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We prove that the medial axis of closed sets is Hausdorff stable in thefollowing 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 sphereinvariant. The map from this space of diffeomorphisms (endowed with some Banachnorm) to the space of closed subsets of $\mathbb{R}^d$ (endowed with theHausdorff distance), mapping a diffeomorphism $F$ to the closure of the medialaxis of $F(\mathcal{S})$, is Lipschitz. This extends a previous stabilityresult of Chazal and Soufflet on the stability of the medial axis of $C^2$manifolds under $C^2$ ambient diffeomorphisms.

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