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

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...

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Bibliographic Details
Published in:arXiv.org 2012-05
Main Authors: Ghezzi, Roberta, Jean, Frédéric
Format: Article
Language:English
Subjects:
Euclidean geometry
Interpolation
Metric space
Online Access:Get full text
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Summary: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 H\"older 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.
ISSN:2331-8422