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
U2 - 10.1073/pnas.1202500109
DO - 10.1073/pnas.1202500109
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AN - SCOPUS:84888351935
SN - 0027-8424
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
IS - 48
ER -