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Comment: 15 pages, 6 figures
Comment: 17 pages, 3 Figures, corrected several typos, revised proofs
Comment: 20 pages, 22 figures
In this paper I explain what is a pair of dilatation structures, one looking down to another. Such a pair of dilatation structures leads to the intrinsic definition of a distribution as a field of topological filters. To any pair of dilatation structures there is an associated notion of differentiability which generalizes the Pansu differentiability. This allows the introduction of the Radon-Nikodym property for dilatation structures, which is the straightforward generalization of the Radon-Nikodym property for Banach spaces. After an introducting section about length metric spaces and metric derivatives, is proved that for a dilatation structure with the Radon-Nikodym property the length of absolutely continuous curves expresses as an integral of the norms of the tangents to the curve, as in Riemannian geometry. Further it is ...
Comment: 10 pages, 3 figures. revised version correctly attributes the idea of Section 3 to Tverberg; and replaced k-sets by "linearly separable sets" in the paper and the title. Accepted for publication in Israel Journal of Mathematics
Comment: 6 pages, 1 figure, 3 tables
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