The Kuratowski convergence of medial axes and conflict sets

TL;DR

This paper investigates how medial axes and conflict sets behave under continuous deformations in Riemannian manifolds using Kuratowski convergence, without requiring regularity in the deformation process.

Contribution

It introduces a new approach to study deformations of medial axes and conflict sets via Kuratowski convergence, enabling analysis without regularity assumptions.

Findings

Proves medial axis inner semi-continuity under deformation.

Provides a method to compute tangent cones of medial axes.

Applies results to singularity theory.

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.