scholarly journals A new approach to low-distortion embeddings of finite metric spaces into non-superreflexive Banach spaces

2017 ◽  
Vol 273 (2) ◽  
pp. 598-651 ◽  
Author(s):  
Mikhail I. Ostrovskii ◽  
Beata Randrianantoanina
Keyword(s):  
Banach Spaces ◽  
Metric Spaces ◽  
New Approach ◽  
Low Distortion ◽  
Finite Metric Spaces

scholarly journals Metric Spaces Admitting Low-distortion Embeddings into All n-dimensional Banach Spaces

2016 ◽  
Vol 68 (4) ◽  
pp. 876-907 ◽  
Author(s):  
Mikhail Ostrovskii ◽  
Beata Randrianantoanina
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Spaces ◽  
Euclidean Spaces ◽  
Dimensional Banach Space ◽  
Low Distortion ◽  
Bounded Distortion ◽  
Finite Metric Spaces ◽  
Uniformly Bounded

AbstractFor a fixed K > 1 and n ∈ ℕ, n ≫ 1, we study metric spaces which admit embeddings with distortion ≤ K into each n-dimensional Banach space. Classical examples include spaces embeddable into log n-dimensional Euclidean spaces, and equilateral spaces.We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that n-point ultrametrics can be embedded with uniformly bounded distortions into arbitrary Banach spaces of dimension log n.The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension n. This partially answers a question of G. Schechtman.

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 Gromov minimal fillings for finite metric spaces

2013 ◽  
Vol 94 (108) ◽  
pp. 3-15
Author(s):  
Alexander Ivanov ◽  
Alexey Tuzhilin
Keyword(s):  
Metric Spaces ◽  
State University ◽  
Steiner Ratio ◽  
Chip Design ◽  
New Approach ◽  
One Dimensional ◽  
Design Evolution ◽  
Steiner Minimal Trees ◽  
Finite Metric Spaces ◽  
Moscow State University

The problem discussed in this paper was stated by Alexander O. Ivanov and Alexey A. Tuzhilin in 2009. It stands at the intersection of the theories of Gromov minimal fillings and Steiner minimal trees. Thus, it can be considered as one-dimensional stratified version of the Gromov minimal fillings problem. Here we state the problem; discuss various properties of one-dimensional minimal fillings, including a formula calculating their weights in terms of some special metrics characteristics of the metric spaces they join (it was obtained by A.Yu. Eremin after many fruitful discussions with participants of Ivanov-Tuzhilin seminar at Moscow State University); show various examples illustrating how one can apply the developed theory to get nontrivial results; discuss the connection with additive spaces appearing in bioinformatics and classical Steiner minimal trees having many applications, say, in transportation problem, chip design, evolution theory etc. In particular, we generalize the concept of Steiner ratio and get a few of its modifications defined by means of minimal fillings, which could give a new approach to attack the long standing Gilbert-Pollack Conjecture on the Steiner ratio of the Euclidean plane.

scholarly journals An Intrinsic Characterization of Five Points in a CAT(0) Space

2020 ◽  
Vol 8 (1) ◽  
pp. 114-165
Author(s):  
Tetsu Toyoda
Keyword(s):  
Metric Space ◽  
Isometric Embedding ◽  
Metric Spaces ◽  
Necessary Conditions ◽  
New Approach ◽  
Intrinsic Characterization ◽  
New Family

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

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