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

TL;DR

This paper investigates metric spaces that can be embedded into all $n$-dimensional Banach spaces with low distortion, introducing new examples and preserving properties under metric composition.

Contribution

It introduces a new family of finite metric spaces that embed with bounded distortion into any $n$-dimensional Banach space, expanding understanding beyond classical examples.

Findings

Ultrametrics embed into $ ext{log } n$-dimensional Banach spaces

Metric composition preserves embeddability properties

New metric spaces embed into all $n$-dimensional Banach spaces

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.