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

  Published:2016-02-29
 Printed: Aug 2016
  • Mikhail Ostrovskii,
    Department of Mathematics and Computer Science, St. John's University, 8000 Utopia Parkway, Queens, NY 11439, USA
  • Beata Randrianantoanina,
    Department of Mathematics, Miami University, Oxford, OH 45056, USA
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Abstract

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: basis constant, bilipschitz embedding, diamond graph, distortion, equilateral set, ultrametric basis constant, bilipschitz embedding, diamond graph, distortion, equilateral set, ultrametric
MSC Classifications: 46B85, 05C12, 30L05, 46B15, 52A21 show english descriptions Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science [See also 05C12, 68Rxx]
Distance in graphs
Geometric embeddings of metric spaces
Summability and bases [See also 46A35]
Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
46B85 - Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science [See also 05C12, 68Rxx]
05C12 - Distance in graphs
30L05 - Geometric embeddings of metric spaces
46B15 - Summability and bases [See also 46A35]
52A21 - Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
 

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