Geodesic
Differentiating maps into $L^1$, and the geometry of $\rm BV$ functions
Jeff Cheeger1 ; Bruce Kleiner1
1 Courant Institute of Mathematical Sciences<br/>251 Mercer Street<br/>New York, NY 10012<br/>United States
Annals of mathematics, Tome 171 (2010) no. 2, pp. 1347-1385.

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This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps $X\to V$ 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 $\mathbb{R}^n$ and $\mathbb{H}$, the Heisenberg group with its Carnot-Carathéodory metric. It follows that $\mathbb{H}$ 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].
DOI : 10.4007/annals.2010.171.1347

Jeff Cheeger 1 ; Bruce Kleiner 1

1 Courant Institute of Mathematical Sciences<br/>251 Mercer Street<br/>New York, NY 10012<br/>United States 
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Jeff Cheeger; Bruce Kleiner. Differentiating maps into $L^1$, and the geometry of $\rm BV$ functions. Annals of mathematics, Tome 171 (2010) no. 2, pp. 1347-1385. doi : 10.4007/annals.2010.171.1347. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.1347/

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