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Search: MSC category 46B15 ( Summability and bases [See also 46A35] )

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Ostrovskii, Mikhail; Randrianantoanina, Beata
Metric spaces admitting low-distortion embeddings into all $n$-dimensional Banach spaces
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
Categories:46B85, 05C12, 30L05, 46B15, 52A21

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