2011/09/14 by Ghezzi, Roberta, Jean, Frédéric
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1109.3181
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 Hk,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.