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

We prove that the medial axis of a closed set is locally Hausdorff stable in the following sense: Let S \(\subseteq\) R^d be a fixed compact set and S(c,r) some sphere with radius r containing S in its interior. Consider the space of C^1,1 diffeomorphisms of 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 R^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(S \(\cup\) S(c,r)), is Lipschitz. A similar statement holds if S is non-compact but the Hausdorff distance between S and R^d is bounded; in other words, every point in R^d has a point in 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 C² manifolds under C² ambient diffeomorphisms.