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

Published online by Cambridge University Press:  20 November 2018

Mikhail Ostrovskii  and
Beata Randrianantoanina
Mikhail Ostrovskii
Affiliation:
Department of Mathematics and Computer Science, St. John's University, 8000 Utopia Parkway Queens, NY 11439, USA e-mail: ostrovsm@stjohns.edu
Beata Randrianantoanina
Affiliation:
Department of Mathematics, Miami University, Oxford, OH 45056, USA e-mail: randrib@miamioh.edu
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Abstract

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For a fixed $K\,\gg \,1$ and $n\,\in \,\mathbb{N}$, $n\,\gg \,1$ we study metric spaces which admit embeddings with distortion $\le \,K$ 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.

Keywords

46B8505C1230L0546B1552A21basis constantbilipschitz embeddingdiamond graphdistortionequilateral setultrametric
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 4 , 01 August 2016 , pp. 876 - 907
Copyright
Copyright © Canadian Mathematical Society 2016

References

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