Generalized differentiation and bi-Lipschitz nonembedding in $ {L}^{1}$

Jeff Cheeger; Bruce Kleiner

doi:10.1016/j.crma.2006.07.001

Mathematical Analysis/Functional Analysis

Generalized differentiation and bi-Lipschitz nonembedding in

L^{1}

Presented by: Jean-Michel Bismut

Jeff Cheeger ¹; Bruce Kleiner ²

¹ Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
² Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, USA

Comptes Rendus. Mathématique, Volume 343 (2006) no. 5, pp. 297-301.

Abstract
Résumé

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, $f : X \to R^{N}$ , remain valid when $R^{N}$ 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.

Nous considérons des applications lipchitziennes, $f : X \to V$ , où X est un espace métrique mesuré tel que l'on contrôle le volume des boules par doublement du rayon et qui satisfait à une inégalité de Poincaré, et où V est un espace de Banach. On montre que des résultats antérieurs de différentiabilité et de non plongement bilipschitzien pour des applications $f : X \to R^{N}$ , restent valables quand on suppose que V est un dual séparable. Nous donnons des exemples d'espaces plongés de manière bilipschitzienne dans $L^{1}$ , mais qui ne sont plongeables dans aucun dual séparable. Pour certains espaces, dont le groupe d'Heisenberg muni de la métrique de Carnot–Caratheodory, on établit une nouvelle notion de différentiabilité pour des applications dans $L^{1}$ . Ceci implique que le groupe de Heisenberg ne possède aucun plongement bilipschitzien dans $L^{1}$ , un résultat conjecturé par J. Lee et A. Naor. Quand il est combiné avec des résultats de ces deux auteurs, notre travail a des applications en informatique théorique.

Received: 2006-06-02
Accepted: 2006-06-07
Published online: 2006-08-10

DOI: 10.1016/j.crma.2006.07.001

Author's affiliations:

Jeff Cheeger ¹; Bruce Kleiner ²

¹ Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
² Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, USA

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     author = {Jeff Cheeger and Bruce Kleiner},
     title = {Generalized differentiation and {bi-Lipschitz} nonembedding in $ {L}^{1}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {297--301},
     publisher = {Elsevier},
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     number = {5},
     year = {2006},
     doi = {10.1016/j.crma.2006.07.001},
     language = {en},
}

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Jeff Cheeger; Bruce Kleiner. Generalized differentiation and bi-Lipschitz nonembedding in $ {L}^{1}$. Comptes Rendus. Mathématique, Volume 343 (2006) no. 5, pp. 297-301. doi : 10.1016/j.crma.2006.07.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.07.001/

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