An Algorithm for Computing Virtual Cut Points in Finite Metric Spaces

2007 ◽  
pp. 4-10 ◽  
Author(s):  
Andreas W. M. Dress ◽  
Katharina T. Huber ◽  
Jacobus Koolen ◽  
Vincent Moulton
Keyword(s):  
Metric Spaces ◽  
Cut Points ◽  
Finite Metric Spaces

Main metric invariants of finite metric spaces. II

2016 ◽  
Vol 60 (6) ◽  
pp. 75-78 ◽  
Author(s):  
E. N. Sosov
Keyword(s):  
Metric Spaces ◽  
Metric Invariants ◽  
Finite Metric Spaces

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 On embeddings of locally finite metric spaces into ℓ

2019 ◽  
Vol 474 (1) ◽  
pp. 666-673 ◽  
Author(s):  
Sofiya Ostrovska ◽  
Mikhail I. Ostrovskii
Keyword(s):  
Metric Spaces ◽  
Locally Finite ◽  
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.

Main Metric invariants of finite metric spaces. III

2017 ◽  
Vol 61 (8) ◽  
pp. 55-58
Author(s):  
E. N. Sosov
Keyword(s):  
Metric Spaces ◽  
Metric Invariants ◽  
Finite Metric Spaces
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