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A new class of $$(\mathcal H^k, 1)$$-rectifiable subsets of metric spaces. (English) Zbl 1268.28005
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 Hölder but not Lipschitz continuous when $$k > 1$$. Replacing the Lipschitz curves by this kind of curves, we define $$(\mathcal H^k, 1)$$-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.

##### MSC:
 28A78 Hausdorff and packing measures 30L99 Analysis on metric spaces 53C17 Sub-Riemannian geometry
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