1. Metric spaces admitting low-distortion embeddings into all $n$-dimensional Banach spaces
- Author
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Ostrovskii, Mikhail I. and Randrianantoanina, Beata
- Subjects
Mathematics - Functional Analysis ,Mathematics - Metric Geometry ,Primary: 46B85, Secondary: 05C12, 30L05, 46B15, 52A21 - 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., Comment: 37 pages, 5 figures, some small improvements of presentation
- Published
- 2014
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