We prove that the medial axis of a closed set is locally Hausdorff stable in the following sense: Let \({\mathcal {S}}\subseteq \mathbb {R}^d\) be a fixed compact set and \(S(c,r)\) some sphere with radius r containing \({\mathcal {S}}\) in its interior. Consider the space of \(C^{1,1}\) diffeomorphisms of \(\mathbb {R}^d\) to itself, which keep the exterior of \(S(c,r)\) 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 \(F({\mathcal {S}}\cup S(c,r))\) , is Lipschitz. A similar statement holds if \({\mathcal {S}}\) is non-compact but the Hausdorff distance between \({\mathcal {S}}\) and \(\mathbb {R}^d\) is bounded; in other words, every point in \(\mathbb {R}^d\) has a point in \({\mathcal {S}}\) nearby. The latter statement can further be localized at the cost of having to consider two one-sided Hausdorff distances. Our result extends result of Chazal and Soufflet on the stability of the medial axis of \(C^2\) manifolds under \(C^2\) ambient diffeomorphisms.
