  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.
