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.
All Science Journal Classification (ASJC) codes
- Geometry and Topology
- Metric trees
- Uniform convexity
- bi-Lipschitz embeddings