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.

翻译：我们证明闭集的中心轴在以下意义下是豪斯多夫稳定的：设 $\mathcal{S} \subseteq \mathbb{R}^d$ 是（固定的）闭集（包含一个包围球）。考虑 $\mathbb{R}^d$ 到自身的 $C^{1,1}$ 微分同胚空间，这些微分同胚保持包围球不变。从该微分同胚空间（配备某种巴拿赫范数）到 $\mathbb{R}^d$ 闭子集空间（配备豪斯多夫距离）的映射，即将微分同胚 $F$ 映射到 $F(\mathcal{S})$ 的中心轴的闭包，是利普希茨的。这推广了 Chazal 和 Soufflet 先前关于 $C^2$ 流形在 $C^2$ 环境微分同胚下中心轴稳定性的结果。