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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 (Bd(x, r)) 6 (μ(Bd(x,Cεr)) 1−ε for all x ∈ X and r ∈ (0,∞). The dependen...
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ژورنال
عنوان ژورنال: Nature
سال: 1884
ISSN: 0028-0836,1476-4687
DOI: 10.1038/030078a0