TY - JOUR
T1 - Generalized differentiation and bi-Lipschitz nonembedding in L1
AU - Cheeger, Jeff
AU - Kleiner, Bruce
N1 - Funding Information:
✩ The first author was partially supported by NSF Grant DMS 0105128 and the second by NSF Grant DMS 0505610. E-mail addresses: [email protected] (J. Cheeger), [email protected] (B. Kleiner).
PY - 2006/9/1
Y1 - 2006/9/1
N2 - We consider Lipschitz mappings, f : X → 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 → RN, remain valid when RN is replaced by any separable dual space. We exhibit spaces which bi-Lipschitz embed in L1, 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 L1. This implies that the Heisenberg group does not bi-Lipschitz embed in L1, 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).
AB - We consider Lipschitz mappings, f : X → 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 → RN, remain valid when RN is replaced by any separable dual space. We exhibit spaces which bi-Lipschitz embed in L1, 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 L1. This implies that the Heisenberg group does not bi-Lipschitz embed in L1, 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).
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U2 - 10.1016/j.crma.2006.07.001
DO - 10.1016/j.crma.2006.07.001
M3 - Article
AN - SCOPUS:33747350815
SN - 1631-073X
VL - 343
SP - 297
EP - 301
JO - Comptes Rendus Mathematique
JF - Comptes Rendus Mathematique
IS - 5
ER -