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

12/02/2022

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by   Hana Dal Poz Kouřimská, et al.

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We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮⊆ℝ^d be (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 some 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.