A new class of $({\cal H}^k,1)$-rectifiable subsets of metric spaces

Abstract : 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.
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Pré-publication, Document de travail
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Contributeur : Roberta Ghezzi <>
Soumis le : jeudi 24 mai 2012 - 15:23:36
Dernière modification le : vendredi 10 février 2017 - 01:12:39
Document(s) archivé(s) le : samedi 25 août 2012 - 02:38:33


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  • HAL Id : hal-00623647, version 2
  • ARXIV : 1109.3181



Roberta Ghezzi, Frédéric Jean. A new class of $({\cal H}^k,1)$-rectifiable subsets of metric spaces. 2012. <hal-00623647v2>



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