### Abstract

We prove that for every ε ∈ (0,1) there exists C_{ε} ∈ (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(B_{d}(x,C_{ε}r)) ^{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 language | English (US) |
---|---|

Pages (from-to) | 19256-19262 |

Number of pages | 7 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 110 |

Issue number | 48 |

DOIs | |

State | Published - Nov 26 2013 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- General

### Keywords

- Bi-Lipschitz embeddings
- Majorizing measures
- Metric geometry

## Fingerprint Dive into the research topics of 'Ultrametric skeletons'. Together they form a unique fingerprint.

## Cite this

Mendel, M., & Naor, A. (2013). Ultrametric skeletons.

*Proceedings of the National Academy of Sciences of the United States of America*,*110*(48), 19256-19262. https://doi.org/10.1073/pnas.1202500109