The medial axis of closed bounded sets is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms
arxiv(2022)
摘要
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.
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