HAL : hal-00623647, version 2
 arXiv : 1109.3181
 Versions disponibles : v1 (14-09-2011) v2 (24-05-2012)
 A new class of $({\cal H}^k,1)$-rectifiable subsets of metric spaces
 (13/04/2012)
 The main motivation of this paper arises from the study of Carnot--Carathéodory spaces, where the class of $1$-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree $k$, which are Hölder but not Lipschitz continuous when $k>1$. Replacing Lipschitz curves by this kind of curves we define $({\cal H}^k,1)$-rectifiable sets and show a density result generalizing the corresponding one in Euclidean geometry. This theorem is a consequence of computations of Hausdorff measures along curves, for which we give an integral formula. In particular, we show that both spherical and usual Hausdorff measures along curves coincide with a class of dimensioned lengths and are related to an interpolation complexity, for which estimates have already been obtained in Carnot--Carathéodory spaces.
 1 : Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP) Polytechnique - X – CNRS : UMR7641 2 : Unité de Mathématiques Appliquées (UMA) ENSTA ParisTech
 Domaine : Mathématiques/Géométrie métrique
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 rectifiable_reviewed.pdf(228.2 KB)
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 rectifiable_reviewed.ps(806.7 KB)
 hal-00623647, version 2 http://hal.archives-ouvertes.fr/hal-00623647 oai:hal.archives-ouvertes.fr:hal-00623647 Contributeur : Roberta Ghezzi <> Soumis le : Jeudi 24 Mai 2012, 15:23:36 Dernière modification le : Jeudi 24 Mai 2012, 21:28:03