This is an announcement for the paper "Metric spaces admitting low-distortion embeddings into all $n$-dimensional Banach spaces" by Mikhail I. Ostrovskii and Beata Randrianantoanina.
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 any $n$-point ultrametric can be embedded with uniformly bounded distortion into any Banach space 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.
Archive classification: math.FA math.MG
Mathematics Subject Classification: Primary: 46B85, Secondary: 05C12, 30L05, 46B15, 52A21
Remarks: 35 pages, 4 figures
Submitted from: randrib@miamioh.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1412.7670
or