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Topology"],"published-print":{"date-parts":[[2025,9]]},"abstract":"Abstract<\/jats:title>\n We prove that the medial axis of a closed set is locally Hausdorff stable in the following sense: Let \n \n $${\\mathcal {S}}\\subseteq \\mathbb {R}^d$$<\/jats:tex-math>\n \n \n S<\/mml:mi>\n \u2286<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> be a fixed compact set and \n \n $$S(c,r)$$<\/jats:tex-math>\n \n \n S<\/mml:mi>\n (<\/mml:mo>\n c<\/mml:mi>\n ,<\/mml:mo>\n r<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> some sphere with radius r<\/jats:italic> containing \n \n $${\\mathcal {S}}$$<\/jats:tex-math>\n \n S<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in its interior. Consider the space of \n \n $$C^{1,1}$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n \n 1<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\u00a0diffeomorphisms of\u00a0\n \n $$\\mathbb {R}^d$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> to itself, which keep the exterior of \n \n $$S(c,r)$$<\/jats:tex-math>\n \n \n S<\/mml:mi>\n (<\/mml:mo>\n c<\/mml:mi>\n ,<\/mml:mo>\n r<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of \n \n $$\\mathbb {R}^d$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> (endowed with the Hausdorff distance), mapping a diffeomorphism F<\/jats:italic> to the closure of the medial axis of \n \n $$F({\\mathcal {S}}\\cup S(c,r))$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n (<\/mml:mo>\n S<\/mml:mi>\n \u222a<\/mml:mo>\n S<\/mml:mi>\n (<\/mml:mo>\n c<\/mml:mi>\n ,<\/mml:mo>\n r<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, is Lipschitz. A similar statement holds if \n \n $${\\mathcal {S}}$$<\/jats:tex-math>\n \n S<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is non-compact but the Hausdorff distance between \n \n $${\\mathcal {S}}$$<\/jats:tex-math>\n \n S<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$\\mathbb {R}^d$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is bounded; in other words, every point in \n \n $$\\mathbb {R}^d$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> has a point in \n \n $${\\mathcal {S}}$$<\/jats:tex-math>\n \n S<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> 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 \n \n $$C^2$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\u00a0manifolds under \n \n $$C^2$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> ambient diffeomorphisms.<\/jats:p>","DOI":"10.1007\/s41468-025-00216-w","type":"journal-article","created":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T13:44:00Z","timestamp":1753278240000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The medial axis of any closed bounded set is locally Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms"],"prefix":"10.1007","volume":"9","author":[{"given":"Hana","family":"Dal Poz Kou\u0159imsk\u00e1","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andr\u00e9","family":"Lieutier","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mathijs","family":"Wintraecken","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,7,23]]},"reference":[{"key":"216_CR1","doi-asserted-by":"publisher","unstructured":"Aamari, E., Knop, A.: Statistical query complexity of manifold estimation. 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