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For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaceswhich admit embeddings with distortion $\le K$ into each $n$-dimensional Banachspace. Classical examples include spaces embeddable into $\log n$-dimensionalEuclidean spaces, and equilateral spaces.  We prove that good embeddability properties are preserved under the operationof metric composition of metric spaces. In particular, we prove that any$n$-point ultrametric can be embedded with uniformly bounded distortion intoany Banach space of dimension $\log n$.  The main result of the paper is a new example of a family of finite metricspaces which are not metric compositions of classical examples and which doembed with uniformly bounded distortion into any Banach space of dimension $n$.This partially answers a question of G. Schechtman.

Metric spaces admitting low-distortion embeddings into all $n$-dimensional Banach spaces