We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let S ⊆ R d be a fixed closed set that contains a bounding sphere. Consider the space of C 1,1 diffeomorphisms of R d to itself, which keep the bounding sphere 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), 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.