Asymptotic coarse Lipschitz equivalence
[Équivalence asymptotiquement grossièrement Lipschitz]
Braga, Bruno M.1 ; Lancien, Gilles2
1IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro (Brazil)
2Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623), 25000 Besançon (France)
Annales de l'Institut Fourier, Online first, 34 p.

We introduce the notion of asymptotic coarse Lipschitz equivalence of metric spaces. We show that it is strictly weaker than coarse Lipschitz equivalence. We study its impact on the asymptotic dimension of metric spaces. Then we focus on Banach spaces. We prove that, for $2\le p< \infty $, being linearly isomorphic to $\ell _p$ is stable under asymptotic coarse Lipschitz equivalences. Finally, we establish a version of the Gorelik principle in this setting and apply it to prove the stability of various properties of asymptotic uniform smoothness of Banach spaces under asymptotic coarse Lipschitz equivalences.

Nous introduisons la notion d’équivalence asymptotiquement grossièrement Lipschitz entre espaces métriques. Nous étudions son impact sur la dimension asymptotique des espaces métriques. Ensuite, nous nous concentrons sur le cas des espaces de Banach. Nous montrons que, pour $2\le p< \infty $, être linéairement isomorphe à $\ell _p$ est stable par équivalence asymptotiquement grossièrement Lipschitz. Enfin, nous établissons une version du principe de Gorelik dans ce cadre et l’appliquons pour prouver la stabilité de plusieurs propriétés de lissité asymptotique uniforme des espaces de Banach par équivalence asymptotiquement grossièrement Lipschitz.

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Révisé le :
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DOI : 10.5802/aif.3775
Classification : 46B80
Keywords: Banach spaces, nonlinear geometry, coarse Lipschitz equivalence
Mots-clés : espaces de Banach, géométrie non linéaire, équivalence grossièrement Lipschitz

Braga, Bruno M.  1   ; Lancien, Gilles  2

1 IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro (Brazil)
2 Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623), 25000 Besançon (France) 
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Braga, Bruno M.; Lancien, Gilles. Asymptotic coarse Lipschitz equivalence. Annales de l'Institut Fourier, Online first, 34 p.

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