1. CJM 2016 (vol 68 pp. 876)
 Ostrovskii, Mikhail; Randrianantoanina, Beata

Metric Spaces Admitting Lowdistortion 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 

2. CJM 2007 (vol 59 pp. 1029)
 Kalton, N. J.; Koldobsky, A.; Yaskin, V.; Yaskina, M.

The Geometry of $L_0$
Suppose that we have the unit Euclidean ball in
$\R^n$ and construct new bodies using three operations  linear
transformations, closure in the radial metric, and multiplicative
summation defined by $\x\_{K+_0L} = \sqrt{\x\_K\x\_L}.$ We prove
that in dimension $3$ this procedure gives all originsymmetric convex
bodies, while this is no longer true in dimensions $4$ and higher. We
introduce the concept of embedding of a normed space in $L_0$ that
naturally extends the corresponding properties of $L_p$spaces with
$p\ne0$, and show that the procedure described above gives exactly the
unit balls of subspaces of $L_0$ in every dimension. We provide
Fourier analytic and geometric characterizations of spaces embedding
in $L_0$, and prove several facts confirming the place of $L_0$ in the
scale of $L_p$spaces.
Categories:52A20, 52A21, 46B20 
