On the distortion required for embedding finite metric spaces into normed spaces

1996 ◽  
Vol 93 (1) ◽  
pp. 333-344 ◽  
Author(s):  
Jiří Matoušek
Keyword(s):  
Metric Spaces ◽  
Normed Spaces ◽  
Finite Metric Spaces

Fuzzy Normed Spaces and Fuzzy Metric Spaces

2018 ◽  
pp. 11-43
Author(s):  
Yeol Je Cho ◽  
Themistocles M. Rassias ◽  
Reza Saadati
Keyword(s):  
Metric Spaces ◽  
Normed Spaces ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric

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.

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