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Ultrametric skeletons
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DOI：10.1073/pnas.1202500109.png)
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Abstract
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We prove that for every epsilon is an element of (0,1) there exists C-epsilon is an element of (0,infinity) with the following property. If (X,d) is a compact metric space and mu is a Borel probability measure on X then there exists a compact subset S subset of X that embeds into an ultrametric space with distortion O(1/epsilon), and a probability measure nu supported on S satisfying nu(B-d(x,r))<=(mu(B-d(x,C(epsilon)r))(1-epsilon) for all x is an element of X and r is an element of (0,infinity). The dependence of the distortion on e is sharp. We discuss an extension of this statement to multiple measures, as well as how it implies Talagrand's majorizing measure theorem.
Keywords:
bi-Lipschitz embeddings
majorizing measures
metric geometry
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