A new class of \((\mathcal H^k, 1)\)-rectifiable subsets of metric spaces. (English) Zbl 1268.28005

Ghezzi, Roberta; Jean, Frédéric

Commun. Pure Appl. Anal. 12, No. 2, 881-898 (2013).

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

Keywords:

rectifiability; Hausdorff measures; non Euclidean metric spaces; Hölder maps; sub-Riemannian geometry