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

2009 ◽  
Vol 80 (3) ◽  
pp. 486-497 ◽  
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.

scholarly journals ROUNDNESS PROPERTIES OF ULTRAMETRIC SPACES

2013 ◽  
Vol 56 (3) ◽  
pp. 519-535 ◽  
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, ∞].

scholarly journals Lipschitz-free Spaces on Finite Metric Spaces

2019 ◽  
Vol 72 (3) ◽  
pp. 774-804 ◽  
Author(s):  
Stephen J. Dilworth ◽  
Denka Kutzarova ◽  
Mikhail I. Ostrovskii
Keyword(s):  
Metric Space ◽  
Free Space ◽  
Metric Spaces ◽  
Large Classes ◽  
Complemented Subspace ◽  
Finite Metric Space ◽  
Free Spaces ◽  
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.

scholarly journals TESTING EMBEDDABILITY BETWEEN METRIC SPACES

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.

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

2018 ◽  
Vol 61 (1) ◽  
pp. 33-47 ◽  
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).

scholarly journals On statistical convergence in quasi-metric spaces

2019 ◽  
Vol 52 (1) ◽  
pp. 225-236 ◽  
Author(s):  
Merve İlkhan ◽  
Emrah Evren Kara
Keyword(s):  
Functional Analysis ◽  
Computer Science ◽  
Distance Function ◽  
Metric Space ◽  
Statistical Convergence ◽  
Triangle Inequality ◽  
Metric Spaces ◽  
Applied Mathematics ◽  
Comprehensive Investigation ◽  
Cauchy Sequences

AbstractA quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric in general. Quasi-metrics are a subject of comprehensive investigation both in pure and applied mathematics in areas such as in functional analysis, topology and computer science. The main purpose of this paper is to extend the convergence and Cauchy conditions in a quasi-metric space by using the notion of asymptotic density. Furthermore, some results obtained are related to completeness, compactness and precompactness in this setting using statistically Cauchy sequences.

MODAL LOGICS OF METRIC SPACES

2014 ◽  
Vol 8 (1) ◽  
pp. 178-191 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
DAVID GABELAIA ◽  
JOEL LUCERO-BRYAN
Keyword(s):  
Modal Logic ◽  
Metric Space ◽  
Metric Spaces ◽  
Modal Logics ◽  
Finite Model ◽  
Finite Model Property ◽  
Classic Result ◽  
Topological Closure ◽  
Image Position

AbstractIt is a classic result (McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the ⊃ relation) to the ordinal ω + 3:$S4.Gr{z_1} \supset S4.Gr{z_2} \supset S4.Gr{z_3} \supset \cdots \,S4.Grz \supset S4.1 \supset S4.$It follows that the modal logic of an arbitrary metric space is finitely axiomatizable, has the finite model property, and hence is decidable.

Distance Sets of Urysohn Metric Spaces

2013 ◽  
Vol 65 (1) ◽  
pp. 222-240 ◽  
Author(s):  
N.W. Sauer
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Finite Metric Space ◽  
Distance Set ◽  
Distance Sets ◽  
Separable Metric Space

Abstract.A metric space M = (M; d) is homogeneous if for every isometry f of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending f . The space M is universal if it isometrically embeds every finite metric space F with dist(F) ⊆ dist(M) (with dist(M) being the set of distances between points in M).A metric space U is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space U isometrically embeds every separable metric space M with dist(M) ⊆ dist(U).)The main results are: (1) A characterization of the sets dist(U) for Urysohn metric spaces U. (2) If R is the distance set of a Urysohn metric space and M and N are two metric spaces, of any cardinality with distances in R, then they amalgamate disjointly to a metric space with distances in R. (3) The completion of every homogeneous, universal, separable metric space M is homogeneous.

scholarly journals A fixed point theorem in generalized metric spaces

2013 ◽  
Vol 46 (1) ◽  
Author(s):  
Luljeta Kikina ◽  
Kristaq Kikina
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Triangle Inequality ◽  
Metric Spaces ◽  
Generalized Metric Space ◽  
Generalized Metric Spaces ◽  
Generalized Metric

AbstractA generalized metric space has been defined by Branciari as a metric space in which the triangle inequality is replaced by a more general inequality. Subsequently, some classical metric fixed point theorems have been transferred to such a space. In this paper, we continue in this direction and prove a version of Fisher’s fixed point theorem in generalized metric spaces.

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