Ultrametric Skeletons

We prove that for every ε ∈ (0, 1) there exists Cε ∈ (0,∞) with the following property. If (X, d) is a compact metric space and μ is a Borel probability measure on X then there exists a compact subset S ⊆ X that embeds into an ultrametric space with distortion O(1/ε), and a probability measure ν supported on S satisfying ν (Bd(x, r)) 6 (μ(Bd(x,Cεr)) 1−ε for all x ∈ X and r ∈ (0,∞). The dependence of the distortion on ε is sharp. We discuss an extension of this statement to multiple measures, as well as how it implies Talagrand’s majorizing measures theorem.