2010/01/18 by Quentin Merigot, Merigot, Quentin
Computer Science · Mathematics · #28A78 #35F21 #52A39 #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG) #cs.CG #math.MG #msc:28A78 #msc:35F21 #msc:52A39
paper · pdf · doi:10.48550/arxiv.1001.2961
15 pages, 2 figures
arxiv created 2010/01/18 · arxiv updated 2010/02/26
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 L1 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.