Generalized differentiation and bi-Lipschitz nonembedding in L1

We consider Lipschitz mappings, $f:X\to V$, where X is a doubling metric measure space which satisfies a Poincaré inequality, and V is a Banach space. We show that earlier differentiability and bi-Lipschitz nonembedding results for maps, $f:X\to {\mathbb{R}}^N$, remain valid when ${\mathbb{R}}^N$ is replaced by any separable dual space. We exhibit spaces which bi-Lipschitz embed in $L^1$, but not in any separable dual V. For certain domains, including the Heisenberg group with its Carnot–Caratheodory metric, we establish a new notion of differentiability for maps into $L^1$. This implies that the Heisenberg group does not bi-Lipschitz embed in $L^1$, thereby proving a conjecture of J. Lee and A. Naor. When combined with their work, this has implications for theoretical computer science. To cite this article: J. Cheeger, B. Kleiner, C. R. Acad. Sci. Paris, Ser. I 343 (2006).