The Kuratowski convergence of medial axes and conflict sets

Abstract

This paper consists of two parts. In the first one we study the behaviour of medial axes (skeletons) of closed sets in a connected complete Riemannian manifold $\mathcal{M}$ under deformations. The second one is devoted to a similar study of conflict sets. We apply a new approach to the deformation process. Instead of seeing it as a `jump' from the initial to the final state, we perceive it as a continuous process, expressed using the Kuratowski convergence of sets (hence, unlike other authors, we do not require any regularity of the deformation). Our main `medial axis inner semi-continuity' result has already proved useful, as it was used to compute the tangent cone of the medial axis with application in singularity theory.