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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].
@article{10_4007_annals_2010_171_1347,
author = {Jeff Cheeger and Bruce Kleiner},
title = {Differentiating maps into $L^1$, and the geometry of $\rm BV$ functions},
journal = {Annals of mathematics},
pages = {1347--1385},
publisher = {mathdoc},
volume = {171},
number = {2},
year = {2010},
doi = {10.4007/annals.2010.171.1347},
mrnumber = {2630066},
zbl = {1194.22009},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.1347/}
}
TY - JOUR AU - Jeff Cheeger AU - Bruce Kleiner TI - Differentiating maps into $L^1$, and the geometry of $\rm BV$ functions JO - Annals of mathematics PY - 2010 SP - 1347 EP - 1385 VL - 171 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.1347/ DO - 10.4007/annals.2010.171.1347 LA - en ID - 10_4007_annals_2010_171_1347 ER -
%0 Journal Article %A Jeff Cheeger %A Bruce Kleiner %T Differentiating maps into $L^1$, and the geometry of $\rm BV$ functions %J Annals of mathematics %D 2010 %P 1347-1385 %V 171 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.1347/ %R 10.4007/annals.2010.171.1347 %G en %F 10_4007_annals_2010_171_1347
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
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