Metric Dichotomies

TL;DR

This paper surveys dichotomies in finite metric spaces, showing that for any host space, either all metrics embed almost isometrically or some have unbounded distortion.

Contribution

It introduces and discusses the concept of metric dichotomies, providing a survey of their properties and implications in geometry and algorithms.

Findings

Identification of metric dichotomies in finite metric spaces

Conditions under which embeddings are almost isometric or unbounded

Connections to group theory and computer science applications

Abstract

These are notes from talks given at ICMS, Edinburgh, 4/2007 ("Geometry and Algorithms workshop") and at Bernoulli Center, Lausanne 5/2007 ("Limits of graphs in group theory and computer science"). We survey the following type of dichotomies exhibited by certain classes X of finite metric spaces: For every host space H, either all metrics in X embed almost isometrically in H, or the distortion of embedding some metrics of X in H is unbounded.