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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
18/1999

Rectifiable sets in metric and Banach spaces

Luigi Ambrosio and Bernd Kirchheim

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.

Received:
Mar 12, 1999
Published:
Mar 12, 1999

Related publications

inJournal
2000 Repository Open Access
Luigi Ambrosio and Bernd Kirchheim

Rectifiable sets in metric and Banach spaces

In: Mathematische Annalen, 318 (2000) 3, pp. 527-555