### 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 = L^{1}, 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 L^{1}, 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 L^{1} and functions of bounded variation, which permits us to exploit results on the structure of BV functions on the Heisenberg group [FSSC01].

Original language | English (US) |
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Pages (from-to) | 1347-1385 |

Number of pages | 39 |

Journal | Annals of Mathematics |

Volume | 171 |

Issue number | 2 |

DOIs | |

State | Published - Jul 6 2010 |

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty