TY - JOUR

T1 - Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1

AU - Cheeger, Jeff

AU - Kleiner, Bruce

N1 - Funding Information:
J. Cheeger was supported by NSF grants DMS-0704404 and DMS-1005552. B. Kleiner was supported by NSF grants DMS-1007508 and DMS-1105656.

PY - 2013/2

Y1 - 2013/2

N2 - We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map u: X → ℝ such that for every interval I ⊂ ℝ, the connected components of u-1(I) have diameter ≤ const} ̇ diam(I), then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111-123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon-Nikodym property (e. g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1.

AB - We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map u: X → ℝ such that for every interval I ⊂ ℝ, the connected components of u-1(I) have diameter ≤ const} ̇ diam(I), then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111-123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon-Nikodym property (e. g. the space ℓ1 of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L1.

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U2 - 10.1007/s00039-012-0201-8

DO - 10.1007/s00039-012-0201-8

M3 - Article

AN - SCOPUS:84875682464

VL - 23

SP - 96

EP - 133

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 1

ER -