Ultrametric skeletons

Manor Mendel, Assaf Naor

Research output: Contribution to journalArticlepeer-review


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(B d(x,r))≤(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 measure theorem.

Original languageEnglish
Pages (from-to)19256-19262
Number of pages7
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number48
StatePublished - 26 Nov 2013


  • Bi-Lipschitz embeddings
  • Majorizing measures
  • Metric geometry


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