Trees and Markov convexity

TL;DR

This paper characterizes when infinite weighted trees can be embedded into Hilbert space based on the absence of large binary subtrees and introduces Markov convexity as a new invariant to measure Euclidean distortion.

Contribution

It provides a complete characterization of Hilbert space embeddability for infinite trees and introduces Markov convexity as a tool for distortion analysis.

Findings

Characterization of Hilbert space embeddability for infinite trees

Introduction of Markov convexity as a metric invariant

Method to compute Euclidean distortion of metric trees

Abstract

We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors.