Local Geometry of the Gromov–Hausdorff Metric Space and Totally Asymmetric Finite Metric Spaces

AbstractMain results of the paper are as follows:(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

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TESTING EMBEDDABILITY BETWEEN METRIC SPACES

International Journal of Foundations of Computer Science ◽

10.1142/s0129054109006577 ◽

2009 ◽

Vol 20 (02) ◽

pp. 313-329

Author(s):

CHING-LUEH CHANG ◽

YUH-DAUH LYUU ◽

YEN-WU TI

Keyword(s):

Metric Space ◽

Upper Bound ◽

Metric Spaces ◽

Graph Isomorphism ◽

Upper And Lower Bounds ◽

Wrong Answer ◽

Probability Of Error ◽

Finite Metric Space ◽

Finite Metric Spaces ◽

Special Case

Let L ≥ 1, ε > 0 be real numbers, (M, d) be a finite metric space and (N, ρ) be a metric space. A query to a metric space consists of a pair of points and asks for the distance between these points. We study the number of queries to metric spaces (M, d) and (N, ρ) needed to decide whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into N, ρ). When (M, d) is ∊-far from being L-bilipschitz embeddable into (N, ρ), we allow an o(1) probability of error (i.e., returning the wrong answer "L-bilipschitz embeddable"). However, no error is allowed when (M, d) is L-bilipschitz embeddable into (N, ρ). That is, algorithms with only one-sided errors are studied in this paper. When |M| ≤ |N| are both finite, we give an upper bound of [Formula: see text] on the number of queries for determining with one-sided error whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into (N, ρ). For the special case of finite |M| = |N|, the above upper bound evaluates to [Formula: see text]. We also prove a lower bound of Ω(|N|3/2) for the special case when |M| = |N| are finite and L = 1, which coincides with testing isometry between finite metric spaces. For finite |M| = |N|, the upper and lower bounds thus match up to a multiplicative factor of at most [Formula: see text], which depends only sublogarithmically in |N|. We also investigate the case when (N, ρ) is not necessarily finite. Our results are based on techniques developed in an earlier work on testing graph isomorphism.

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ROUNDNESS PROPERTIES OF ULTRAMETRIC SPACES

Glasgow Mathematical Journal ◽

10.1017/s0017089513000438 ◽

2013 ◽

Vol 56 (3) ◽

pp. 519-535 ◽

Cited By ~ 4

Author(s):

TIMOTHY FAVER ◽

KATELYNN KOCHALSKI ◽

MATHAV KISHORE MURUGAN ◽

HEIDI VERHEGGEN ◽

ELIZABETH WESSON ◽

...

Keyword(s):

Euclidean Space ◽

Metric Space ◽

Metric Spaces ◽

Independent Set ◽

Ultrametric Space ◽

Ultrametric Spaces ◽

Negative Type ◽

Finite Metric Space ◽

Finite Metric Spaces ◽

Generalized Roundness

AbstractMotivated by a classical theorem of Schoenberg, we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space $\mathbb{R}^{n}$ of dimension n but it cannot be isometrically embedded in any Euclidean space $\mathbb{R}^{r}$ of dimension r < n. We use this result as a technical tool to study ‘roundness’ properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X,d): (1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0 and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class $\mathcal{M}$ of all finite metric spaces that may be isometrically embedded into ℓ2 as an affinely independent set. The results of this paper show that Shkarin's class $\mathcal{M}$ consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].

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Coarse infinite-dimensionality of hyperspaces of finite subsets

European Journal of Mathematics ◽

10.1007/s40879-021-00515-3 ◽

2021 ◽

Author(s):

Thomas Weighill ◽

Takamitsu Yamauchi ◽

Nicolò Zava

Keyword(s):

Banach Space ◽

Metric Space ◽

Metric Spaces ◽

Hausdorff Metric ◽

Convex Banach Space ◽

Uniformly Convex Banach Space ◽

Uniformly Convex ◽

Dimensional Properties ◽

Bounded Geometry ◽

Infinite Dimensional

AbstractWe consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

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Paths in hyperspaces

Applied General Topology ◽

10.4995/agt.2003.2040 ◽

2003 ◽

Vol 4 (2) ◽

pp. 377 ◽

Cited By ~ 5

Author(s):

Camillo Constantini ◽

Wieslaw Kubís

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Hausdorff Metric ◽

Absolute Retract ◽

Almost Convex ◽

Convex Metric Spaces ◽

Metric Topology ◽

Hausdorff Metric Topology ◽

Bounded Sets ◽

Separable Metric Space

<p>We prove that the hyperspace of closed bounded sets with the Hausdor_ topology, over an almost convex metric space, is an absolute retract. Dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. We give some necessary conditions for the path-wise connectedness of the Hausdorff metric topology on closed bounded sets. Finally, we describe properties of a separable metric space, under which its hyperspace with the Wijsman topology is path-wise connected.</p>

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SIFAT KELENGKAPAN DAN KEKOMPAKAN PADA RUANG METRIK HAUSDORFF

Jurnal Silogisme Kajian Ilmu Matematika dan Pembelajarannya ◽

10.24269/silogisme.v3i2.1469 ◽

2019 ◽

Vol 3 (2) ◽

pp. 78

Author(s):

Dewanti Inesia Putri ◽

Arta Ekayanti

Keyword(s):

Metric Space ◽

Hausdorff Metric ◽

Compact Set ◽

Definition Of ◽

Hausdorff Metric Space

In this paper, will be discuss the definition of the Hausdorff metric space, completeness of the Hausdorff metric space, and compactness of the Hausdorff metric space. By used the theory of the metric space, the compact set was given the definition of the Hausdorff metric space. By used the completeness of the metric space, it is shown that the Hausdorff metric space was complete if the metric space was complete. Furthermore, used the compactness of the metric space was shown the Hausdorff metric space was compact if the metric space was compact

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OPTIMAL LOWER BOUND ON THE SUPREMAL STRICT p-NEGATIVE TYPE OF A FINITE METRIC SPACE

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709000604 ◽

2009 ◽

Vol 80 (3) ◽

pp. 486-497 ◽

Cited By ~ 4

Author(s):

ANTHONY WESTON

Keyword(s):

Lower Bound ◽

Metric Space ◽

Triangle Inequality ◽

Metric Spaces ◽

Elementary Approach ◽

Negative Type ◽

Finite Metric Space ◽

Image Position ◽

Finite Metric Spaces ◽

Optimal Lower Bound

AbstractDetermining meaningful lower bounds on the supremal strict p-negative type of classes of finite metric spaces is a difficult nonlinear problem. In this paper we use an elementary approach to obtain the following result: given a finite metric space (X,d) there is a constant ζ>0, dependent only on n=∣X∣ and the scaled diameter 𝔇=(diamX)/min{d(x,y)∣x⁄=y} of X (which we may assume is >1), such that (X,d) has p-negative type for all p∈[0,ζ] and strict p-negative type for all p∈[0,ζ). In fact, we obtain A consideration of basic examples shows that our value of ζ is optimal provided that 𝔇≤2. In other words, for each 𝔇∈(1,2] and natural number n≥3, there exists an n-point metric space of scaled diameter 𝔇 whose supremal strict p-negative type is exactly ζ. The results of this paper hold more generally for all finite semi-metric spaces since the triangle inequality is not used in any of the proofs. Moreover, ζ is always optimal in the case of finite semi-metric spaces.

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On Multivalued Fuzzy Contractions in Extended b -Metric Spaces

Journal of Mathematics ◽

10.1155/2021/5579991 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Nayab Alamgir ◽

Quanita Kiran ◽

Hassen Aydi ◽

Yaé Ulrich Gaba

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Metric Space ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Hausdorff Metric ◽

Contraction Mappings ◽

Nadler’S Fixed Point Theorem ◽

The Family ◽

Fuzzy Fixed Point

In this paper, we establish a Hausdorff metric over the family of nonempty closed subsets of an extended b -metric space. Thereafter, we introduce the concept of multivalued fuzzy contraction mappings and prove related α -fuzzy fixed point theorems in the context of extended b -metric spaces that generalize Nadler’s fixed point theorem as well as many preexisting results in the literature. Further, we establish α -fuzzy fixed point theorems for Ćirić type fuzzy contraction mappings as a generalization of previous results. Moreover, we give some examples to support the obtained results.

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Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra

10.29007/pw5g ◽

2018 ◽

Author(s):

Larry Moss ◽

Jayampathy Ratnayake ◽

Robert Rose

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Topological Spaces ◽

Fractal Sets ◽

Cauchy Completion ◽

Sweeping Generalization ◽

Initial Algebra ◽

Finite Metric Spaces ◽

Set Of Points

This paper is a contribution to the presentation of fractal sets in terms of final coalgebras.The first result on this topic was Freyd's Theorem: the unit interval [0,1] is the final coalgebra ofa certain functor on the category of bipointed sets. Leinster 2011 offersa sweeping generalization of this result. He is able to represent many of what would be intuitivelycalled "self-similar" spaces using (a) bimodules (also called profunctors or distributors),(b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction offinal coalgebras for the types of functors of interest using a notion of resolution. In addition to thecharacterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces.Our major contribution is to suggest that in many cases of interest, point (c) above on resolutionsis not needed in the construction of final coalgebras. Instead, one may obtain a number of spaces ofinterest as the Cauchy completion of an initial algebra,and this initial algebra is the set of points in a colimit of an omega-sequence of finite metric spaces.This generalizes Hutchinson's 1981 characterization of fractal attractors asclosures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is ``computationally related'' to the overall fractal. For example, when applied to Freyd's construction, our method yields the metric space.of dyadic rational numbers in [0,1].Our second contribution is not completed at this time, but it is a set of results on \emph{metric space}characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui 2010,and our interest in quotient metrics comes from their paper. So in terms of (a)--(c) above, our workdevelops (a) and (b) in metric settings while dropping (c).

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DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES

Glasgow Mathematical Journal ◽

10.1017/s0017089518000022 ◽

2018 ◽

Vol 61 (1) ◽

pp. 33-47 ◽

Cited By ~ 2

Author(s):

S. OSTROVSKA ◽

M. I. OSTROVSKII

Keyword(s):

Banach Space ◽

Real Number ◽

Banach Spaces ◽

Metric Space ◽

Metric Spaces ◽

Universal Constant ◽

Locally Finite ◽

Finite Dimensional ◽

Finite Metric Space ◽

Finite Metric Spaces

AbstractGiven a Banach spaceXand a real number α ≥ 1, we write: (1)D(X) ≤ α if, for any locally finite metric spaceA, all finite subsets of which admit bilipschitz embeddings intoXwith distortions ≤C, the spaceAitself admits a bilipschitz embedding intoXwith distortion ≤ α ⋅C; (2)D(X) = α+if, for every ϵ > 0, the conditionD(X) ≤ α + ϵ holds, whileD(X) ≤ α does not; (3)D(X) ≤ α+ifD(X) = α+orD(X) ≤ α. It is known thatD(X) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1)D((⊕n=1∞Xn)p) ≤ 1+for every nested family of finite-dimensional Banach spaces {Xn}n=1∞and every 1 ≤p≤ ∞. (2)D((⊕n=1∞ℓ∞n)p) = 1+for 1 <p< ∞. (3)D(X) ≤ 4+for every Banach spaceXwith no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008).

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