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arXiv · 2212.01118

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

Hana Dal Poz Kouřimská·André Lieutier·Mathijs Wintraecken
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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.

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Hana Dal Poz Kouřimská, André Lieutier, Mathijs Wintraecken. 2022-12-02. The medial axis of closed bounded sets is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms. https://arxiv.org/abs/2212.01118

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