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

TL;DR

This paper proves that the medial axis of closed sets remains stable under small perturbations of ambient space via Lipschitz continuous transformations, extending previous stability results to broader classes of sets.

Contribution

It establishes Lipschitz stability of the medial axis for closed sets under $C^{1,1}$ ambient diffeomorphisms, generalizing prior $C^2$ manifold results.

Findings

Medial axis is Lipschitz stable under ambient $C^{1,1}$ diffeomorphisms.

Stability extends previous $C^2$ manifold results to more general closed sets.

Provides a quantitative measure of medial axis stability in the Hausdorff metric.

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 (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 some 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.