Realization of metric spaces as inverse limits, and bilipschitz\n embedding in L₁

2011/10/11 by Jeff Cheeger, Bruce Kleiner, Cheeger, Jeff +1 · 1 citation
Mathematics · Computer Science · #advanced mathematical theories #Topological and Geometric Data Analysis #Advanced Banach Space Theory

paper · pdf · doi:10.48550/arxiv.1110.2406

Abstract

We give sufficient conditions for a metric space to bilipschitz embed in L₁.\nIn particular, if X is a length space and there is a Lipschitz map u:X--->R\nsuch that for every interval I in R, the connected components of the inverse\nimage f^-1(I) have diameter at most a constant time the diameter of I, then X\nadmits a bilipschitz embedding in L₁. As a corollary, well-known examples of\nLaakso bilipschitz embed in L₁, though they do not embed in any any Banach\nspace with the Radon-Nikodym property (e.g. the space l₁ of summable\nsequences).\n The spaces appearing the statement of the bilipschitz embedding theorem have\nan alternate characterization as inverse limits of systems of metric graphs\nsatisfying certain additional conditions. This representation, which may be of\nindependent interest, is the initial part of the proof of the bilipschitz\nembedding theorem. The rest of the proof uses the combinatorial structure of\nthe inverse system of graphs and a diffusion construction, to produce the\nembedding in L₁.\n

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