Trees and Markov convexity

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.

💡 Research Summary

The paper addresses the long‑standing problem of characterising which infinite weighted trees admit a bi‑Lipschitz embedding into a Hilbert space. The authors prove a precise dichotomy: an infinite weighted tree can be embedded with uniformly bounded distortion if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. In other words, the presence of large, low‑distortion binary sub‑trees is the sole obstruction to a Hilbert‑space embedding.

To obtain this result the authors introduce a novel metric invariant called Markov convexity. For a metric space ((X,d)) and any reversible Markov chain ({X_t}) on (X), the Markov convexity constant (M) is the smallest number such that
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