### 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 |

### Fingerprint

### Keywords

- Bi-Lipschitz embeddings
- Majorizing measures
- Metric geometry

### ASJC Scopus subject areas

- General

### Cite this

*Proceedings of the National Academy of Sciences of the United States of America*,

*110*(48), 19256-19262. https://doi.org/10.1073/pnas.1202500109

**Ultrametric skeletons.** / Mendel, Manor; Naor, Assaf.

Research output: Contribution to journal › Article

*Proceedings of the National Academy of Sciences of the United States of America*, vol. 110, no. 48, pp. 19256-19262. https://doi.org/10.1073/pnas.1202500109

}

TY - JOUR

T1 - Ultrametric skeletons

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2013/11/26

Y1 - 2013/11/26

N2 - 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.

AB - 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.

KW - Bi-Lipschitz embeddings

KW - Majorizing measures

KW - Metric geometry

UR - http://www.scopus.com/inward/record.url?scp=84888351935&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84888351935&partnerID=8YFLogxK

U2 - 10.1073/pnas.1202500109

DO - 10.1073/pnas.1202500109

M3 - Article

VL - 110

SP - 19256

EP - 19262

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 48

ER -