This is an announcement for the paper "Ultrametric skeletons" by Manor Mendel and Assaf Naor.
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 $\nu\left(B_d(x,r)\right)\le \left(\mu(B_d(x,C_\epsilon r)\right)^{1-\epsilon}$ 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.
Archive classification: math.MG math.FA math.PR
Submitted from: naor@cims.nyu.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1112.3416
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