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

In this article, we study the (d − 1)-volume and the covering numbers of the medial axis of a compact subset of R. 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 ε from the compact set can be explicitely bounded. The behaviour of the bound we obtain with respect to μ, ε and the covering numbers of K is optimal. From this result we deduce that the projection function on a compact subsetK of R depends continuously on the compact set K, in the L sense. This implies in particular that Federer’s curvature measure of a compact subset of R with positive reach can be reliably estimated from a Hausdorff approximation of this subset, regardless of any regularity assumption on the approximating subset.