Metric differentiation, monotonicity and maps to L^1

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Cheeger, Jeff
Kleiner, Bruce
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Abstract
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We give a new approach to the infinitesimal structure of Lipschitz maps into L^1. As a first application, we give an alternative proof of the main theorem from an earlier paper, that the Heisenberg group does not admit a bi-Lipschitz embedding in L^1. The proof uses the metric differentiation theorem of Pauls and the cut metric decomposition to reduce the nonembedding argument to a classification of monotone subsets of the Heisenberg group.
Comment: Added a missing condition to Definition 5.3. Also made a number of minor corrections, and added a reference to a paper by Lee-Raghavendra
Keywords
Mathematics - Metric Geometry, Mathematics - Differential Geometry, Mathematics - Functional Analysis, Mathematics - Group Theory
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