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

Abstract

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 μ-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 μ, 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 L¹ sense. This implies in particular that Federer's curvature measure of a compact subset with positive reach can be reliably estimated from a Hausdorff approximation of this subset, regardless of any regularity assumption on the approximating subset.