The medial axis of closed bounded sets is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms

Hana Dal Poz Kourimska, André Lieutier, Mathijs Wintraecken

Keywords: Medial axis, Hausdorff distance, Lipschitz continuity

Abstract:

We prove that the medial axis of closed sets is Hausdorff stable in the following sense:
Let $\mathcal{S} \subseteq \mathbb{R}^d$ be a fixed closed set that contains a bounding sphere. Consider the space of $C^{1,1}$~diffeomorphisms of $\mathbb{R}^d$ to itself, which keep the bounding sphere invariant.
The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of $\mathbb{R}^d$ (endowed with the Hausdorff distance), mapping a diffeomorphism $F$ to the closure of the medial axis of $F(\mathcal{S})$, is Lipschitz.

This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of $C^2$~manifolds under $C^2$ ambient diffeomorphisms.

TLDR: We prove that the medial axis of ANY closed bounded set is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphism

Submission Number: 202