TY - JOUR

T1 - Ultrametric skeletons

AU - Mendel, Manor

AU - Naor, Assaf

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

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

C2 - 22652571

AN - SCOPUS:84888351935

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 -