We investigate the hyperspace GH(Rn) of the isometry classes of all non-empty compact subsets of a Euclidean space in the Gromov-Hausdorff metric. It is proved that for any n≥1, GH(Rn) is ...homeomorphic to the orbit space 2Rn/E(n) of the hyperspace 2Rn of all non-empty compact subsets of a Euclidean space Rn equipped with the Hausdorff metric and the natural action of the Euclidean group E(n). This is further applied to prove that 2Rn/E(n) is homeomorphic to the open cone OCone(Ch(Bn)/O(n)), where Ch(Bn) stands for the set of all A∈2Rn for which the closed Euclidean unit ball Bn is the least circumscribed ball (the Chebyshev ball). These results lead to determine the complete topological structure of GH(Rn) for n≤2, namely, we prove that GH(Rn) is homeomorphic to the Hilbert cube with a removed point. We also prove that for n≤2, GH(Bn) is homeomorphic to the Hilbert cube.
In this paper we investigate the hyperspace GH(Rn) of the isometry classes of all non-empty compact subsets of a Euclidean space Rn in the Gromov-Hausdorff metric. It is continuation of our previous ...paper 5, where it was established that for any n≥1, GH(Rn) is homeomorphic to the orbit space 2Rn/E(n) of the hyperspace 2Rn of all non-empty compact subsets of Rn equipped with the Hausdorff metric and the natural action of the Euclidean group E(n). This model is further applied to determine the complete topological structure of GH(Rn) for all n≥1, namely, we prove that GH(Rn) is homeomorphic to the Hilbert cube with a removed point. We also prove that for n≥1, GH(Bn) is homeomorphic to the Hilbert cube, where Bn denotes the closed unit ball of Rn.
We calculate the Gromov–Hausdorff distance between an interval and a circle in the Euclidean plane. To do that, we introduced a few new notions like round spaces and nonlinearity degree of a metric ...space.
The main aim of this paper is to study average Hewitt-Stromberg and box dimensions of typical compact metric spaces belonging to the Gromov-Hausdorff space equipped with the Gromov-Hausdorff metric.
We study average Hewitt–Stromberg measures of typical compact metric spaces belonging to the Gromov–Hausdorff space (of all compact metric spaces) equipped with the Gromov–Hausdorff metric.
In this work, it is proved that the set of boundedly-compact pointed metric spaces, equipped with the Gromov–Hausdorff topology, is a Polish space. The same is done for the Gromov–Hausdorff–Prokhorov ...topology. This extends previous works which consider only length spaces or discrete metric spaces. This is a measure theoretic requirement to study random boundedly-compact pointed (measured) metric spaces, which is the main motivation of this work. In particular, this provides a unified framework for studying random graphs, random discrete spaces and random length spaces. The proofs use a generalization of the classical theorem of Strassen, presented here, which is of independent interest. This generalization provides an equivalent formulation of the Prokhorov distance of two finite measures, having possibly different total masses, in terms of approximate couplings. A Strassen-type result is also provided for the Gromov–Hausdorff–Prokhorov metric for compact spaces.
We study the Hausdorff and packing measures of typical compact metric spaces belonging to the Gromov–Hausdorff space (of all compact metric spaces) equipped with the Gromov–Hausdorff metric.
We combine the pointed Gromov-Hausdorff metric with the locally
C
0
-distance to obtain the pointed
C
0
-Gromov-Hausdorff distance between maps of possibly different non-compact pointed metric ...spaces. The latter is combined with Walters’s locally topological stability proposed by Lee–Nguyen–Yang, and
GH
-stability from Arbieto-Morales to obtain the notion of topologically
GH
-stable pointed homeomorphism. We give one example to show the difference between the distance when taking different base points in a pointed metric space.
We apply the Gromov–Hausdorff metric
d
G
for characterization of certain generalized manifolds. Previously, we have proven that with respect to the metric
d
G
,
generalized
n
-manifolds are limits of ...spaces which are obtained by gluing two topological
n
-manifolds by a controlled homotopy equivalence (the so-called 2-patch spaces). In the present paper, we consider the so-called
manifold-like
generalized
n
-manifolds
X
n
,
introduced in 1966 by Mardeić and Segal, which are characterized by the existence of
δ
-mappings
f
δ
of
X
n
onto closed manifolds
M
δ
n
,
for arbitrary small
δ
>
0
, i.e., there exist onto maps
f
δ
:
X
n
→
M
δ
n
such that for every
u
∈
M
δ
n
,
f
δ
-
1
(
u
)
has diameter less than
δ
. We prove that with respect to the metric
d
G
,
manifold-like generalized
n
-manifolds
X
n
are limits of topological
n
-manifolds
M
i
n
. Moreover, if topological
n
-manifolds
M
i
n
satisfy a certain local contractibility condition
M
(
ϱ
,
n
)
, we prove that generalized
n
-manifold
X
n
is resolvable.
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