Boundary measures for geometric inference

Abstract

We study the boundary measures of compact subsets of the d-dimensional Euclidean space, which are closely related to the Federer's curvature measures. We show that they can be computed efficiently for point clouds and suggest these measures can be used for geometric inference. The main contribution of this work is the proof of a quantitative stability theorem for boundary measures using tools of convex analysis and geometric measure theory. As a corollary we obtain a stability result for Federer's curvature measures of a compact set, showing that they can be reliably estimated from point-cloud approximations.