A new class of \((\mathcal{H}^k,1)\)-rectifiable subsets of metric spaces
speaker: Roberta Ghezzi (Università di Roma Tor Vergata)abstract: The main motivation of this talk 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 \((\mathcal{H}^k,1)\)-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.
timetable:
Tue 11 Oct, 17:10 - 17:40, Aula Dini
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