TY - JOUR

T1 - Differentiating maps into L1, and the geometry of BV functions

AU - Cheeger, Jeff

AU - Kleiner, Bruce

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2010

Y1 - 2010

N2 - 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].

AB - 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].

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U2 - 10.4007/annals.2010.171.1347

DO - 10.4007/annals.2010.171.1347

M3 - Article

AN - SCOPUS:77954158932

VL - 171

SP - 1347

EP - 1385

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 2

ER -