Differentiating maps into L1, and the geometry of BV functions

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Abstract

This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps X →V and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V = L1, where differentiability fails. We establish another kind of differentiability for certain X, including R{double-struck}n and H{double-struck}, the Heisenberg group with its Carnot-Carathéodory metric. It follows that H{double-struck} does not bi-Lipschitz embed into L1, as conjectured by J. Lee and A. Naor. When combined with their work, this provides a natural counterexample to the Goemans-Linial conjecture in theoretical computer science; the first such counterexample was found by Khot-Vishnoi [KV05]. A key ingredient in the proof of our main theorem is a new connection between Lipschitz maps to L1 and functions of bounded variation, which permits us to exploit results on the structure of BV functions on the Heisenberg group [FSSC01].

Original languageEnglish (US)
Pages (from-to)1347-1385
Number of pages39
JournalAnnals of Mathematics
Volume171
Issue number2
DOIs
StatePublished - 2010

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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