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

TL;DR

This paper investigates the size and stability of Federer's curvature measures by analyzing the medial axis of compact sets in Euclidean space, providing bounds and continuity results that facilitate reliable estimation from approximations.

Contribution

It introduces explicit bounds on the medial axis' volume and covering numbers, and proves the continuity of Federer's curvature measures with respect to Hausdorff approximations.

Findings

Bounded the (d-1)-volume and covering numbers of the filtered medial axis.

Established the continuous dependence of Federer's curvature measures on set approximations.

Demonstrated that curvature measures can be reliably estimated from Hausdorff approximations.

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 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.