Ultrametric skeletons

TL;DR

This paper proves the existence of ultrametric skeletons within any compact metric space, providing a new structural insight and implications for majorizing measures, with sharp dependence on a parameter.

Contribution

It establishes the existence of ultrametric skeletons with sharp distortion bounds and extends the concept to multiple measures, impacting the understanding of metric space structures.

Findings

Existence of ultrametric skeletons with distortion O(1/ε)

Extension to multiple measures

Implication for Talagrand's majorizing measures theorem

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.