Ultrametric skeletons

Abstract

We prove that for every $\epsilon \in (0,1)$ there exists $C_{\epsilon }\in (0,\infty )$ with the following property. If $(X,d)$ is a compact metric space and $\mu$ is a Borel probability measure on $X$ then there exists a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1/\epsilon )$, and a probability measure $\nu$ supported on $S$ satisfying $\nu (B_d(x,r))\le (\mu (B_d(x,C_{\epsilon }r){)}^{1-\epsilon }$ for all $x\in X$ and $r\in (0,\infty )$. The dependence of the distortion on $\epsilon$ is sharp. We discuss an extension of this statement to multiple measures, as well as how it implies Talagrand's majorizing measures theorem.

typos fixed

References in corpus (3)

• On metric Ramsey-type phenomena
• Ramsey partitions and proximity data structures
• Euclidean quotients of finite metric spaces

Cited by in corpus (3)

• A geometric study of Wasserstein spaces: ultrametrics
• Dvoretzky-type theorem for Ahlfors regular spaces
• A simple proof of Dvoretzky-type theorem for Hausdorff dimension in doubling spaces