[{"publication_status":"published","corr_author":"1","language":[{"iso":"eng"}],"date_created":"2025-10-12T22:01:26Z","external_id":{"arxiv":["2212.03121"],"isi":["001584166900001"]},"department":[{"_id":"HeEd"}],"quality_controlled":"1","volume":75,"abstract":[{"text":"Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.","lang":"eng"}],"license":"https://creativecommons.org/licenses/by/4.0/","isi":1,"intvolume":"        75","ec_funded":1,"file_date_updated":"2026-01-05T13:21:20Z","publication":"Discrete and Computational Geometry","author":[{"orcid":"0000-0002-5372-7890","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","last_name":"Biswas","full_name":"Biswas, Ranita"},{"full_name":"Cultrera di Montesano, Sebastiano","last_name":"Cultrera di Montesano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"orcid":"0000-0003-0464-3823","last_name":"Draganov","full_name":"Draganov, Ondrej","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","first_name":"Ondrej"},{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","last_name":"Saghafian"}],"scopus_import":"1","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"oa":1,"file":[{"success":1,"access_level":"open_access","relation":"main_file","file_name":"2026_DiscreteCompGeom_Biswas.pdf","creator":"dernst","file_id":"20952","date_created":"2026-01-05T13:21:20Z","file_size":570922,"content_type":"application/pdf","checksum":"0addb5c1b78142f9fb453bfa04695400","date_updated":"2026-01-05T13:21:20Z"}],"acknowledgement":"The fourth author thanks Boris Aronov for insightful discussions on the size of the overlay of Voronoi tessellations. Open access funding provided by Institute of Science and Technology (IST Austria). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35.","ddc":["510"],"arxiv":1,"OA_type":"hybrid","article_type":"original","doi":"10.1007/s00454-025-00778-7","OA_place":"publisher","month":"01","title":"On the size of chromatic Delaunay mosaics","_id":"20456","PlanS_conform":"1","related_material":{"record":[{"id":"15090","status":"public","relation":"earlier_version"}]},"citation":{"chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s00454-025-00778-7\">https://doi.org/10.1007/s00454-025-00778-7</a>.","ista":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. 75, 24–47.","ama":"Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. <i>Discrete and Computational Geometry</i>. 2026;75:24-47. doi:<a href=\"https://doi.org/10.1007/s00454-025-00778-7\">10.1007/s00454-025-00778-7</a>","apa":"Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (2026). On the size of chromatic Delaunay mosaics. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00778-7\">https://doi.org/10.1007/s00454-025-00778-7</a>","ieee":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” <i>Discrete and Computational Geometry</i>, vol. 75. Springer Nature, pp. 24–47, 2026.","short":"R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 75 (2026) 24–47.","mla":"Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>, vol. 75, Springer Nature, 2026, pp. 24–47, doi:<a href=\"https://doi.org/10.1007/s00454-025-00778-7\">10.1007/s00454-025-00778-7</a>."},"article_processing_charge":"Yes (via OA deal)","date_updated":"2026-01-05T13:21:56Z","oa_version":"Published Version","year":"2026","has_accepted_license":"1","date_published":"2026-01-01T00:00:00Z","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"page":"24-47","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","publisher":"Springer Nature","type":"journal_article","day":"01"},{"corr_author":"1","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"external_id":{"isi":["001610592600001"],"arxiv":["2310.14801"]},"date_created":"2025-11-19T09:44:58Z","publication_status":"epub_ahead","abstract":[{"text":"The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². ","lang":"eng"}],"quality_controlled":"1","ec_funded":1,"isi":1,"author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János","last_name":"Pach","full_name":"Pach, János"}],"scopus_import":"1","publication":"Discrete & Computational Geometry","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00454-025-00796-5"}],"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"oa":1,"acknowledgement":"The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. The second author is supported by the European Research Council (ERC), grant “GeoScape” and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.","doi":"10.1007/s00454-025-00796-5","OA_place":"publisher","OA_type":"hybrid","article_type":"original","arxiv":1,"ddc":["510"],"citation":{"ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. 2025. doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>","apa":"Edelsbrunner, H., &#38; Pach, J. (2025). Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>.","ista":"Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete &#38; Computational Geometry.","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>.","short":"H. Edelsbrunner, J. Pach, Discrete &#38; Computational Geometry (2025).","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025."},"related_material":{"record":[{"id":"17146","relation":"earlier_version","status":"public"}]},"PlanS_conform":"1","article_processing_charge":"Yes (via OA deal)","title":"Maximum Betti numbers of Čech complexes","month":"11","_id":"20657","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025","oa_version":"Published Version","date_updated":"2025-12-01T15:19:21Z","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"}],"has_accepted_license":"1","date_published":"2025-11-10T00:00:00Z","publisher":"Springer Nature","type":"journal_article","day":"10","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public"},{"scopus_import":"1","file_date_updated":"2025-04-23T07:31:32Z","publication":"Discrete & Computational Geometry","author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"orcid":"0000-0002-0659-3201","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton","full_name":"Nikitenko, Anton","last_name":"Nikitenko"}],"ec_funded":1,"intvolume":"        73","isi":1,"file":[{"file_name":"2025_DiscreteComputGeom_EdelsbrunnerHe.pdf","creator":"dernst","date_created":"2025-04-23T07:31:32Z","file_id":"19610","file_size":283443,"content_type":"application/pdf","checksum":"ffb0c818222138f9f113f4bbea41e834","date_updated":"2025-04-23T07:31:32Z","success":1,"access_level":"open_access","relation":"main_file"}],"acknowledgement":"The authors thank Ranita Biswas and Tatiana Ezubova for the collaboration on computational experiments that motivated the work reported in this paper. The authors also thank Daniel Bonnema for proofreading and noticing an issue with the original proof of Lemma 4.3.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).\r\nThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.","pmid":1,"oa":1,"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"publication_status":"published","external_id":{"pmid":["39974750"],"arxiv":["2012.03350"],"isi":["001238566200004"]},"department":[{"_id":"HeEd"}],"date_created":"2024-06-16T22:01:07Z","corr_author":"1","language":[{"iso":"eng"}],"volume":73,"quality_controlled":"1","abstract":[{"lang":"eng","text":"The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about 4/Pi. We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes."}],"has_accepted_license":"1","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"}],"date_published":"2025-03-01T00:00:00Z","year":"2025","oa_version":"Published Version","date_updated":"2026-02-16T12:18:50Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"490-499","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"day":"01","type":"journal_article","publisher":"Springer Nature","arxiv":1,"ddc":["510"],"OA_place":"publisher","doi":"10.1007/s00454-024-00660-y","OA_type":"hybrid","article_type":"original","_id":"17149","title":"Average and expected distortion of Voronoi paths and scapes","month":"03","article_processing_charge":"Yes (via OA deal)","citation":{"ama":"Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. 2025;73:490-499. doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>","apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>.","ista":"Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete &#38; Computational Geometry. 73, 490–499.","short":"H. Edelsbrunner, A. Nikitenko, Discrete &#38; Computational Geometry 73 (2025) 490–499.","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>, vol. 73, Springer Nature, 2025, pp. 490–99, doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>.","ieee":"H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi paths and scapes,” <i>Discrete &#38; Computational Geometry</i>, vol. 73. Springer Nature, pp. 490–499, 2025."}},{"article_processing_charge":"Yes (via OA deal)","citation":{"ista":"Aronov B, Basit A, Ramesh I, Tasinato G, Wagner U. 2025. Eight-partitioning points in 3D, and efficiently too. Discrete &#38; Computational Geometry.","chicago":"Aronov, Boris, Abdul Basit, Indu Ramesh, Gianluca Tasinato, and Uli Wagner. “Eight-Partitioning Points in 3D, and Efficiently Too.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-025-00739-0\">https://doi.org/10.1007/s00454-025-00739-0</a>.","apa":"Aronov, B., Basit, A., Ramesh, I., Tasinato, G., &#38; Wagner, U. (2025). Eight-partitioning points in 3D, and efficiently too. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00739-0\">https://doi.org/10.1007/s00454-025-00739-0</a>","ama":"Aronov B, Basit A, Ramesh I, Tasinato G, Wagner U. Eight-partitioning points in 3D, and efficiently too. <i>Discrete &#38; Computational Geometry</i>. 2025. doi:<a href=\"https://doi.org/10.1007/s00454-025-00739-0\">10.1007/s00454-025-00739-0</a>","ieee":"B. Aronov, A. Basit, I. Ramesh, G. Tasinato, and U. Wagner, “Eight-partitioning points in 3D, and efficiently too,” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025.","mla":"Aronov, Boris, et al. “Eight-Partitioning Points in 3D, and Efficiently Too.” <i>Discrete &#38; Computational Geometry</i>, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00454-025-00739-0\">10.1007/s00454-025-00739-0</a>.","short":"B. Aronov, A. Basit, I. Ramesh, G. Tasinato, U. Wagner, Discrete &#38; Computational Geometry (2025)."},"related_material":{"link":[{"url":"https://doi.org/10.1007/s00454-025-00759-w","relation":"erratum"}],"record":[{"id":"18917","relation":"earlier_version","status":"public"},{"id":"20339","status":"public","relation":"dissertation_contains"}]},"_id":"19860","title":"Eight-partitioning points in 3D, and efficiently too","month":"06","doi":"10.1007/s00454-025-00739-0","OA_place":"publisher","article_type":"original","OA_type":"hybrid","arxiv":1,"publisher":"Springer Nature","type":"journal_article","day":"12","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2025-06-12T00:00:00Z","oa_version":"Published Version","year":"2025","date_updated":"2026-04-07T12:36:50Z","abstract":[{"text":"An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in R^3\r\n consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in R^3 admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: any mass distribution (or point set) in R^3 admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of n points in R^3 (with prescribed normal direction of one of the planes) in time O(n^7/3). A preliminary version of this work appeared in SoCG’24 (Aronov et al., 40th International Symposium on Computational Geometry, 2024).","lang":"eng"}],"quality_controlled":"1","external_id":{"arxiv":["2403.02627"],"isi":["001506904300001"]},"department":[{"_id":"UlWa"}],"date_created":"2025-06-22T22:02:07Z","language":[{"iso":"eng"}],"publication_status":"epub_ahead","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00454-025-00739-0"}],"acknowledgement":"BA and AB would like to thank William Steiger for insightful initial discussions of the problems addressed in this work. Open Access funding enabled and organized by CAUL and its Member Institutions.","oa":1,"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"author":[{"first_name":"Boris","last_name":"Aronov","full_name":"Aronov, Boris"},{"first_name":"Abdul","full_name":"Basit, Abdul","last_name":"Basit"},{"last_name":"Ramesh","full_name":"Ramesh, Indu","first_name":"Indu"},{"id":"0433290C-AF8F-11E9-A4C7-F729E6697425","first_name":"Gianluca","full_name":"Tasinato, Gianluca","last_name":"Tasinato"},{"orcid":"0000-0002-1494-0568","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","full_name":"Wagner, Uli","last_name":"Wagner"}],"scopus_import":"1","publication":"Discrete & Computational Geometry","isi":1},{"month":"09","title":"The crossing Tverberg theorem","_id":"13974","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"6647"}]},"citation":{"ista":"Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. 2024. The crossing Tverberg theorem. Discrete and Computational Geometry. 72, 831–848.","chicago":"Fulek, Radoslav, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli Wagner. “The Crossing Tverberg Theorem.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00454-023-00532-x\">https://doi.org/10.1007/s00454-023-00532-x</a>.","apa":"Fulek, R., Gärtner, B., Kupavskii, A., Valtr, P., &#38; Wagner, U. (2024). The crossing Tverberg theorem. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00532-x\">https://doi.org/10.1007/s00454-023-00532-x</a>","ama":"Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. The crossing Tverberg theorem. <i>Discrete and Computational Geometry</i>. 2024;72:831-848. doi:<a href=\"https://doi.org/10.1007/s00454-023-00532-x\">10.1007/s00454-023-00532-x</a>","ieee":"R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, and U. Wagner, “The crossing Tverberg theorem,” <i>Discrete and Computational Geometry</i>, vol. 72. Springer Nature, pp. 831–848, 2024.","mla":"Fulek, Radoslav, et al. “The Crossing Tverberg Theorem.” <i>Discrete and Computational Geometry</i>, vol. 72, Springer Nature, 2024, pp. 831–48, doi:<a href=\"https://doi.org/10.1007/s00454-023-00532-x\">10.1007/s00454-023-00532-x</a>.","short":"R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, U. Wagner, Discrete and Computational Geometry 72 (2024) 831–848."},"article_processing_charge":"No","arxiv":1,"article_type":"original","OA_type":"green","doi":"10.1007/s00454-023-00532-x","OA_place":"repository","status":"public","day":"01","type":"journal_article","publisher":"Springer Nature","date_updated":"2025-04-14T13:52:36Z","year":"2024","oa_version":"Preprint","project":[{"_id":"261FA626-B435-11E9-9278-68D0E5697425","grant_number":"M02281","call_identifier":"FWF","name":"Eliminating intersections in drawings of graphs"}],"date_published":"2024-09-01T00:00:00Z","page":"831-848","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","volume":72,"abstract":[{"lang":"eng","text":"The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r−1)+1 points in Rd, one can find a partition X=X1∪⋯∪Xr of X, such that the convex hulls of the Xi, i=1,…,r, all share a common point. In this paper, we prove a trengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span ⌊n/3⌋ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Álvarez-Rebollar et al. guarantees ⌊n/6⌋pairwise crossing triangles. Our result generalizes to a result about simplices in Rd, d≥2."}],"publication_status":"published","language":[{"iso":"eng"}],"date_created":"2023-08-06T22:01:12Z","department":[{"_id":"UlWa"}],"external_id":{"isi":["001038546500001"],"arxiv":["1812.04911"]},"oa":1,"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"acknowledgement":"Part of the research leading to this paper was done during the 16th Gremo Workshop on Open Problems (GWOP), Waltensburg, Switzerland, June 12–16, 2018. We thank Patrick Schnider for suggesting the problem, and Stefan Felsner, Malte Milatz, and Emo Welzl for fruitful discussions during the workshop. We also thank Stefan Felsner and Manfred Scheucher for finding, communicating the example from Sect. 3.3, and the kind permission to include their visualization of the point set. We thank Dömötör Pálvölgyi, the SoCG reviewers, and DCG reviewers for various helpful comments.\r\nR. Fulek gratefully acknowledges support from Austrian Science Fund (FWF), Project  M2281-N35. A. Kupavskii was supported by the Advanced Postdoc.Mobility Grant no. P300P2_177839 of the Swiss National Science Foundation. Research by P. Valtr was supported by the Grant no. 18-19158 S of the Czech Science Foundation (GAČR).","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1812.04911","open_access":"1"}],"isi":1,"intvolume":"        72","scopus_import":"1","author":[{"orcid":"0000-0001-8485-1774","full_name":"Fulek, Radoslav","last_name":"Fulek","first_name":"Radoslav","id":"39F3FFE4-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Bernd","last_name":"Gärtner","full_name":"Gärtner, Bernd"},{"full_name":"Kupavskii, Andrey","last_name":"Kupavskii","first_name":"Andrey"},{"first_name":"Pavel","full_name":"Valtr, Pavel","last_name":"Valtr"},{"orcid":"0000-0002-1494-0568","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","last_name":"Wagner","full_name":"Wagner, Uli"}],"publication":"Discrete and Computational Geometry"},{"year":"2024","oa_version":"Published Version","date_updated":"2025-04-23T08:41:59Z","date_published":"2024-07-01T00:00:00Z","has_accepted_license":"1","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes"}],"page":"29-48","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","day":"01","type":"journal_article","publisher":"Springer Nature","arxiv":1,"ddc":["510"],"doi":"10.1007/s00454-023-00566-1","article_type":"original","title":"On angles in higher order Brillouin tessellations and related tilings in the plane","month":"07","_id":"14345","citation":{"chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>.","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. 72, 29–48.","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. 2024;72:29-48. doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2024). On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles in higher order Brillouin tessellations and related tilings in the plane,” <i>Discrete and Computational Geometry</i>, vol. 72. Springer Nature, pp. 29–48, 2024.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry 72 (2024) 29–48.","mla":"Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>, vol. 72, Springer Nature, 2024, pp. 29–48, doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>."},"article_processing_charge":"Yes (via OA deal)","ec_funded":1,"intvolume":"        72","isi":1,"file_date_updated":"2024-07-22T09:43:19Z","publication":"Discrete and Computational Geometry","author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"last_name":"Garber","full_name":"Garber, Alexey","first_name":"Alexey"},{"first_name":"Mohadese","full_name":"Ghafari, Mohadese","last_name":"Ghafari"},{"orcid":"0000-0002-1780-2689","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","first_name":"Teresa","full_name":"Heiss, Teresa","last_name":"Heiss"},{"full_name":"Saghafian, Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","first_name":"Morteza"}],"scopus_import":"1","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"oa":1,"acknowledgement":"Work by all authors but A. Garber is 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. Work by A. Garber is partially supported by the Alexander von Humboldt Foundation.","file":[{"access_level":"open_access","relation":"main_file","success":1,"date_updated":"2024-07-22T09:43:19Z","checksum":"b207b4e00f904e8ea8a30e24f0251f79","content_type":"application/pdf","file_size":892019,"creator":"dernst","file_name":"2024_DiscreteComputGeom_Edelsbrunner.pdf","file_id":"17301","date_created":"2024-07-22T09:43:19Z"}],"pmid":1,"publication_status":"published","language":[{"iso":"eng"}],"corr_author":"1","external_id":{"isi":["001060727600004"],"pmid":["39610762"],"arxiv":["2204.01076"]},"department":[{"_id":"HeEd"}],"date_created":"2023-09-17T22:01:10Z","quality_controlled":"1","volume":72,"abstract":[{"text":"For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2  is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970)).","lang":"eng"}]},{"oa_version":"Published Version","year":"2023","date_updated":"2025-04-14T07:43:59Z","has_accepted_license":"1","date_published":"2023-04-01T00:00:00Z","project":[{"name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411"}],"page":"745–770","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","publisher":"Springer Nature","day":"01","type":"journal_article","arxiv":1,"ddc":["510"],"doi":"10.1007/s00454-022-00394-9","article_type":"original","title":"Inserting one edge into a simple drawing is hard","month":"04","_id":"11999","citation":{"ieee":"A. M. Arroyo Guevara, F. Klute, I. Parada, B. Vogtenhuber, R. Seidel, and T. Wiedera, “Inserting one edge into a simple drawing is hard,” <i>Discrete and Computational Geometry</i>, vol. 69. Springer Nature, pp. 745–770, 2023.","short":"A.M. Arroyo Guevara, F. Klute, I. Parada, B. Vogtenhuber, R. Seidel, T. Wiedera, Discrete and Computational Geometry 69 (2023) 745–770.","mla":"Arroyo Guevara, Alan M., et al. “Inserting One Edge into a Simple Drawing Is Hard.” <i>Discrete and Computational Geometry</i>, vol. 69, Springer Nature, 2023, pp. 745–770, doi:<a href=\"https://doi.org/10.1007/s00454-022-00394-9\">10.1007/s00454-022-00394-9</a>.","ista":"Arroyo Guevara AM, Klute F, Parada I, Vogtenhuber B, Seidel R, Wiedera T. 2023. Inserting one edge into a simple drawing is hard. Discrete and Computational Geometry. 69, 745–770.","chicago":"Arroyo Guevara, Alan M, Fabian Klute, Irene Parada, Birgit Vogtenhuber, Raimund Seidel, and Tilo Wiedera. “Inserting One Edge into a Simple Drawing Is Hard.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00454-022-00394-9\">https://doi.org/10.1007/s00454-022-00394-9</a>.","apa":"Arroyo Guevara, A. M., Klute, F., Parada, I., Vogtenhuber, B., Seidel, R., &#38; Wiedera, T. (2023). Inserting one edge into a simple drawing is hard. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-022-00394-9\">https://doi.org/10.1007/s00454-022-00394-9</a>","ama":"Arroyo Guevara AM, Klute F, Parada I, Vogtenhuber B, Seidel R, Wiedera T. Inserting one edge into a simple drawing is hard. <i>Discrete and Computational Geometry</i>. 2023;69:745–770. doi:<a href=\"https://doi.org/10.1007/s00454-022-00394-9\">10.1007/s00454-022-00394-9</a>"},"article_processing_charge":"Yes (in subscription journal)","ec_funded":1,"isi":1,"intvolume":"        69","file_date_updated":"2022-08-29T11:23:15Z","scopus_import":"1","publication":"Discrete and Computational Geometry","author":[{"full_name":"Arroyo Guevara, Alan M","last_name":"Arroyo Guevara","id":"3207FDC6-F248-11E8-B48F-1D18A9856A87","first_name":"Alan M","orcid":"0000-0003-2401-8670"},{"first_name":"Fabian","last_name":"Klute","full_name":"Klute, Fabian"},{"full_name":"Parada, Irene","last_name":"Parada","first_name":"Irene"},{"last_name":"Vogtenhuber","full_name":"Vogtenhuber, Birgit","first_name":"Birgit"},{"last_name":"Seidel","full_name":"Seidel, Raimund","first_name":"Raimund"},{"full_name":"Wiedera, Tilo","last_name":"Wiedera","first_name":"Tilo"}],"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"oa":1,"acknowledgement":"This work was started during the 6th Austrian–Japanese–Mexican–Spanish Workshop on Discrete Geometry in June 2019 in Austria. We thank all the participants for the good atmosphere as well as discussions on the topic. Also, we thank Jan Kynčl for sending us remarks on a preliminary version of this work and an anonymous referee for further helpful comments.Alan Arroyo was funded by the Marie Skłodowska-Curie grant agreement No 754411. Fabian Klute was partially supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 612.001.651 and by the Austrian Science Fund (FWF): J-4510. Irene Parada and Birgit Vogtenhuber were partially supported by the Austrian Science Fund (FWF): W1230 and within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35. Irene Parada was also partially supported by the Independent Research Fund Denmark grant 2020-2023 (9131-00044B) Dynamic Network Analysis and by the Margarita Salas Fellowship funded by the Ministry of Universities of Spain and the European Union (NextGenerationEU). Tilo Wiedera was supported by the German Research Foundation (DFG) grant CH 897/2-2.","file":[{"success":1,"access_level":"open_access","relation":"main_file","file_size":1002218,"creator":"alisjak","file_name":"2022_DiscreteandComputionalGeometry_Arroyo.pdf","date_created":"2022-08-29T11:23:15Z","file_id":"12006","date_updated":"2022-08-29T11:23:15Z","checksum":"def7ae3b28d9fd6aec16450e40090302","content_type":"application/pdf"}],"publication_status":"published","language":[{"iso":"eng"}],"external_id":{"isi":["000840292800001"],"arxiv":["1909.07347"]},"department":[{"_id":"UlWa"}],"date_created":"2022-08-28T22:02:01Z","quality_controlled":"1","volume":69,"abstract":[{"lang":"eng","text":"A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G+e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment σ, it can be decided in polynomial time whether there exists a pseudocircle Φσ extending σ for which A∪{Φσ} is again an arrangement of pseudocircles."}]},{"publisher":"Springer Nature","day":"01","type":"journal_article","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","page":"156-191","date_published":"2023-01-01T00:00:00Z","has_accepted_license":"1","project":[{"grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships"},{"grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses"}],"oa_version":"Published Version","year":"2023","date_updated":"2025-04-14T07:44:00Z","article_processing_charge":"No","citation":{"apa":"Boissonnat, J.-D., Dyer, R., Ghosh, A., &#38; Wintraecken, M. (2023). Local criteria for triangulating general manifolds. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-022-00431-7\">https://doi.org/10.1007/s00454-022-00431-7</a>","ama":"Boissonnat J-D, Dyer R, Ghosh A, Wintraecken M. Local criteria for triangulating general manifolds. <i>Discrete &#38; Computational Geometry</i>. 2023;69:156-191. doi:<a href=\"https://doi.org/10.1007/s00454-022-00431-7\">10.1007/s00454-022-00431-7</a>","ista":"Boissonnat J-D, Dyer R, Ghosh A, Wintraecken M. 2023. Local criteria for triangulating general manifolds. Discrete &#38; Computational Geometry. 69, 156–191.","chicago":"Boissonnat, Jean-Daniel, Ramsay Dyer, Arijit Ghosh, and Mathijs Wintraecken. “Local Criteria for Triangulating General Manifolds.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00454-022-00431-7\">https://doi.org/10.1007/s00454-022-00431-7</a>.","short":"J.-D. Boissonnat, R. Dyer, A. Ghosh, M. Wintraecken, Discrete &#38; Computational Geometry 69 (2023) 156–191.","mla":"Boissonnat, Jean-Daniel, et al. “Local Criteria for Triangulating General Manifolds.” <i>Discrete &#38; Computational Geometry</i>, vol. 69, Springer Nature, 2023, pp. 156–91, doi:<a href=\"https://doi.org/10.1007/s00454-022-00431-7\">10.1007/s00454-022-00431-7</a>.","ieee":"J.-D. Boissonnat, R. Dyer, A. Ghosh, and M. Wintraecken, “Local criteria for triangulating general manifolds,” <i>Discrete &#38; Computational Geometry</i>, vol. 69. Springer Nature, pp. 156–191, 2023."},"_id":"12287","title":"Local criteria for triangulating general manifolds","month":"01","doi":"10.1007/s00454-022-00431-7","article_type":"original","ddc":["510"],"acknowledgement":"This work has been funded by the European Research Council under the European Union’s ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions). Arijit Ghosh is supported by Ramanujan Fellowship (No. SB/S2/RJN-064/2015). Part of this work was done when Arijit Ghosh was a Researcher at Max-Planck-Institute for Informatics, Germany, supported by the IndoGerman Max Planck Center for Computer Science (IMPECS). Mathijs Wintraecken also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411 and the Austrian Science Fund (FWF): M-3073. A part of the results described in this paper were presented at SoCG 2018 and in [3]. \r\nOpen access funding provided by the Austrian Science Fund (FWF).","file":[{"file_size":582850,"file_name":"2023_DiscreteCompGeometry_Boissonnat.pdf","creator":"dernst","date_created":"2023-02-02T11:01:10Z","file_id":"12488","date_updated":"2023-02-02T11:01:10Z","checksum":"46352e0ee71e460848f88685ca852681","content_type":"application/pdf","success":1,"access_level":"open_access","relation":"main_file"}],"oa":1,"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"author":[{"full_name":"Boissonnat, Jean-Daniel","last_name":"Boissonnat","first_name":"Jean-Daniel"},{"full_name":"Dyer, Ramsay","last_name":"Dyer","first_name":"Ramsay"},{"last_name":"Ghosh","full_name":"Ghosh, Arijit","first_name":"Arijit"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","full_name":"Wintraecken, Mathijs","last_name":"Wintraecken","orcid":"0000-0002-7472-2220"}],"keyword":["Computational Theory and Mathematics","Discrete Mathematics and Combinatorics","Geometry and Topology","Theoretical Computer Science"],"scopus_import":"1","publication":"Discrete & Computational Geometry","file_date_updated":"2023-02-02T11:01:10Z","ec_funded":1,"isi":1,"intvolume":"        69","abstract":[{"lang":"eng","text":"We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use."}],"volume":69,"quality_controlled":"1","department":[{"_id":"HeEd"}],"external_id":{"isi":["000862193600001"]},"date_created":"2023-01-16T10:04:06Z","language":[{"iso":"eng"}],"corr_author":"1","publication_status":"published"},{"oa":1,"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"file":[{"checksum":"71ce7e59f7ee4620acc704fecca620c2","content_type":"application/pdf","date_updated":"2023-03-07T14:40:14Z","file_id":"12715","date_created":"2023-03-07T14:40:14Z","file_name":"2023_DisCompGeo_Corbet.pdf","creator":"cchlebak","file_size":1359323,"relation":"main_file","access_level":"open_access","success":1}],"acknowledgement":"We thank the anonymous reviewers for many helpful comments and suggestions, which led to substantial improvements of the paper. The first two authors were supported by the Austrian Science Fund (FWF) grant number P 29984-N35 and W1230. The first author was partly supported by an Austrian Marshall Plan Scholarship, and by the Brummer & Partners MathDataLab. A conference version of this paper was presented at the 37th International Symposium on Computational Geometry (SoCG 2021). Open access funding provided by the Royal Institute of Technology.","pmid":1,"intvolume":"        70","isi":1,"publication":"Discrete and Computational Geometry","file_date_updated":"2023-03-07T14:40:14Z","scopus_import":"1","author":[{"full_name":"Corbet, René","last_name":"Corbet","first_name":"René"},{"first_name":"Michael","id":"36E4574A-F248-11E8-B48F-1D18A9856A87","full_name":"Kerber, Michael","last_name":"Kerber","orcid":"0000-0002-8030-9299"},{"first_name":"Michael","last_name":"Lesnick","full_name":"Lesnick, Michael"},{"first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","full_name":"Osang, Georg F","last_name":"Osang","orcid":"0000-0002-8882-5116"}],"abstract":[{"lang":"eng","text":"Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness."}],"quality_controlled":"1","volume":70,"language":[{"iso":"eng"}],"external_id":{"arxiv":["2103.07823"],"pmid":["37581017"],"isi":["000936496800001"]},"department":[{"_id":"HeEd"}],"date_created":"2023-03-05T23:01:06Z","publication_status":"published","publisher":"Springer Nature","day":"01","type":"journal_article","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","page":"376-405","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","year":"2023","date_updated":"2025-07-10T12:01:57Z","has_accepted_license":"1","date_published":"2023-09-01T00:00:00Z","citation":{"short":"R. Corbet, M. Kerber, M. Lesnick, G.F. Osang, Discrete and Computational Geometry 70 (2023) 376–405.","mla":"Corbet, René, et al. “Computing the Multicover Bifiltration.” <i>Discrete and Computational Geometry</i>, vol. 70, Springer Nature, 2023, pp. 376–405, doi:<a href=\"https://doi.org/10.1007/s00454-022-00476-8\">10.1007/s00454-022-00476-8</a>.","ieee":"R. Corbet, M. Kerber, M. Lesnick, and G. F. Osang, “Computing the multicover bifiltration,” <i>Discrete and Computational Geometry</i>, vol. 70. Springer Nature, pp. 376–405, 2023.","ama":"Corbet R, Kerber M, Lesnick M, Osang GF. Computing the multicover bifiltration. <i>Discrete and Computational Geometry</i>. 2023;70:376-405. doi:<a href=\"https://doi.org/10.1007/s00454-022-00476-8\">10.1007/s00454-022-00476-8</a>","apa":"Corbet, R., Kerber, M., Lesnick, M., &#38; Osang, G. F. (2023). Computing the multicover bifiltration. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-022-00476-8\">https://doi.org/10.1007/s00454-022-00476-8</a>","chicago":"Corbet, René, Michael Kerber, Michael Lesnick, and Georg F Osang. “Computing the Multicover Bifiltration.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00454-022-00476-8\">https://doi.org/10.1007/s00454-022-00476-8</a>.","ista":"Corbet R, Kerber M, Lesnick M, Osang GF. 2023. Computing the multicover bifiltration. Discrete and Computational Geometry. 70, 376–405."},"related_material":{"record":[{"id":"9605","relation":"earlier_version","status":"public"}]},"article_processing_charge":"Yes (via OA deal)","title":"Computing the multicover bifiltration","month":"09","_id":"12709","doi":"10.1007/s00454-022-00476-8","article_type":"original","arxiv":1,"ddc":["000"]},{"article_processing_charge":"Yes (via OA deal)","citation":{"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>.","short":"H. Kourimska, Discrete and Computational Geometry 70 (2023) 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>","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>","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>.","ista":"Kourimska H. 2023. Discrete yamabe problem for polyhedral surfaces. Discrete and Computational Geometry. 70, 123–153."},"_id":"12764","month":"07","title":"Discrete yamabe problem for polyhedral surfaces","article_type":"original","doi":"10.1007/s00454-023-00484-2","ddc":["510"],"type":"journal_article","publisher":"Springer Nature","day":"01","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"123-153","project":[{"_id":"26AD5D90-B435-11E9-9278-68D0E5697425","grant_number":"I04245","call_identifier":"FWF","name":"Algebraic Footprints of Geometric Features in Homology"}],"has_accepted_license":"1","date_published":"2023-07-01T00:00:00Z","date_updated":"2025-04-23T08:59:15Z","oa_version":"Published Version","year":"2023","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"}],"volume":70,"quality_controlled":"1","date_created":"2023-03-26T22:01:09Z","department":[{"_id":"HeEd"}],"external_id":{"pmid":["37292248"],"isi":["000948148000001"]},"corr_author":"1","language":[{"iso":"eng"}],"publication_status":"published","pmid":1,"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.","file":[{"file_size":1026683,"file_name":"2023_DiscreteGeometry_Kourimska.pdf","creator":"dernst","date_created":"2023-10-04T11:46:24Z","file_id":"14396","date_updated":"2023-10-04T11:46:24Z","content_type":"application/pdf","checksum":"cdbf90ba4a7ddcb190d37b9e9d4cb9d3","success":1,"access_level":"open_access","relation":"main_file"}],"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"oa":1,"author":[{"orcid":"0000-0001-7841-0091","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","first_name":"Hana","last_name":"Kourimska","full_name":"Kourimska, Hana"}],"scopus_import":"1","publication":"Discrete and Computational Geometry","file_date_updated":"2023-10-04T11:46:24Z","intvolume":"        70","isi":1},{"date_updated":"2024-10-09T21:06:01Z","oa_version":"Published Version","year":"2023","has_accepted_license":"1","date_published":"2023-07-05T00:00:00Z","page":"1059-1089","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","issue":"3","day":"05","type":"journal_article","publisher":"Springer Nature","ddc":["510"],"arxiv":1,"article_type":"original","doi":"10.1007/s00454-023-00500-5","month":"07","title":"Iterated medial triangle subdivision in surfaces of constant curvature","_id":"13270","citation":{"apa":"Brunck, F. R. (2023). Iterated medial triangle subdivision in surfaces of constant curvature. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00500-5\">https://doi.org/10.1007/s00454-023-00500-5</a>","ama":"Brunck FR. Iterated medial triangle subdivision in surfaces of constant curvature. <i>Discrete and Computational Geometry</i>. 2023;70(3):1059-1089. doi:<a href=\"https://doi.org/10.1007/s00454-023-00500-5\">10.1007/s00454-023-00500-5</a>","ista":"Brunck FR. 2023. Iterated medial triangle subdivision in surfaces of constant curvature. Discrete and Computational Geometry. 70(3), 1059–1089.","chicago":"Brunck, Florestan R. “Iterated Medial Triangle Subdivision in Surfaces of Constant Curvature.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00454-023-00500-5\">https://doi.org/10.1007/s00454-023-00500-5</a>.","mla":"Brunck, Florestan R. “Iterated Medial Triangle Subdivision in Surfaces of Constant Curvature.” <i>Discrete and Computational Geometry</i>, vol. 70, no. 3, Springer Nature, 2023, pp. 1059–89, doi:<a href=\"https://doi.org/10.1007/s00454-023-00500-5\">10.1007/s00454-023-00500-5</a>.","short":"F.R. Brunck, Discrete and Computational Geometry 70 (2023) 1059–1089.","ieee":"F. R. Brunck, “Iterated medial triangle subdivision in surfaces of constant curvature,” <i>Discrete and Computational Geometry</i>, vol. 70, no. 3. Springer Nature, pp. 1059–1089, 2023."},"article_processing_charge":"Yes (via OA deal)","intvolume":"        70","isi":1,"publication":"Discrete and Computational Geometry","file_date_updated":"2024-01-29T11:15:22Z","author":[{"full_name":"Brunck, Florestan R","last_name":"Brunck","id":"6ab6e556-f394-11eb-9cf6-9dfb78f00d8d","first_name":"Florestan R"}],"scopus_import":"1","oa":1,"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"acknowledgement":"Open access funding provided by the Institute of Science and Technology (IST Austria).","file":[{"date_updated":"2024-01-29T11:15:22Z","checksum":"865e68daafdd4edcfc280172ec50f5ea","content_type":"application/pdf","file_size":1466020,"creator":"dernst","file_name":"2023_DiscreteComputGeometry_Brunck.pdf","date_created":"2024-01-29T11:15:22Z","file_id":"14897","access_level":"open_access","relation":"main_file","success":1}],"publication_status":"published","language":[{"iso":"eng"}],"corr_author":"1","date_created":"2023-07-23T22:01:14Z","external_id":{"arxiv":["2107.04112"],"isi":["001023742800003"]},"department":[{"_id":"UlWa"}],"quality_controlled":"1","volume":70,"abstract":[{"lang":"eng","text":"Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into four triangles by joining the midpoints of its edges. We show the existence of a uniform δ>0\r\n such that, at any step of the subdivision, all the triangle angles lie in the interval (δ,π−δ)\r\n. Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses."}]},{"abstract":[{"text":"A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z2 -genus of a graph G is the minimum g such that G has an independently even drawing on the orientable surface of genus g. An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective t×t grid or one of the following so-called t -Kuratowski graphs: K3,t, or t copies of K5 or K3,3 sharing at most two common vertices. We show that the Z2-genus of graphs in these families is unbounded in t; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its Z2-genus, solving a problem posed by Schaefer and Štefankovič, and giving an approximate version of the Hanani–Tutte theorem on orientable surfaces. We also obtain an analogous result for Euler genus and Euler Z2-genus of graphs.","lang":"eng"}],"quality_controlled":"1","volume":68,"language":[{"iso":"eng"}],"date_created":"2022-07-17T22:01:56Z","department":[{"_id":"UlWa"}],"external_id":{"isi":["000825014500001"],"arxiv":["1803.05085"]},"publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1803.05085"}],"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"oa":1,"acknowledgement":"We thank Zdeněk Dvořák, Xavier Goaoc, and Pavel Paták for helpful discussions. We also thank Bojan Mohar, Paul Seymour, Gelasio Salazar, Jim Geelen, and John Maharry for information about their unpublished results related to Conjecture 3.1. Finally we thank the reviewers for corrections and suggestions for improving the presentation.\r\nSupported by Austrian Science Fund (FWF): M2281-N35. Supported by project 19-04113Y of the Czech Science Foundation (GAČR), by the Czech-French collaboration project EMBEDS II (CZ: 7AMB17FR029, FR: 38087RM), and by Charles University project UNCE/SCI/004.","isi":1,"intvolume":"        68","author":[{"first_name":"Radoslav","id":"39F3FFE4-F248-11E8-B48F-1D18A9856A87","full_name":"Fulek, Radoslav","last_name":"Fulek","orcid":"0000-0001-8485-1774"},{"full_name":"Kynčl, Jan","last_name":"Kynčl","first_name":"Jan"}],"scopus_import":"1","publication":"Discrete and Computational Geometry","related_material":{"record":[{"id":"186","status":"public","relation":"earlier_version"}]},"citation":{"ieee":"R. Fulek and J. Kynčl, “The Z2-Genus of Kuratowski minors,” <i>Discrete and Computational Geometry</i>, vol. 68. Springer Nature, pp. 425–447, 2022.","mla":"Fulek, Radoslav, and Jan Kynčl. “The Z2-Genus of Kuratowski Minors.” <i>Discrete and Computational Geometry</i>, vol. 68, Springer Nature, 2022, pp. 425–47, doi:<a href=\"https://doi.org/10.1007/s00454-022-00412-w\">10.1007/s00454-022-00412-w</a>.","short":"R. Fulek, J. Kynčl, Discrete and Computational Geometry 68 (2022) 425–447.","ista":"Fulek R, Kynčl J. 2022. The Z2-Genus of Kuratowski minors. Discrete and Computational Geometry. 68, 425–447.","chicago":"Fulek, Radoslav, and Jan Kynčl. “The Z2-Genus of Kuratowski Minors.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00454-022-00412-w\">https://doi.org/10.1007/s00454-022-00412-w</a>.","apa":"Fulek, R., &#38; Kynčl, J. (2022). The Z2-Genus of Kuratowski minors. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-022-00412-w\">https://doi.org/10.1007/s00454-022-00412-w</a>","ama":"Fulek R, Kynčl J. The Z2-Genus of Kuratowski minors. <i>Discrete and Computational Geometry</i>. 2022;68:425-447. doi:<a href=\"https://doi.org/10.1007/s00454-022-00412-w\">10.1007/s00454-022-00412-w</a>"},"article_processing_charge":"No","month":"09","title":"The Z2-Genus of Kuratowski minors","_id":"11593","article_type":"original","doi":"10.1007/s00454-022-00412-w","arxiv":1,"type":"journal_article","publisher":"Springer Nature","day":"01","status":"public","page":"425-447","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2025-04-14T13:52:37Z","oa_version":"Preprint","year":"2022","project":[{"_id":"261FA626-B435-11E9-9278-68D0E5697425","grant_number":"M02281","name":"Eliminating intersections in drawings of graphs","call_identifier":"FWF"}],"date_published":"2022-09-01T00:00:00Z"},{"_id":"12129","title":"Connectivity of triangulation flip graphs in the plane","month":"11","article_processing_charge":"No","citation":{"ieee":"U. Wagner and E. Welzl, “Connectivity of triangulation flip graphs in the plane,” <i>Discrete &#38; Computational Geometry</i>, vol. 68, no. 4. Springer Nature, pp. 1227–1284, 2022.","mla":"Wagner, Uli, and Emo Welzl. “Connectivity of Triangulation Flip Graphs in the Plane.” <i>Discrete &#38; Computational Geometry</i>, vol. 68, no. 4, Springer Nature, 2022, pp. 1227–84, doi:<a href=\"https://doi.org/10.1007/s00454-022-00436-2\">10.1007/s00454-022-00436-2</a>.","short":"U. Wagner, E. Welzl, Discrete &#38; Computational Geometry 68 (2022) 1227–1284.","chicago":"Wagner, Uli, and Emo Welzl. “Connectivity of Triangulation Flip Graphs in the Plane.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00454-022-00436-2\">https://doi.org/10.1007/s00454-022-00436-2</a>.","ista":"Wagner U, Welzl E. 2022. Connectivity of triangulation flip graphs in the plane. Discrete &#38; Computational Geometry. 68(4), 1227–1284.","ama":"Wagner U, Welzl E. Connectivity of triangulation flip graphs in the plane. <i>Discrete &#38; Computational Geometry</i>. 2022;68(4):1227-1284. doi:<a href=\"https://doi.org/10.1007/s00454-022-00436-2\">10.1007/s00454-022-00436-2</a>","apa":"Wagner, U., &#38; Welzl, E. (2022). Connectivity of triangulation flip graphs in the plane. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-022-00436-2\">https://doi.org/10.1007/s00454-022-00436-2</a>"},"related_material":{"record":[{"relation":"earlier_version","status":"public","id":"7807"},{"status":"public","relation":"earlier_version","id":"7990"}]},"ddc":["510"],"doi":"10.1007/s00454-022-00436-2","article_type":"original","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"day":"14","publisher":"Springer Nature","type":"journal_article","issue":"4","date_published":"2022-11-14T00:00:00Z","has_accepted_license":"1","oa_version":"Published Version","year":"2022","date_updated":"2025-07-10T11:54:56Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","page":"1227-1284","volume":68,"quality_controlled":"1","abstract":[{"text":"Given a finite point set P in general position in the plane, a full triangulation of P is a maximal straight-line embedded plane graph on P. A partial triangulation of P is a full triangulation of some subset P′ of P containing all extreme points in P. A bistellar flip on a partial triangulation either flips an edge (called edge flip), removes a non-extreme point of degree 3, or adds a point in P∖P′ as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The edge flip graph is defined with full triangulations as vertices, and edge flips determining the adjacencies. Lawson showed in the early seventies that these graphs are connected. The goal of this paper is to investigate the structure of these graphs, with emphasis on their vertex connectivity. For sets P of n points in the plane in general position, we show that the edge flip graph is ⌈n/2−2⌉-vertex connected, and the bistellar flip graph is (n−3)-vertex connected; both results are tight. The latter bound matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points to 3-space and projecting back the lower convex hull), where (n−3)-vertex connectivity has been known since the late eighties through the secondary polytope due to Gelfand, Kapranov, & Zelevinsky and Balinski’s Theorem. For the edge flip-graph, we additionally show that the vertex connectivity is at least as large as (and hence equal to) the minimum degree (i.e., the minimum number of flippable edges in any full triangulation), provided that n is large enough. Our methods also yield several other results: (i) The edge flip graph can be covered by graphs of polytopes of dimension ⌈n/2−2⌉ (products of associahedra) and the bistellar flip graph can be covered by graphs of polytopes of dimension n−3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance n−3 in the Hasse diagram of the partial order of partial subdivisions from the trivial subdivision. (iii) All partial triangulations of a point set are regular iff the partial order of partial subdivisions has height n−3. (iv) There are arbitrarily large sets P with non-regular partial triangulations and such that every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular triangulations.","lang":"eng"}],"publication_status":"published","department":[{"_id":"UlWa"}],"external_id":{"isi":["000883222200003"]},"date_created":"2023-01-12T12:02:28Z","corr_author":"1","language":[{"iso":"eng"}],"file":[{"relation":"main_file","access_level":"open_access","success":1,"content_type":"application/pdf","checksum":"307e879d09e52eddf5b225d0aaa9213a","date_updated":"2023-01-23T11:10:03Z","date_created":"2023-01-23T11:10:03Z","file_id":"12345","creator":"dernst","file_name":"2022_DiscreteCompGeometry_Wagner.pdf","file_size":1747581}],"acknowledgement":"This is a full and revised version of [38] (on partial triangulations) in Proceedings of the 36th Annual International Symposium on Computational Geometry (SoCG‘20) and of some of the results in [37] (on full triangulations) in Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA‘20).\r\nThis research started at the 11th Gremo’s Workshop on Open Problems (GWOP), Alp Sellamatt, Switzerland, June 24–28, 2013, motivated by a question posed by Filip Mori´c on full triangulations. Research was supported by the Swiss National Science Foundation within the collaborative DACH project Arrangements and Drawings as SNSF Project 200021E-171681, and by IST Austria and Berlin Free University during a sabbatical stay of the second author. We thank Michael Joswig, Jesús De Loera, and Francisco Santos for helpful discussions on the topics of this paper, and Daniel Bertschinger and Valentin Stoppiello for carefully reading earlier versions and for many helpful comments.\r\nOpen access funding provided by the Swiss Federal Institute of Technology Zürich","oa":1,"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"author":[{"id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","full_name":"Wagner, Uli","last_name":"Wagner","orcid":"0000-0002-1494-0568"},{"first_name":"Emo","last_name":"Welzl","full_name":"Welzl, Emo"}],"keyword":["Computational Theory and Mathematics","Discrete Mathematics and Combinatorics","Geometry and Topology","Theoretical Computer Science"],"scopus_import":"1","publication":"Discrete & Computational Geometry","file_date_updated":"2023-01-23T11:10:03Z","intvolume":"        68","isi":1},{"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","page":"811-842","date_published":"2022-04-01T00:00:00Z","has_accepted_license":"1","date_updated":"2024-10-09T21:01:38Z","year":"2022","oa_version":"Published Version","type":"journal_article","day":"01","publisher":"Springer Nature","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"article_type":"original","doi":"10.1007/s00454-022-00371-2","ddc":["510"],"article_processing_charge":"Yes (via OA deal)","citation":{"mla":"Biswas, Ranita, et al. “Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>, vol. 67, Springer Nature, 2022, pp. 811–42, doi:<a href=\"https://doi.org/10.1007/s00454-022-00371-2\">10.1007/s00454-022-00371-2</a>.","short":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 67 (2022) 811–842.","ieee":"R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics,” <i>Discrete and Computational Geometry</i>, vol. 67. Springer Nature, pp. 811–842, 2022.","ama":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. <i>Discrete and Computational Geometry</i>. 2022;67:811-842. doi:<a href=\"https://doi.org/10.1007/s00454-022-00371-2\">10.1007/s00454-022-00371-2</a>","apa":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M. (2022). Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-022-00371-2\">https://doi.org/10.1007/s00454-022-00371-2</a>","chicago":"Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00454-022-00371-2\">https://doi.org/10.1007/s00454-022-00371-2</a>.","ista":"Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2022. Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. 67, 811–842."},"_id":"10773","month":"04","title":"Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics","scopus_import":"1","file_date_updated":"2022-08-02T06:07:55Z","publication":"Discrete and Computational Geometry","author":[{"orcid":"0000-0002-5372-7890","first_name":"Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita","last_name":"Biswas"},{"full_name":"Cultrera Di Montesano, Sebastiano","last_name":"Cultrera Di Montesano","first_name":"Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832"},{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"first_name":"Morteza","last_name":"Saghafian","full_name":"Saghafian, Morteza"}],"intvolume":"        67","isi":1,"acknowledgement":"Open access funding provided by the Institute of Science and Technology (IST Austria).","file":[{"relation":"main_file","access_level":"open_access","success":1,"date_updated":"2022-08-02T06:07:55Z","content_type":"application/pdf","checksum":"9383d3b70561bacee905e335dc922680","file_size":2518111,"date_created":"2022-08-02T06:07:55Z","file_id":"11718","file_name":"2022_DiscreteCompGeometry_Biswas.pdf","creator":"dernst"}],"oa":1,"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"date_created":"2022-02-20T23:01:34Z","external_id":{"isi":["000752175300002"]},"department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"corr_author":"1","publication_status":"published","abstract":[{"text":"The Voronoi tessellation in Rd is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function.","lang":"eng"}],"volume":67,"quality_controlled":"1"},{"status":"public","type":"journal_article","day":"01","publisher":"Springer Nature","date_updated":"2023-08-02T14:38:58Z","oa_version":"Preprint","year":"2022","date_published":"2022-12-01T00:00:00Z","page":"1133-1154","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","month":"12","title":"Barycentric cuts through a convex body","_id":"10776","citation":{"chicago":"Patakova, Zuzana, Martin Tancer, and Uli Wagner. “Barycentric Cuts through a Convex Body.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00454-021-00364-7\">https://doi.org/10.1007/s00454-021-00364-7</a>.","ista":"Patakova Z, Tancer M, Wagner U. 2022. Barycentric cuts through a convex body. Discrete and Computational Geometry. 68, 1133–1154.","ama":"Patakova Z, Tancer M, Wagner U. Barycentric cuts through a convex body. <i>Discrete and Computational Geometry</i>. 2022;68:1133-1154. doi:<a href=\"https://doi.org/10.1007/s00454-021-00364-7\">10.1007/s00454-021-00364-7</a>","apa":"Patakova, Z., Tancer, M., &#38; Wagner, U. (2022). Barycentric cuts through a convex body. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-021-00364-7\">https://doi.org/10.1007/s00454-021-00364-7</a>","ieee":"Z. Patakova, M. Tancer, and U. Wagner, “Barycentric cuts through a convex body,” <i>Discrete and Computational Geometry</i>, vol. 68. Springer Nature, pp. 1133–1154, 2022.","short":"Z. Patakova, M. Tancer, U. Wagner, Discrete and Computational Geometry 68 (2022) 1133–1154.","mla":"Patakova, Zuzana, et al. “Barycentric Cuts through a Convex Body.” <i>Discrete and Computational Geometry</i>, vol. 68, Springer Nature, 2022, pp. 1133–54, doi:<a href=\"https://doi.org/10.1007/s00454-021-00364-7\">10.1007/s00454-021-00364-7</a>."},"article_processing_charge":"No","arxiv":1,"article_type":"original","doi":"10.1007/s00454-021-00364-7","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"oa":1,"acknowledgement":"The work by Zuzana Patáková has been partially supported by Charles University Research Center Program No. UNCE/SCI/022, and part of it was done during her research stay at IST Austria. The work by Martin Tancer is supported by the GAČR Grant 19-04113Y and by the Charles University Projects PRIMUS/17/SCI/3 and UNCE/SCI/004.","main_file_link":[{"url":"https://arxiv.org/abs/2003.13536","open_access":"1"}],"isi":1,"intvolume":"        68","publication":"Discrete and Computational Geometry","author":[{"orcid":"0000-0002-3975-1683","last_name":"Patakova","full_name":"Patakova, Zuzana","id":"48B57058-F248-11E8-B48F-1D18A9856A87","first_name":"Zuzana"},{"first_name":"Martin","last_name":"Tancer","full_name":"Tancer, Martin"},{"orcid":"0000-0002-1494-0568","full_name":"Wagner, Uli","last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli"}],"scopus_import":"1","quality_controlled":"1","volume":68,"abstract":[{"text":"Let K be a convex body in Rn (i.e., a compact convex set with nonempty interior). Given a point p in the interior of K, a hyperplane h passing through p is called barycentric if p is the barycenter of K∩h. In 1961, Grünbaum raised the question whether, for every K, there exists an interior point p through which there are at least n+1 distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if p=p0 is the point of maximal depth in K. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum’s question. It follows from known results that for n≥2, there are always at least three distinct barycentric cuts through the point p0∈K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through p0 are guaranteed if n≥3.","lang":"eng"}],"publication_status":"published","language":[{"iso":"eng"}],"date_created":"2022-02-20T23:01:35Z","external_id":{"arxiv":["2003.13536"],"isi":["000750681500001"]},"department":[{"_id":"UlWa"}]},{"abstract":[{"text":"Suppose that n is not a prime power and not twice a prime power. We prove that for any Hausdorff compactum X with a free action of the symmetric group Sn, there exists an Sn-equivariant map X→Rn whose image avoids the diagonal {(x,x,…,x)∈Rn∣x∈R}. Previously, the special cases of this statement for certain X were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of Sn-equivariant maps from the boundary ∂Δn−1 of (n−1)-simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser’s conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem.","lang":"eng"}],"quality_controlled":"1","volume":66,"corr_author":"1","language":[{"iso":"eng"}],"external_id":{"arxiv":["1910.12628"]},"date_created":"2022-06-17T08:45:15Z","publication_status":"published","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"acknowledgement":"S. Avvakumov has received funding from the European Research Council under the European Union’s Seventh Framework Programme ERC Grant agreement ERC StG 716424–CASe. S. Kudrya was supported by the Austrian Academic Exchange Service (OeAD), ICM-2019-13577.","intvolume":"        66","keyword":["Computational Theory and Mathematics","Discrete Mathematics and Combinatorics","Geometry and Topology","Theoretical Computer Science"],"author":[{"id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","first_name":"Sergey","last_name":"Avvakumov","full_name":"Avvakumov, Sergey","orcid":"0000-0002-7840-5062"},{"last_name":"Kudrya","full_name":"Kudrya, Sergey","id":"ecf01965-d252-11ea-95a5-8ada5f6c6a67","first_name":"Sergey"}],"scopus_import":"1","publication":"Discrete & Computational Geometry","citation":{"ama":"Avvakumov S, Kudrya S. Vanishing of all equivariant obstructions and the mapping degree. <i>Discrete &#38; Computational Geometry</i>. 2021;66(3):1202-1216. doi:<a href=\"https://doi.org/10.1007/s00454-021-00299-z\">10.1007/s00454-021-00299-z</a>","apa":"Avvakumov, S., &#38; Kudrya, S. (2021). Vanishing of all equivariant obstructions and the mapping degree. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-021-00299-z\">https://doi.org/10.1007/s00454-021-00299-z</a>","chicago":"Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions and the Mapping Degree.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-021-00299-z\">https://doi.org/10.1007/s00454-021-00299-z</a>.","ista":"Avvakumov S, Kudrya S. 2021. Vanishing of all equivariant obstructions and the mapping degree. Discrete &#38; Computational Geometry. 66(3), 1202–1216.","mla":"Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions and the Mapping Degree.” <i>Discrete &#38; Computational Geometry</i>, vol. 66, no. 3, Springer Nature, 2021, pp. 1202–16, doi:<a href=\"https://doi.org/10.1007/s00454-021-00299-z\">10.1007/s00454-021-00299-z</a>.","short":"S. Avvakumov, S. Kudrya, Discrete &#38; Computational Geometry 66 (2021) 1202–1216.","ieee":"S. Avvakumov and S. Kudrya, “Vanishing of all equivariant obstructions and the mapping degree,” <i>Discrete &#38; Computational Geometry</i>, vol. 66, no. 3. Springer Nature, pp. 1202–1216, 2021."},"related_material":{"record":[{"status":"public","relation":"earlier_version","id":"8182"}]},"article_processing_charge":"No","title":"Vanishing of all equivariant obstructions and the mapping degree","month":"10","_id":"11446","doi":"10.1007/s00454-021-00299-z","article_type":"original","arxiv":1,"publisher":"Springer Nature","day":"01","type":"journal_article","issue":"3","status":"public","extern":"1","page":"1202-1216","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","year":"2021","date_updated":"2025-04-14T09:10:06Z","date_published":"2021-10-01T00:00:00Z"},{"quality_controlled":"1","volume":65,"abstract":[{"text":"We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.","lang":"eng"}],"publication_status":"published","language":[{"iso":"eng"}],"corr_author":"1","date_created":"2020-05-30T10:26:04Z","external_id":{"arxiv":["1712.07734"],"isi":["000536324700001"]},"department":[{"_id":"HeEd"}],"oa":1,"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"file":[{"relation":"main_file","access_level":"open_access","success":1,"date_updated":"2020-11-25T09:06:41Z","checksum":"487a84ea5841b75f04f66d7ebd71b67e","content_type":"application/pdf","file_size":1013730,"file_id":"8803","date_created":"2020-11-25T09:06:41Z","file_name":"2020_DiscreteCompGeometry_Brown.pdf","creator":"dernst"}],"acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). This work was partially supported by NSF IIS-1513616 and NSF ABI-1661375. The authors would like to thank the anonymous referees for their insightful comments.","isi":1,"intvolume":"        65","publication":"Discrete and Computational Geometry","file_date_updated":"2020-11-25T09:06:41Z","author":[{"full_name":"Brown, Adam","last_name":"Brown","first_name":"Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425"},{"last_name":"Wang","full_name":"Wang, Bei","first_name":"Bei"}],"scopus_import":"1","month":"06","title":"Sheaf-theoretic stratification learning from geometric and topological perspectives","_id":"7905","citation":{"apa":"Brown, A., &#38; Wang, B. (2021). Sheaf-theoretic stratification learning from geometric and topological perspectives. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00206-y\">https://doi.org/10.1007/s00454-020-00206-y</a>","ama":"Brown A, Wang B. Sheaf-theoretic stratification learning from geometric and topological perspectives. <i>Discrete and Computational Geometry</i>. 2021;65:1166-1198. doi:<a href=\"https://doi.org/10.1007/s00454-020-00206-y\">10.1007/s00454-020-00206-y</a>","ista":"Brown A, Wang B. 2021. Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry. 65, 1166–1198.","chicago":"Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-020-00206-y\">https://doi.org/10.1007/s00454-020-00206-y</a>.","mla":"Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives.” <i>Discrete and Computational Geometry</i>, vol. 65, Springer Nature, 2021, pp. 1166–98, doi:<a href=\"https://doi.org/10.1007/s00454-020-00206-y\">10.1007/s00454-020-00206-y</a>.","short":"A. Brown, B. Wang, Discrete and Computational Geometry 65 (2021) 1166–1198.","ieee":"A. Brown and B. Wang, “Sheaf-theoretic stratification learning from geometric and topological perspectives,” <i>Discrete and Computational Geometry</i>, vol. 65. Springer Nature, pp. 1166–1198, 2021."},"article_processing_charge":"Yes (via OA deal)","ddc":["510"],"arxiv":1,"article_type":"original","doi":"10.1007/s00454-020-00206-y","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","publisher":"Springer Nature","day":"01","type":"journal_article","date_updated":"2025-04-15T06:53:15Z","oa_version":"Published Version","year":"2021","date_published":"2021-06-01T00:00:00Z","project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"has_accepted_license":"1","page":"1166-1198","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87"},{"date_created":"2020-08-11T07:11:51Z","department":[{"_id":"HeEd"}],"external_id":{"isi":["000558119300001"]},"language":[{"iso":"eng"}],"corr_author":"1","publication_status":"published","abstract":[{"text":"We consider the following setting: suppose that we are given a manifold M in Rd with positive reach. Moreover assume that we have an embedded simplical complex A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then A is a triangulation of the manifold, that is, they are homeomorphic.","lang":"eng"}],"volume":66,"quality_controlled":"1","author":[{"first_name":"Jean-Daniel","full_name":"Boissonnat, Jean-Daniel","last_name":"Boissonnat"},{"first_name":"Ramsay","last_name":"Dyer","full_name":"Dyer, Ramsay"},{"first_name":"Arijit","last_name":"Ghosh","full_name":"Ghosh, Arijit"},{"first_name":"Andre","last_name":"Lieutier","full_name":"Lieutier, Andre"},{"full_name":"Wintraecken, Mathijs","last_name":"Wintraecken","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs","orcid":"0000-0002-7472-2220"}],"scopus_import":"1","publication":"Discrete and Computational Geometry","intvolume":"        66","isi":1,"ec_funded":1,"main_file_link":[{"url":"https://doi.org/10.1007/s00454-020-00233-9","open_access":"1"}],"acknowledgement":"Open access funding provided by the Institute of Science and Technology (IST Austria). Arijit Ghosh is supported by the Ramanujan Fellowship (No. SB/S2/RJN-064/2015), India.\r\nThis work has been funded by the European Research Council under the European Union’s ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions). The third author is supported by Ramanujan Fellowship (No. SB/S2/RJN-064/2015), India. The fifth author also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411.","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"oa":1,"article_type":"original","doi":"10.1007/s00454-020-00233-9","ddc":["510"],"article_processing_charge":"Yes (via OA deal)","citation":{"ieee":"J.-D. Boissonnat, R. Dyer, A. Ghosh, A. Lieutier, and M. Wintraecken, “Local conditions for triangulating submanifolds of Euclidean space,” <i>Discrete and Computational Geometry</i>, vol. 66. Springer Nature, pp. 666–686, 2021.","short":"J.-D. Boissonnat, R. Dyer, A. Ghosh, A. Lieutier, M. Wintraecken, Discrete and Computational Geometry 66 (2021) 666–686.","mla":"Boissonnat, Jean-Daniel, et al. “Local Conditions for Triangulating Submanifolds of Euclidean Space.” <i>Discrete and Computational Geometry</i>, vol. 66, Springer Nature, 2021, pp. 666–86, doi:<a href=\"https://doi.org/10.1007/s00454-020-00233-9\">10.1007/s00454-020-00233-9</a>.","ista":"Boissonnat J-D, Dyer R, Ghosh A, Lieutier A, Wintraecken M. 2021. Local conditions for triangulating submanifolds of Euclidean space. Discrete and Computational Geometry. 66, 666–686.","chicago":"Boissonnat, Jean-Daniel, Ramsay Dyer, Arijit Ghosh, Andre Lieutier, and Mathijs Wintraecken. “Local Conditions for Triangulating Submanifolds of Euclidean Space.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-020-00233-9\">https://doi.org/10.1007/s00454-020-00233-9</a>.","apa":"Boissonnat, J.-D., Dyer, R., Ghosh, A., Lieutier, A., &#38; Wintraecken, M. (2021). Local conditions for triangulating submanifolds of Euclidean space. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00233-9\">https://doi.org/10.1007/s00454-020-00233-9</a>","ama":"Boissonnat J-D, Dyer R, Ghosh A, Lieutier A, Wintraecken M. Local conditions for triangulating submanifolds of Euclidean space. <i>Discrete and Computational Geometry</i>. 2021;66:666-686. doi:<a href=\"https://doi.org/10.1007/s00454-020-00233-9\">10.1007/s00454-020-00233-9</a>"},"_id":"8248","month":"09","title":"Local conditions for triangulating submanifolds of Euclidean space","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","page":"666-686","project":[{"grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships"}],"has_accepted_license":"1","date_published":"2021-09-01T00:00:00Z","date_updated":"2025-04-14T07:44:05Z","year":"2021","oa_version":"Published Version","day":"01","type":"journal_article","publisher":"Springer Nature","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"}},{"abstract":[{"lang":"eng","text":"Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory."}],"volume":66,"quality_controlled":"1","date_created":"2020-09-06T22:01:13Z","external_id":{"arxiv":["1908.00856"],"isi":["000564488500002"]},"department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1908.00856"}],"acknowledgement":"This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. W.K.S. was also supported by the Australian Research Council (DP1401000851). A.V.A. was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha).","oa":1,"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"scopus_import":"1","author":[{"orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","full_name":"Akopyan, Arseniy","last_name":"Akopyan"},{"full_name":"Bobenko, Alexander I.","last_name":"Bobenko","first_name":"Alexander I."},{"first_name":"Wolfgang K.","last_name":"Schief","full_name":"Schief, Wolfgang K."},{"first_name":"Jan","full_name":"Techter, Jan","last_name":"Techter"}],"publication":"Discrete and Computational Geometry","intvolume":"        66","isi":1,"ec_funded":1,"article_processing_charge":"No","citation":{"mla":"Akopyan, Arseniy, et al. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” <i>Discrete and Computational Geometry</i>, vol. 66, Springer Nature, 2021, pp. 938–76, doi:<a href=\"https://doi.org/10.1007/s00454-020-00240-w\">10.1007/s00454-020-00240-w</a>.","short":"A. Akopyan, A.I. Bobenko, W.K. Schief, J. Techter, Discrete and Computational Geometry 66 (2021) 938–976.","ieee":"A. Akopyan, A. I. Bobenko, W. K. Schief, and J. Techter, “On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs,” <i>Discrete and Computational Geometry</i>, vol. 66. Springer Nature, pp. 938–976, 2021.","ama":"Akopyan A, Bobenko AI, Schief WK, Techter J. On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. <i>Discrete and Computational Geometry</i>. 2021;66:938-976. doi:<a href=\"https://doi.org/10.1007/s00454-020-00240-w\">10.1007/s00454-020-00240-w</a>","apa":"Akopyan, A., Bobenko, A. I., Schief, W. K., &#38; Techter, J. (2021). On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00240-w\">https://doi.org/10.1007/s00454-020-00240-w</a>","chicago":"Akopyan, Arseniy, Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter. “On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-020-00240-w\">https://doi.org/10.1007/s00454-020-00240-w</a>.","ista":"Akopyan A, Bobenko AI, Schief WK, Techter J. 2021. On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry. 66, 938–976."},"_id":"8338","month":"10","title":"On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs","article_type":"original","doi":"10.1007/s00454-020-00240-w","arxiv":1,"day":"01","publisher":"Springer Nature","type":"journal_article","status":"public","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","page":"938-976","date_published":"2021-10-01T00:00:00Z","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183"}],"date_updated":"2025-04-14T07:48:36Z","year":"2021","oa_version":"Preprint"},{"file_date_updated":"2021-08-06T09:52:29Z","publication":"Discrete & Computational Geometry","keyword":["Theoretical Computer Science","Computational Theory and Mathematics","Geometry and Topology","Discrete Mathematics and Combinatorics"],"author":[{"first_name":"Jean-Daniel","full_name":"Boissonnat, Jean-Daniel","last_name":"Boissonnat"},{"first_name":"Siargey","last_name":"Kachanovich","full_name":"Kachanovich, Siargey"},{"orcid":"0000-0002-7472-2220","last_name":"Wintraecken","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","first_name":"Mathijs"}],"scopus_import":"1","ec_funded":1,"intvolume":"        66","isi":1,"file":[{"file_size":983307,"date_created":"2021-08-06T09:52:29Z","file_id":"9795","file_name":"2021_DescreteCompGeopmetry_Boissonnat.pdf","creator":"kschuh","date_updated":"2021-08-06T09:52:29Z","checksum":"c848986091e56699dc12de85adb1e39c","content_type":"application/pdf","success":1,"relation":"main_file","access_level":"open_access"}],"acknowledgement":"This work has been funded by the European Research Council under the European Union’s ERC Grant Agreement Number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions). The third author also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. Open access funding provided by the Institute of Science and Technology (IST Austria).","oa":1,"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"department":[{"_id":"HeEd"}],"external_id":{"isi":["000597770300001"]},"date_created":"2020-12-12T11:07:02Z","language":[{"iso":"eng"}],"corr_author":"1","publication_status":"published","abstract":[{"text":"We quantise Whitney’s construction to prove the existence of a triangulation for any C^2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.","lang":"eng"}],"volume":66,"quality_controlled":"1","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","page":"386-434","project":[{"_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020"}],"date_published":"2021-07-01T00:00:00Z","has_accepted_license":"1","oa_version":"Published Version","year":"2021","date_updated":"2025-04-14T07:43:50Z","day":"01","type":"journal_article","publisher":"Springer Nature","issue":"1","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"doi":"10.1007/s00454-020-00250-8","article_type":"original","ddc":["516"],"article_processing_charge":"Yes (via OA deal)","citation":{"ieee":"J.-D. Boissonnat, S. Kachanovich, and M. Wintraecken, “Triangulating submanifolds: An elementary and quantified version of Whitney’s method,” <i>Discrete &#38; Computational Geometry</i>, vol. 66, no. 1. Springer Nature, pp. 386–434, 2021.","short":"J.-D. Boissonnat, S. Kachanovich, M. Wintraecken, Discrete &#38; Computational Geometry 66 (2021) 386–434.","mla":"Boissonnat, Jean-Daniel, et al. “Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method.” <i>Discrete &#38; Computational Geometry</i>, vol. 66, no. 1, Springer Nature, 2021, pp. 386–434, doi:<a href=\"https://doi.org/10.1007/s00454-020-00250-8\">10.1007/s00454-020-00250-8</a>.","chicago":"Boissonnat, Jean-Daniel, Siargey Kachanovich, and Mathijs Wintraecken. “Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00454-020-00250-8\">https://doi.org/10.1007/s00454-020-00250-8</a>.","ista":"Boissonnat J-D, Kachanovich S, Wintraecken M. 2021. Triangulating submanifolds: An elementary and quantified version of Whitney’s method. Discrete &#38; Computational Geometry. 66(1), 386–434.","ama":"Boissonnat J-D, Kachanovich S, Wintraecken M. Triangulating submanifolds: An elementary and quantified version of Whitney’s method. <i>Discrete &#38; Computational Geometry</i>. 2021;66(1):386-434. doi:<a href=\"https://doi.org/10.1007/s00454-020-00250-8\">10.1007/s00454-020-00250-8</a>","apa":"Boissonnat, J.-D., Kachanovich, S., &#38; Wintraecken, M. (2021). Triangulating submanifolds: An elementary and quantified version of Whitney’s method. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-020-00250-8\">https://doi.org/10.1007/s00454-020-00250-8</a>"},"_id":"8940","title":"Triangulating submanifolds: An elementary and quantified version of Whitney’s method","month":"07"}]