $H_{o}$.

Show that the automorphism group of the graph $H=(V_{H},E_{H})$ is trivial.
Let $\sigma\colon V_{H}\to V_{H}$ be an automorphism of the graph $H$. Notice
that the set $X\subset V_{H}$ is the set of all vertices from $H$ having
infinite degree, therefore the restriction of $\sigma$ onto $X$ is a bijection
of $X$ onto itself. Further, let $e=(x,y)$ be an oriented edge of the graph
$G_{o}$. Let us show that $\big{(}\sigma(x),\sigma(y)\big{)}$ is also an edge
of the graph $G_{o}$. Assume the contrary, then
$f=\big{(}\sigma(y),\sigma(x)\big{)}$ is an edge of $G_{o}$ because the order
is total. The path $x,\,u_{e},\,v_{e},\,y$ in $H$ is mapped to the path
$\sigma(x),\,\sigma(u_{e}),\,\sigma(v_{e}),\,\sigma(y)$. On the other hand, in
$H$ there exists unique $3$-edge path corresponding to the edge $f$, namely,
the path $\sigma(y),\,u_{f},\,v_{f},\,\sigma(x)$, therefore
$u_{f}=\sigma(v_{e})$ and $v_{f}=\sigma(u_{e})$. But $\deg u_{e}=\deg
u_{f}=2$, and $\deg v_{e}=\deg v_{f}=3$, a contradiction, so
$\big{(}\sigma(x),\sigma(y)\big{)}$ is an edge of $G_{o}$. The latter means
that $\sigma|_{X}$ is an automorphism of the oriented graph $G_{o}$, and
hence, $\sigma|_{X}=\operatorname{id}_{X}$. But then the vertices of each path
$x,\,u_{e},\,v_{e},\,y$ are also fixed, and hence,
$\sigma=\operatorname{id}_{V_{H}}$.

Notice that the cardinality of $X$ and $V_{H}$ are the same. Construct a
metric on $V_{H}$ as follows: put the distances between distinct adjacent
vertices equal $1$, and the distance between non-adjacent vertices equal
$1+\varepsilon$, where $\varepsilon\in(0,1)$. Then the resulting metric space
$V_{H}$ is generic, because $s(M)=1$, $t(M)=1-\varepsilon$, and
$e(M)=\varepsilon$.

This example can be modified easily to obtain an unbounded generic metric
space.

######  Example.

Let $X$ be an arbitrary generic metric space of a finite diameter $d$, in
particular $\\#X\geq 3$. Put $Y=X\sqcup\\{y\\}$ and extend metric from $X$ to
$Y$ as follows: $|xy|=f$, where $f>d$. It is clear that $s(Y)=s(X)$ and $t(Y)$
remains positive, because all new triangles are isosceles with equal sides of
the length $f$ and a smaller side of the length at least $s(X)>0$. Further,
each isometry of $Y$ preserves $y$, and hence, preserves $X$, therefore, $Y$
has no non-trivial isometries. Let $\sigma$ be a bijection of $Y$. If $\sigma$
preserves $y$, then
${\operatorname{dis}}\,\sigma={\operatorname{dis}}\,\sigma|_{X}\geq e(X)>0$.
Otherwise, take a point $x$ such that $\sigma(x)\neq y$ (such point exists,
because $\\#X\geq 3$), then
${\operatorname{dis}}\,\sigma\geq\big{|}|yx|-\big{|}\sigma(y)\sigma(x)\big{|}\big{|}\geq(f-d)>0$.
Thus, $e(Y)>0$ anyway, and hence, $Y$ is a generic metric space also.

This construction permits to obtain an unbounded generic metric space starting
from a bounded metric space by adding point by point and taking the distance
to next new point greater by $1$ than to the previous one. Notice that if the
initial space is infinite, then the resulting space has the same cardinality.

Next example is based on another idea and gives a countable generic metric
space of infinite diameter.

######  Example.

Consider a geometric progression
$\\{q^{n}\\}_{n=0}^{\infty}\subset\mathbb{R}$, where $q>4$, and define a
metric space $X=\\{x_{i}\\}$ with the distances
$|x_{i}x_{j}|=|q^{i}-q^{j}|+1$. It is clear that $s(X)>4$ and $t(X)=1$.
Further, let $\sigma\in S(X)$ be a nontrivial bijection. Then there exists $m$
such that $\sigma(x_{m})=x_{n}$ and $n\neq m$. Now, take $k$ as large that $k$
and $l$, where $\sigma(x_{k})=x_{l}$, are greater than $m$ and $n$. We have:

${\operatorname{dis}}\,\sigma\geq\big{|}|x_{m}x_{k}|-|\sigma(x_{m})\sigma(x_{k})|\big{|}=\big{|}|x_{m}x_{k}|-|x_{n}x_{l}|\big{|}=\big{|}|q^{m}-q^{k}|-|q^{n}-q^{l}|\big{|}.$

If $k\neq l$, then ${\operatorname{dis}}\,\sigma\geq
q^{\max\\{k,l\\}}\big{|}1-3/q\big{|}>q$, because $\max\\{k,l\\}\geq 3$.
Otherwise, ${\operatorname{dis}}\,\sigma\geq|q^{m}-q^{n}|\geq
q^{\max\\{m,n\\}}\big{|}1-1/q\big{|}\geq q-1$, because $\max\\{m,n\\}\geq 1$.
Thus, $e(X)\geq q-1>3$, and so, $X$ is a generic space.

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