Size of the medial axis and stability of Federer's curvature measures

Details Size of the medial axis and stability of Federer's curvature measures Size of the medial axis and stability of Federer's curvature measures

2010-01-18

arxiv.org/abs/1001.2961v1

Mathematics

Metric Geometry

15 pages, 2 figures

Scientific paper

In this article, we study the (d-1)-volume and the covering numbers of the medial axis of a compact set of the Euclidean d-space. In general, this volume is infinite; however, the (d-1)-volume and covering numbers of a filtered medial axis (the mu-medial axis) that is at distance greater than R from the compact set will be explicitely bounded. The behaviour of the bound we obtain with respect to mu, R and the covering numbers of the compact set K are optimal. From this result we deduce that the projection function on a compact subset K of the Euclidean d-space depends continuously on the compact set K, in the L^1 sense. This implies in particular that Federer's curvature measure of a compact set with positive reach can be reliably estimated from a Hausdorff approximation of this set, regardless of any regularity assumption on the approximation.

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