A new class of ( 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 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 (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.