# A new class of $(\mathcal{H}^k,1)$-rectifiable subsets of metric spaces

## Roberta Ghezzi

### (SNS)

Date: Oct 11, 2011, time: 17:10

Place: Centro De Giorgi, Scuola Normale Superiore

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.

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