[{"quality_controlled":"1","conference":{"name":"SoCG: Symposium on Computational Geometry","location":"Athens, Greece","end_date":"2024-06-14"},"oa_version":"Published Version","title":"The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms","article_number":"69","ec_funded":1,"date_published":"2024-06-01T00:00:00Z","status":"public","publication":"40th International Symposium on Computational Geometry","publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.\r\nSupported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Cô d'Azur.\r\nWe are greatly indebted to Fred Chazal for sharing his insights. We further thank Erin Chambers, Christopher Fillmore, and Elizabeth Stephenson for early discussions and all members of the Edelsbrunner group (Institute of Science and Technology Austria) and the Datashape team (Inria) for the atmosphere in which this research was conducted.","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"call_identifier":"H2020","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships"},{"name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","grant_number":"M03073"}],"citation":{"mla":"Kourimska, Hana, et al. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 69, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">10.4230/LIPIcs.SoCG.2024.69</a>.","chicago":"Kourimska, Hana, André Lieutier, and Mathijs Wintraecken. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>.","ama":"Kourimska H, Lieutier A, Wintraecken M. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">10.4230/LIPIcs.SoCG.2024.69</a>","ista":"Kourimska H, Lieutier A, Wintraecken M. 2024. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 69.","ieee":"H. Kourimska, A. Lieutier, and M. Wintraecken, “The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","apa":"Kourimska, H., Lieutier, A., &#38; Wintraecken, M. (2024). The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>","short":"H. Kourimska, A. Lieutier, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024."},"arxiv":1,"date_created":"2024-06-16T22:01:06Z","day":"01","scopus_import":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"type":"conference","author":[{"last_name":"Kourimska","orcid":"0000-0001-7841-0091","first_name":"Hana","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","full_name":"Kourimska, Hana"},{"last_name":"Lieutier","first_name":"André","full_name":"Lieutier, André"},{"orcid":"0000-0002-7472-2220","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs"}],"doi":"10.4230/LIPIcs.SoCG.2024.69","abstract":[{"lang":"eng","text":"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. That is, the bounding sphere is part of the set 𝒮. 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 C² manifolds under C² ambient diffeomorphisms."}],"department":[{"_id":"HeEd"}],"article_processing_charge":"No","external_id":{"arxiv":["2212.01118"]},"language":[{"iso":"eng"}],"has_accepted_license":"1","intvolume":"       293","oa":1,"ddc":["510"],"_id":"17144","date_updated":"2025-04-15T07:16:58Z","file":[{"file_size":1612558,"checksum":"b40ff456c19294adb5d9613fcfd751c6","access_level":"open_access","date_updated":"2024-06-17T08:33:40Z","file_name":"2024_LIPICS_Kourimska.pdf","relation":"main_file","success":1,"creator":"dernst","file_id":"17150","content_type":"application/pdf","date_created":"2024-06-17T08:33:40Z"}],"file_date_updated":"2024-06-17T08:33:40Z","volume":293,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","month":"06","year":"2024","alternative_title":["LIPIcs"],"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.69"},{"author":[{"last_name":"Attali","first_name":"Dominique","full_name":"Attali, Dominique"},{"full_name":"Kourimska, Hana","orcid":"0000-0001-7841-0091","first_name":"Hana","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","last_name":"Kourimska"},{"last_name":"Fillmore","first_name":"Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","full_name":"Fillmore, Christopher D"},{"first_name":"Ishika","id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff","last_name":"Ghosh","full_name":"Ghosh, Ishika"},{"full_name":"Lieutier, André","last_name":"Lieutier","first_name":"André"},{"first_name":"Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-6862-208X","last_name":"Stephenson","full_name":"Stephenson, Elizabeth R"},{"last_name":"Wintraecken","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs"}],"page":"11:1-11:19","doi":"10.4230/LIPIcs.SoCG.2024.11","article_processing_charge":"No","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"abstract":[{"text":"In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of C² manifolds with positive reach embedded in ℝ^d. We extend their results in the following ways: - As the ambient space we consider both ℝ^d and Riemannian manifolds with lower bounded sectional curvature. - In both types of ambient spaces, we study sets of positive reach - a significantly more general setting than C² manifolds - as well as general manifolds of positive reach. - The sample P of a set (or a manifold) 𝒮 of positive reach may be noisy. We work with two one-sided Hausdorff distances - ε and δ - between P and 𝒮. We provide tight bounds in terms of ε and δ, that guarantee that there exists a parameter r such that the union of balls of radius r centred at the sample P deformation-retracts to 𝒮. We exhibit their tightness by an explicit construction. We carefully distinguish the roles of δ and ε. This is not only essential to achieve tight bounds, but also sensible in practical situations, since it allows one to adapt the bound according to sample density and the amount of noise present in the sample separately.","lang":"eng"}],"language":[{"iso":"eng"}],"external_id":{"arxiv":["2206.10485"]},"ddc":["516"],"intvolume":"       293","has_accepted_license":"1","oa":1,"_id":"17170","date_updated":"2025-04-15T07:16:57Z","file":[{"date_created":"2024-06-25T11:47:26Z","file_id":"17171","creator":"cfillmor","content_type":"application/pdf","relation":"main_file","file_name":"LIPIcs.SoCG.2024.11.pdf","success":1,"date_updated":"2024-06-25T11:47:26Z","access_level":"open_access","checksum":"6a2ddc8b51aa58f197a8b294750f1f8d","file_size":20886142}],"file_date_updated":"2024-06-25T11:47:26Z","month":"06","year":"2024","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","volume":293,"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.11","alternative_title":["LIPIcs"],"conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11","end_date":"2024-06-14","location":"Athens, Greece"},"quality_controlled":"1","ec_funded":1,"title":"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds","oa_version":"Published Version","date_published":"2024-06-06T00:00:00Z","publication":"40th International Symposium on Computational Geometry","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"published","status":"public","publication_identifier":{"isbn":["9783959773164"],"eissn":["1868-8969"]},"acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.\r\nWintraecken, Mathijs: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Côte d'Azur.","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","name":"Alpha Shape Theory Extended","call_identifier":"H2020"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF"},{"name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","grant_number":"M03073"}],"citation":{"chicago":"Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, André Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” In <i>40th International Symposium on Computational Geometry</i>, 293:11:1-11:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>.","mla":"Attali, Dominique, et al. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>.","ama":"Attali D, Kourimska H, Fillmore CD, et al. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024:11:1-11:19. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>","ieee":"D. Attali <i>et al.</i>, “Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293, p. 11:1-11:19.","ista":"Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken M. 2024. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 11:1-11:19.","short":"D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19.","apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., &#38; Wintraecken, M. (2024). Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293, p. 11:1-11:19). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>"},"date_created":"2024-06-25T11:45:58Z","day":"06","scopus_import":"1","arxiv":1,"type":"conference","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"}},{"corr_author":"1","day":"06","date_created":"2024-09-19T10:29:48Z","type":"conference","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I02979-N35. Mathijs Wintraecken: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Côte d’Azur.\r\nWe thank Jean-Daniel Boissonnat, Herbert Edelsbrunner, and Mariette Yvinec for discussion.","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342","name":"Mathematics, Computer Science"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"},{"call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425"},{"_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","grant_number":"M03073","name":"Learning and triangulating manifolds via collapses"}],"citation":{"chicago":"Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, Andre Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">https://doi.org/10.4230/LIPIcs.SoCG.2024.87</a>.","mla":"Attali, Dominique, et al. “The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 87, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">10.4230/LIPIcs.SoCG.2024.87</a>.","ama":"Attali D, Kourimska H, Fillmore CD, et al. The ultimate frontier: An optimality construction for homotopy inference (media exposition). In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">10.4230/LIPIcs.SoCG.2024.87</a>","ieee":"D. Attali <i>et al.</i>, “The ultimate frontier: An optimality construction for homotopy inference (media exposition),” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","ista":"Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken M. 2024. The ultimate frontier: An optimality construction for homotopy inference (media exposition). 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 87.","short":"D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., &#38; Wintraecken, M. (2024). The ultimate frontier: An optimality construction for homotopy inference (media exposition). In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.87\">https://doi.org/10.4230/LIPIcs.SoCG.2024.87</a>"},"publication":"40th International Symposium on Computational Geometry","publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","publication_identifier":{"isbn":["9783959773164"]},"conference":{"location":"Athens, Greece","end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11"},"quality_controlled":"1","ec_funded":1,"article_number":"87","oa_version":"Published Version","title":"The ultimate frontier: An optimality construction for homotopy inference (media exposition)","date_published":"2024-06-06T00:00:00Z","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","month":"06","year":"2024","volume":293,"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.87","alternative_title":["LIPIcs"],"date_updated":"2025-04-15T07:16:58Z","file":[{"date_created":"2024-09-19T10:30:37Z","file_name":"2024_LIPICs_Attali.pdf","relation":"main_file","success":1,"file_id":"18098","creator":"dernst","content_type":"application/pdf","access_level":"open_access","date_updated":"2024-09-19T10:30:37Z","checksum":"9355c2e60b8ec285e1b22719c5b73f1a","file_size":3507177}],"file_date_updated":"2024-09-19T10:30:37Z","language":[{"iso":"eng"}],"ddc":["000"],"has_accepted_license":"1","intvolume":"       293","oa":1,"_id":"18097","author":[{"full_name":"Attali, Dominique","last_name":"Attali","first_name":"Dominique"},{"full_name":"Kourimska, Hana","orcid":"0000-0001-7841-0091","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana","last_name":"Kourimska"},{"last_name":"Fillmore","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","first_name":"Christopher D","full_name":"Fillmore, Christopher D"},{"last_name":"Ghosh","first_name":"Ishika","id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff","full_name":"Ghosh, Ishika"},{"full_name":"Lieutier, Andre","last_name":"Lieutier","first_name":"Andre"},{"id":"2D04F932-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-6862-208X","first_name":"Elizabeth R","last_name":"Stephenson","full_name":"Stephenson, Elizabeth R"},{"full_name":"Wintraecken, Mathijs","last_name":"Wintraecken","first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220"}],"doi":"10.4230/LIPIcs.SoCG.2024.87","article_processing_charge":"Yes","abstract":[{"lang":"eng","text":"In our companion paper \"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds\" we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample P of an input shape 𝒮 (either manifold or general set with positive reach) such that one can infer the homotopy of 𝒮 from the union of balls with some radius centred at P, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video is to visualize and thus elucidate said construction in the Euclidean setting."}],"department":[{"_id":"HeEd"}]},{"fulldoi":"https://doi.org/10.1007/s00454-023-00484-2","pmid":1,"year":"2023","publisher":"Springer Nature","month":"07","volume":70,"file_date_updated":"2023-10-04T11:46:24Z","file":[{"file_size":1026683,"checksum":"cdbf90ba4a7ddcb190d37b9e9d4cb9d3","date_updated":"2023-10-04T11:46:24Z","access_level":"open_access","content_type":"application/pdf","creator":"dernst","file_id":"14396","success":1,"file_name":"2023_DiscreteGeometry_Kourimska.pdf","relation":"main_file","date_created":"2023-10-04T11:46:24Z"}],"date_updated":"2025-04-23T08:59:15Z","_id":"12764","ddc":["510"],"intvolume":"        70","has_accepted_license":"1","oa":1,"language":[{"iso":"eng"}],"external_id":{"pmid":["37292248"],"isi":["000948148000001"]},"article_processing_charge":"Yes (via OA deal)","abstract":[{"text":"We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique.","lang":"eng"}],"department":[{"_id":"HeEd"}],"doi":"10.1007/s00454-023-00484-2","author":[{"first_name":"Hana","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","orcid":"0000-0001-7841-0091","last_name":"Kourimska","full_name":"Kourimska, Hana"}],"page":"123-153","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"day":"01","date_created":"2023-03-26T22:01:09Z","scopus_import":"1","article_type":"original","isi":1,"corr_author":"1","citation":{"apa":"Kourimska, H. (2023). Discrete yamabe problem for polyhedral surfaces. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00484-2\">https://doi.org/10.1007/s00454-023-00484-2</a>","short":"H. Kourimska, Discrete and Computational Geometry 70 (2023) 123–153.","ista":"Kourimska H. 2023. Discrete yamabe problem for polyhedral surfaces. Discrete and Computational Geometry. 70, 123–153.","ieee":"H. Kourimska, “Discrete yamabe problem for polyhedral surfaces,” <i>Discrete and Computational Geometry</i>, vol. 70. Springer Nature, pp. 123–153, 2023.","ama":"Kourimska H. Discrete yamabe problem for polyhedral surfaces. <i>Discrete and Computational Geometry</i>. 2023;70:123-153. doi:<a href=\"https://doi.org/10.1007/s00454-023-00484-2\">10.1007/s00454-023-00484-2</a>","mla":"Kourimska, Hana. “Discrete Yamabe Problem for Polyhedral Surfaces.” <i>Discrete and Computational Geometry</i>, vol. 70, Springer Nature, 2023, pp. 123–53, doi:<a href=\"https://doi.org/10.1007/s00454-023-00484-2\">10.1007/s00454-023-00484-2</a>.","chicago":"Kourimska, Hana. “Discrete Yamabe Problem for Polyhedral Surfaces.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00454-023-00484-2\">https://doi.org/10.1007/s00454-023-00484-2</a>."},"project":[{"call_identifier":"FWF","name":"Algebraic Footprints of Geometric Features in Homology","grant_number":"I04245","_id":"26AD5D90-B435-11E9-9278-68D0E5697425"}],"acknowledgement":"Open access funding provided by the Austrian Science Fund (FWF). This research was supported by the FWF grant, Project number I4245-N35, and by the Deutsche Forschungsgemeinschaft (DFG - German Research Foundation) - Project-ID 195170736 - TRR109.","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"publication":"Discrete and Computational Geometry","publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","date_published":"2023-07-01T00:00:00Z","title":"Discrete yamabe problem for polyhedral surfaces","oa_version":"Published Version","quality_controlled":"1"},{"author":[{"first_name":"Henry","last_name":"Adams","full_name":"Adams, Henry"},{"last_name":"Kourimska","orcid":"0000-0001-7841-0091","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana","full_name":"Kourimska, Hana"},{"first_name":"Teresa","orcid":"0000-0002-1780-2689","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","last_name":"Heiss","full_name":"Heiss, Teresa"},{"last_name":"Percival","first_name":"Sarah","full_name":"Percival, Sarah"},{"full_name":"Ziegelmeier, Lori","first_name":"Lori","last_name":"Ziegelmeier"}],"page":"1511-1514","doi":"10.1090/noti2349","article_processing_charge":"No","department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"ddc":["500"],"intvolume":"        68","oa":1,"_id":"10071","issue":"9","date_updated":"2026-06-18T08:36:13Z","publisher":"American Mathematical Society","year":"2021","month":"10","volume":68,"fulldoi":"https://doi.org/10.1090/noti2349","alternative_title":["Early Career"],"main_file_link":[{"open_access":"1","url":"http://www.ams.org/notices/"}],"quality_controlled":"1","oa_version":"Published Version","title":"How to tutorial-a-thon","date_published":"2021-10-01T00:00:00Z","publication":"Notices of the American Mathematical Society","publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","publication_identifier":{"eissn":["1088-9477"],"issn":["0002-9920"]},"citation":{"ama":"Adams H, Kourimska H, Heiss T, Percival S, Ziegelmeier L. How to tutorial-a-thon. <i>Notices of the American Mathematical Society</i>. 2021;68(9):1511-1514. doi:<a href=\"https://doi.org/10.1090/noti2349\">10.1090/noti2349</a>","chicago":"Adams, Henry, Hana Kourimska, Teresa Heiss, Sarah Percival, and Lori Ziegelmeier. “How to Tutorial-a-Thon.” <i>Notices of the American Mathematical Society</i>. American Mathematical Society, 2021. <a href=\"https://doi.org/10.1090/noti2349\">https://doi.org/10.1090/noti2349</a>.","mla":"Adams, Henry, et al. “How to Tutorial-a-Thon.” <i>Notices of the American Mathematical Society</i>, vol. 68, no. 9, American Mathematical Society, 2021, pp. 1511–14, doi:<a href=\"https://doi.org/10.1090/noti2349\">10.1090/noti2349</a>.","short":"H. Adams, H. Kourimska, T. Heiss, S. Percival, L. Ziegelmeier, Notices of the American Mathematical Society 68 (2021) 1511–1514.","apa":"Adams, H., Kourimska, H., Heiss, T., Percival, S., &#38; Ziegelmeier, L. (2021). How to tutorial-a-thon. <i>Notices of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/noti2349\">https://doi.org/10.1090/noti2349</a>","ieee":"H. Adams, H. Kourimska, T. Heiss, S. Percival, and L. Ziegelmeier, “How to tutorial-a-thon,” <i>Notices of the American Mathematical Society</i>, vol. 68, no. 9. American Mathematical Society, pp. 1511–1514, 2021.","ista":"Adams H, Kourimska H, Heiss T, Percival S, Ziegelmeier L. 2021. How to tutorial-a-thon. Notices of the American Mathematical Society. 68(9), 1511–1514."},"date_created":"2021-10-03T22:01:22Z","scopus_import":"1","day":"01","article_type":"letter_note","type":"journal_article"}]