Rectifiable sets in metric and Banach spaces

Luigi Ambrosio and Bernd Kirchheim

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Submission date: 12. Mar. 1999
published in: Mathematische Annalen, 318 (2000) 3, p. 527-555 
DOI number (of the published article): 10.1007/s002080000122
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Abstract:
In this paper we study k-rectifiable sets in metric spaces, i.e. sets which can be covered up to a set of Hausdorff measure zero a countable family of Lipschitz images of subsets of k-dimensional Euclidean space.

We prove the existence of an k-dimensional approximate tangent space together with a corresponding local norm at almost each point of such sets. These objects represent the geometry of the considered set only in a "metric sense", however they exist also in cases where the classical differentiablity results for Lipschitz maps fail badly. Based on this analysis we can derive several rectifiablity criteria as well as an area and coarea formula.