Abstract view
Metric Spaces Admitting Lowdistortion Embeddings into All $n$dimensional Banach Spaces


Published:20160229
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
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] Finitedimensional 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  Finitedimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
