[{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"ista":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. 2020. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. Topological Data Analysis. , Abel Symposia, vol. 15, 181–218.","chicago":"Edelsbrunner, Herbert, Anton Nikitenko, Katharina Ölsböck, and Peter Synak. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” In Topological Data Analysis, 15:181–218. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-43408-3_8.","ieee":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, and P. Synak, “Radius functions on Poisson–Delaunay mosaics and related complexes experimentally,” in Topological Data Analysis, 2020, vol. 15, pp. 181–218.","short":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, P. Synak, in:, Topological Data Analysis, Springer Nature, 2020, pp. 181–218.","ama":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In: Topological Data Analysis. Vol 15. Springer Nature; 2020:181-218. doi:10.1007/978-3-030-43408-3_8","apa":"Edelsbrunner, H., Nikitenko, A., Ölsböck, K., & Synak, P. (2020). Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In Topological Data Analysis (Vol. 15, pp. 181–218). Springer Nature. https://doi.org/10.1007/978-3-030-43408-3_8","mla":"Edelsbrunner, Herbert, et al. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” Topological Data Analysis, vol. 15, Springer Nature, 2020, pp. 181–218, doi:10.1007/978-3-030-43408-3_8."},"title":"Radius functions on Poisson–Delaunay mosaics and related complexes experimentally","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton","last_name":"Nikitenko","full_name":"Nikitenko, Anton"},{"full_name":"Ölsböck, Katharina","last_name":"Ölsböck","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","first_name":"Katharina"},{"first_name":"Peter","id":"331776E2-F248-11E8-B48F-1D18A9856A87","last_name":"Synak","full_name":"Synak, Peter"}],"article_processing_charge":"No","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"2533E772-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Efficient Simulation of Natural Phenomena at Extremely Large Scales","grant_number":"638176"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"day":"22","publication":"Topological Data Analysis","has_accepted_license":"1","year":"2020","date_published":"2020-06-22T00:00:00Z","doi":"10.1007/978-3-030-43408-3_8","date_created":"2020-07-19T22:00:59Z","page":"181-218","acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No 78818 Alpha and No 638176). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","quality_controlled":"1","publisher":"Springer Nature","oa":1,"ddc":["510"],"date_updated":"2021-01-12T08:17:06Z","department":[{"_id":"HeEd"}],"file_date_updated":"2020-10-08T08:56:14Z","_id":"8135","status":"public","type":"conference","file":[{"file_size":2207071,"date_updated":"2020-10-08T08:56:14Z","creator":"dernst","file_name":"2020-B-01-PoissonExperimentalSurvey.pdf","date_created":"2020-10-08T08:56:14Z","content_type":"application/pdf","relation":"main_file","access_level":"open_access","success":1,"checksum":"7b5e0de10675d787a2ddb2091370b8d8","file_id":"8628"}],"language":[{"iso":"eng"}],"publication_identifier":{"eissn":["21978549"],"isbn":["9783030434076"],"issn":["21932808"]},"publication_status":"published","volume":15,"ec_funded":1,"oa_version":"Submitted Version","abstract":[{"lang":"eng","text":"Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics."}],"month":"06","intvolume":" 15","scopus_import":"1","alternative_title":["Abel Symposia"]},{"ddc":["510"],"date_updated":"2021-03-22T09:01:50Z","file_date_updated":"2021-03-22T08:56:37Z","department":[{"_id":"HeEd"}],"_id":"9249","status":"public","article_type":"original","type":"journal_article","tmp":{"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)","short":"CC BY (4.0)"},"file":[{"file_size":3668725,"date_updated":"2021-03-22T08:56:37Z","creator":"dernst","file_name":"2020_MathMorpholTheoryAppl_Biswas.pdf","date_created":"2021-03-22T08:56:37Z","content_type":"application/pdf","relation":"main_file","access_level":"open_access","success":1,"checksum":"4a1043fa0548a725d464017fe2483ce0","file_id":"9272"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["2353-3390"]},"publication_status":"published","volume":4,"issue":"1","ec_funded":1,"oa_version":"Published Version","abstract":[{"lang":"eng","text":"Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system."}],"month":"11","intvolume":" 4","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"chicago":"Biswas, Ranita, Gaëlle Largeteau-Skapin, Rita Zrour, and Eric Andres. “Digital Objects in Rhombic Dodecahedron Grid.” Mathematical Morphology - Theory and Applications. De Gruyter, 2020. https://doi.org/10.1515/mathm-2020-0106.","ista":"Biswas R, Largeteau-Skapin G, Zrour R, Andres E. 2020. Digital objects in rhombic dodecahedron grid. Mathematical Morphology - Theory and Applications. 4(1), 143–158.","mla":"Biswas, Ranita, et al. “Digital Objects in Rhombic Dodecahedron Grid.” Mathematical Morphology - Theory and Applications, vol. 4, no. 1, De Gruyter, 2020, pp. 143–58, doi:10.1515/mathm-2020-0106.","short":"R. Biswas, G. Largeteau-Skapin, R. Zrour, E. Andres, Mathematical Morphology - Theory and Applications 4 (2020) 143–158.","ieee":"R. Biswas, G. Largeteau-Skapin, R. Zrour, and E. Andres, “Digital objects in rhombic dodecahedron grid,” Mathematical Morphology - Theory and Applications, vol. 4, no. 1. De Gruyter, pp. 143–158, 2020.","apa":"Biswas, R., Largeteau-Skapin, G., Zrour, R., & Andres, E. (2020). Digital objects in rhombic dodecahedron grid. Mathematical Morphology - Theory and Applications. De Gruyter. https://doi.org/10.1515/mathm-2020-0106","ama":"Biswas R, Largeteau-Skapin G, Zrour R, Andres E. Digital objects in rhombic dodecahedron grid. Mathematical Morphology - Theory and Applications. 2020;4(1):143-158. doi:10.1515/mathm-2020-0106"},"title":"Digital objects in rhombic dodecahedron grid","author":[{"last_name":"Biswas","orcid":"0000-0002-5372-7890","full_name":"Biswas, Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","first_name":"Ranita"},{"first_name":"Gaëlle","last_name":"Largeteau-Skapin","full_name":"Largeteau-Skapin, Gaëlle"},{"full_name":"Zrour, Rita","last_name":"Zrour","first_name":"Rita"},{"first_name":"Eric","full_name":"Andres, Eric","last_name":"Andres"}],"article_processing_charge":"No","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"day":"17","publication":"Mathematical Morphology - Theory and Applications","has_accepted_license":"1","year":"2020","date_published":"2020-11-17T00:00:00Z","doi":"10.1515/mathm-2020-0106","date_created":"2021-03-16T08:55:19Z","page":"143-158","acknowledgement":"This work has been partially supported by the European Research Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation programme, grant no. 788183, and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. ","quality_controlled":"1","publisher":"De Gruyter","oa":1},{"date_updated":"2021-04-06T11:32:32Z","department":[{"_id":"HeEd"}],"series_title":"LNCS","_id":"9299","type":"conference","conference":{"end_date":"2020-09-18","location":"Virtual, Online","start_date":"2020-09-16","name":"GD: Graph Drawing and Network Visualization"},"status":"public","publication_identifier":{"eissn":["1611-3349"],"isbn":["9783030687656"],"issn":["0302-9743"]},"publication_status":"published","language":[{"iso":"eng"}],"volume":12590,"abstract":[{"lang":"eng","text":"We call a multigraph non-homotopic if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. It is easy to see that a non-homotopic multigraph on n>1 vertices can have arbitrarily many edges. We prove that the number of crossings between the edges of a non-homotopic multigraph with n vertices and m>4n edges is larger than cm2n for some constant c>0 , and that this bound is tight up to a polylogarithmic factor. We also show that the lower bound is not asymptotically sharp as n is fixed and m⟶∞ ."}],"oa_version":"Preprint","scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2006.14908"}],"month":"09","intvolume":" 12590","citation":{"ieee":"J. Pach, G. Tardos, and G. Tóth, “Crossings between non-homotopic edges,” in 28th International Symposium on Graph Drawing and Network Visualization, Virtual, Online, 2020, vol. 12590, pp. 359–371.","short":"J. Pach, G. Tardos, G. Tóth, in:, 28th International Symposium on Graph Drawing and Network Visualization, Springer Nature, 2020, pp. 359–371.","apa":"Pach, J., Tardos, G., & Tóth, G. (2020). Crossings between non-homotopic edges. In 28th International Symposium on Graph Drawing and Network Visualization (Vol. 12590, pp. 359–371). Virtual, Online: Springer Nature. https://doi.org/10.1007/978-3-030-68766-3_28","ama":"Pach J, Tardos G, Tóth G. Crossings between non-homotopic edges. In: 28th International Symposium on Graph Drawing and Network Visualization. Vol 12590. LNCS. Springer Nature; 2020:359-371. doi:10.1007/978-3-030-68766-3_28","mla":"Pach, János, et al. “Crossings between Non-Homotopic Edges.” 28th International Symposium on Graph Drawing and Network Visualization, vol. 12590, Springer Nature, 2020, pp. 359–71, doi:10.1007/978-3-030-68766-3_28.","ista":"Pach J, Tardos G, Tóth G. 2020. Crossings between non-homotopic edges. 28th International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network VisualizationLNCS vol. 12590, 359–371.","chicago":"Pach, János, Gábor Tardos, and Géza Tóth. “Crossings between Non-Homotopic Edges.” In 28th International Symposium on Graph Drawing and Network Visualization, 12590:359–71. LNCS. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-68766-3_28."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"full_name":"Pach, János","last_name":"Pach","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János"},{"full_name":"Tardos, Gábor","last_name":"Tardos","first_name":"Gábor"},{"first_name":"Géza","full_name":"Tóth, Géza","last_name":"Tóth"}],"external_id":{"arxiv":["2006.14908"]},"article_processing_charge":"No","title":"Crossings between non-homotopic edges","project":[{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"The Wittgenstein Prize","grant_number":"Z00342"}],"year":"2020","day":"20","publication":"28th International Symposium on Graph Drawing and Network Visualization","page":"359-371","date_published":"2020-09-20T00:00:00Z","doi":"10.1007/978-3-030-68766-3_28","date_created":"2021-03-28T22:01:44Z","acknowledgement":"Supported by the National Research, Development and Innovation Office, NKFIH, KKP-133864, K-131529, K-116769, K-132696, by the Higher Educational Institutional Excellence Program 2019 NKFIH-1158-6/2019, the Austrian Science Fund (FWF), grant Z 342-N31, by the Ministry of Education and Science of the Russian Federation MegaGrant No. 075-15-2019-1926, and by the ERC Synergy Grant “Dynasnet” No. 810115. A full version can be found at https://arxiv.org/abs/2006.14908.","quality_controlled":"1","publisher":"Springer Nature","oa":1},{"project":[{"_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Discretization in Geometry and Dynamics","grant_number":"I4887"}],"citation":{"ista":"Edelsbrunner H, Virk Z, Wagner H. 2020. Topological data analysis in information space. Journal of Computational Geometry. 11(2), 162–182.","chicago":"Edelsbrunner, Herbert, Ziga Virk, and Hubert Wagner. “Topological Data Analysis in Information Space.” Journal of Computational Geometry. Carleton University, 2020. https://doi.org/10.20382/jocg.v11i2a7.","ieee":"H. Edelsbrunner, Z. Virk, and H. Wagner, “Topological data analysis in information space,” Journal of Computational Geometry, vol. 11, no. 2. Carleton University, pp. 162–182, 2020.","short":"H. Edelsbrunner, Z. Virk, H. Wagner, Journal of Computational Geometry 11 (2020) 162–182.","ama":"Edelsbrunner H, Virk Z, Wagner H. Topological data analysis in information space. Journal of Computational Geometry. 2020;11(2):162-182. doi:10.20382/jocg.v11i2a7","apa":"Edelsbrunner, H., Virk, Z., & Wagner, H. (2020). Topological data analysis in information space. Journal of Computational Geometry. Carleton University. https://doi.org/10.20382/jocg.v11i2a7","mla":"Edelsbrunner, Herbert, et al. “Topological Data Analysis in Information Space.” Journal of Computational Geometry, vol. 11, no. 2, Carleton University, 2020, pp. 162–82, doi:10.20382/jocg.v11i2a7."},"user_id":"6785fbc1-c503-11eb-8a32-93094b40e1cf","article_processing_charge":"Yes","author":[{"orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"id":"2E36B656-F248-11E8-B48F-1D18A9856A87","first_name":"Ziga","full_name":"Virk, Ziga","last_name":"Virk"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","full_name":"Wagner, Hubert","last_name":"Wagner"}],"title":"Topological data analysis in information space","acknowledgement":"This research is partially supported by the Office of Naval Research, through grant no. N62909-18-1-2038, and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","oa":1,"quality_controlled":"1","publisher":"Carleton University","year":"2020","has_accepted_license":"1","publication":"Journal of Computational Geometry","day":"14","page":"162-182","date_created":"2021-07-04T22:01:26Z","date_published":"2020-12-14T00:00:00Z","doi":"10.20382/jocg.v11i2a7","_id":"9630","tmp":{"short":"CC BY (3.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/3.0/legalcode","name":"Creative Commons Attribution 3.0 Unported (CC BY 3.0)"},"type":"journal_article","article_type":"original","status":"public","date_updated":"2021-08-11T12:26:34Z","ddc":["510","000"],"department":[{"_id":"HeEd"}],"file_date_updated":"2021-08-11T11:55:11Z","abstract":[{"text":"Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context.","lang":"eng"}],"oa_version":"Published Version","scopus_import":"1","intvolume":" 11","month":"12","publication_status":"published","publication_identifier":{"eissn":["1920180X"]},"language":[{"iso":"eng"}],"file":[{"content_type":"application/pdf","access_level":"open_access","relation":"main_file","checksum":"f02d0b2b3838e7891a6c417fc34ffdcd","file_id":"9882","success":1,"date_updated":"2021-08-11T11:55:11Z","file_size":1449234,"creator":"asandaue","date_created":"2021-08-11T11:55:11Z","file_name":"2020_JournalOfComputationalGeometry_Edelsbrunner.pdf"}],"license":"https://creativecommons.org/licenses/by/3.0/","issue":"2","volume":11},{"month":"09","scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/2001.02934","open_access":"1"}],"oa_version":"Preprint","abstract":[{"text":"We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.","lang":"eng"}],"ec_funded":1,"language":[{"iso":"eng"}],"publication_identifier":{"eissn":["2199-6768"],"issn":["2199-675X"]},"publication_status":"published","status":"public","article_type":"original","type":"journal_article","_id":"8538","department":[{"_id":"HeEd"}],"date_updated":"2021-12-02T15:10:17Z","publisher":"Springer Nature","quality_controlled":"1","oa":1,"acknowledgement":" This paper would not be written if not for Dan Reznik’s curiosity and persistence; we are very grateful to him. We also thank R. Garcia and J. Koiller for interesting discussions. It is a pleasure to thank the Mathematical Institute of the University of Heidelberg for its stimulating atmosphere. ST thanks M. Bialy for interesting discussions and the Tel Aviv\r\nUniversity for its invariable hospitality. AA was supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 78818 Alpha). RS is supported by NSF Grant DMS-1807320. ST was supported by NSF grant DMS-1510055 and SFB/TRR 191.","doi":"10.1007/s40879-020-00426-9","date_published":"2020-09-09T00:00:00Z","date_created":"2020-09-20T22:01:38Z","day":"09","publication":"European Journal of Mathematics","year":"2020","project":[{"name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"title":"Billiards in ellipses revisited","author":[{"full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","last_name":"Akopyan","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Schwartz","full_name":"Schwartz, Richard","first_name":"Richard"},{"first_name":"Serge","last_name":"Tabachnikov","full_name":"Tabachnikov, Serge"}],"external_id":{"arxiv":["2001.02934"]},"article_processing_charge":"No","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","citation":{"ista":"Akopyan A, Schwartz R, Tabachnikov S. 2020. Billiards in ellipses revisited. European Journal of Mathematics.","chicago":"Akopyan, Arseniy, Richard Schwartz, and Serge Tabachnikov. “Billiards in Ellipses Revisited.” European Journal of Mathematics. Springer Nature, 2020. https://doi.org/10.1007/s40879-020-00426-9.","short":"A. Akopyan, R. Schwartz, S. Tabachnikov, European Journal of Mathematics (2020).","ieee":"A. Akopyan, R. Schwartz, and S. Tabachnikov, “Billiards in ellipses revisited,” European Journal of Mathematics. Springer Nature, 2020.","apa":"Akopyan, A., Schwartz, R., & Tabachnikov, S. (2020). Billiards in ellipses revisited. European Journal of Mathematics. Springer Nature. https://doi.org/10.1007/s40879-020-00426-9","ama":"Akopyan A, Schwartz R, Tabachnikov S. Billiards in ellipses revisited. European Journal of Mathematics. 2020. doi:10.1007/s40879-020-00426-9","mla":"Akopyan, Arseniy, et al. “Billiards in Ellipses Revisited.” European Journal of Mathematics, Springer Nature, 2020, doi:10.1007/s40879-020-00426-9."}},{"ec_funded":1,"related_material":{"record":[{"id":"9649","status":"public","relation":"later_version"}]},"volume":164,"publication_status":"published","publication_identifier":{"isbn":["978-3-95977-143-6"],"issn":["1868-8969"]},"language":[{"iso":"eng"}],"file":[{"content_type":"application/pdf","relation":"main_file","access_level":"open_access","file_id":"7969","checksum":"38cbfa4f5d484d267a35d44d210df044","file_size":1009739,"date_updated":"2020-07-14T12:48:06Z","creator":"dernst","file_name":"2020_LIPIcsSoCG_Boissonnat.pdf","date_created":"2020-06-17T10:13:34Z"}],"alternative_title":["LIPIcs"],"scopus_import":"1","intvolume":" 164","month":"06","abstract":[{"lang":"eng","text":"Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f: ℝ^d → ℝ^(d-n). A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation 𝒯. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary. "}],"oa_version":"Published Version","department":[{"_id":"HeEd"}],"file_date_updated":"2020-07-14T12:48:06Z","date_updated":"2023-08-02T06:49:16Z","ddc":["510"],"conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2020-06-22","end_date":"2020-06-26","location":"Zürich, Switzerland"},"tmp":{"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)","short":"CC BY (4.0)"},"type":"conference","status":"public","_id":"7952","date_created":"2020-06-09T07:24:11Z","doi":"10.4230/LIPIcs.SoCG.2020.20","date_published":"2020-06-01T00:00:00Z","year":"2020","has_accepted_license":"1","publication":"36th International Symposium on Computational Geometry","day":"01","oa":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","quality_controlled":"1","article_processing_charge":"No","author":[{"last_name":"Boissonnat","full_name":"Boissonnat, Jean-Daniel","first_name":"Jean-Daniel"},{"first_name":"Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs"}],"title":"The topological correctness of PL-approximations of isomanifolds","citation":{"apa":"Boissonnat, J.-D., & Wintraecken, M. (2020). The topological correctness of PL-approximations of isomanifolds. In 36th International Symposium on Computational Geometry (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2020.20","ama":"Boissonnat J-D, Wintraecken M. The topological correctness of PL-approximations of isomanifolds. In: 36th International Symposium on Computational Geometry. Vol 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:10.4230/LIPIcs.SoCG.2020.20","short":"J.-D. Boissonnat, M. Wintraecken, in:, 36th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.","ieee":"J.-D. Boissonnat and M. Wintraecken, “The topological correctness of PL-approximations of isomanifolds,” in 36th International Symposium on Computational Geometry, Zürich, Switzerland, 2020, vol. 164.","mla":"Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL-Approximations of Isomanifolds.” 36th International Symposium on Computational Geometry, vol. 164, 20:1-20:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020, doi:10.4230/LIPIcs.SoCG.2020.20.","ista":"Boissonnat J-D, Wintraecken M. 2020. The topological correctness of PL-approximations of isomanifolds. 36th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 164, 20:1-20:18.","chicago":"Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL-Approximations of Isomanifolds.” In 36th International Symposium on Computational Geometry, Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. https://doi.org/10.4230/LIPIcs.SoCG.2020.20."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411"}],"article_number":"20:1-20:18"},{"editor":[{"first_name":"Bo'az","last_name":"Klartag","full_name":"Klartag, Bo'az"},{"first_name":"Emanuel","last_name":"Milman","full_name":"Milman, Emanuel"}],"title":"Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures","author":[{"orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy"},{"last_name":"Karasev","full_name":"Karasev, Roman","first_name":"Roman"}],"article_processing_charge":"No","external_id":{"isi":["000557689300003"],"arxiv":["1808.07350"]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"chicago":"Akopyan, Arseniy, and Roman Karasev. “Gromov’s Waist of Non-Radial Gaussian Measures and Radial Non-Gaussian Measures.” In Geometric Aspects of Functional Analysis, edited by Bo’az Klartag and Emanuel Milman, 2256:1–27. LNM. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-36020-7_1.","ista":"Akopyan A, Karasev R. 2020.Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In: Geometric Aspects of Functional Analysis. vol. 2256, 1–27.","mla":"Akopyan, Arseniy, and Roman Karasev. “Gromov’s Waist of Non-Radial Gaussian Measures and Radial Non-Gaussian Measures.” Geometric Aspects of Functional Analysis, edited by Bo’az Klartag and Emanuel Milman, vol. 2256, Springer Nature, 2020, pp. 1–27, doi:10.1007/978-3-030-36020-7_1.","apa":"Akopyan, A., & Karasev, R. (2020). Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In B. Klartag & E. Milman (Eds.), Geometric Aspects of Functional Analysis (Vol. 2256, pp. 1–27). Springer Nature. https://doi.org/10.1007/978-3-030-36020-7_1","ama":"Akopyan A, Karasev R. Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In: Klartag B, Milman E, eds. Geometric Aspects of Functional Analysis. Vol 2256. LNM. Springer Nature; 2020:1-27. doi:10.1007/978-3-030-36020-7_1","ieee":"A. Akopyan and R. Karasev, “Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures,” in Geometric Aspects of Functional Analysis, vol. 2256, B. Klartag and E. Milman, Eds. Springer Nature, 2020, pp. 1–27.","short":"A. Akopyan, R. Karasev, in:, B. Klartag, E. Milman (Eds.), Geometric Aspects of Functional Analysis, Springer Nature, 2020, pp. 1–27."},"project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"716117","name":"Optimal Transport and Stochastic Dynamics"}],"date_published":"2020-06-21T00:00:00Z","doi":"10.1007/978-3-030-36020-7_1","date_created":"2018-12-11T11:44:29Z","page":"1-27","day":"21","publication":"Geometric Aspects of Functional Analysis","isi":1,"year":"2020","quality_controlled":"1","publisher":"Springer Nature","oa":1,"department":[{"_id":"HeEd"},{"_id":"JaMa"}],"date_updated":"2023-08-17T13:48:31Z","status":"public","type":"book_chapter","series_title":"LNM","_id":"74","volume":2256,"ec_funded":1,"language":[{"iso":"eng"}],"publication_identifier":{"eisbn":["9783030360207"],"issn":["00758434"],"eissn":["16179692"],"isbn":["9783030360191"]},"publication_status":"published","month":"06","intvolume":" 2256","scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/1808.07350","open_access":"1"}],"oa_version":"Preprint","abstract":[{"text":"We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class\r\nof compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2.\r\nWe use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian\r\nmeasures.","lang":"eng"}]},{"_id":"7554","article_type":"original","type":"journal_article","status":"public","date_updated":"2023-08-18T06:45:48Z","department":[{"_id":"HeEd"}],"abstract":[{"lang":"eng","text":"Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in ${R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in ${R}^n$."}],"oa_version":"Preprint","scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/1705.08735","open_access":"1"}],"month":"02","intvolume":" 64","publication_identifier":{"issn":["0040585X"],"eissn":["10957219"]},"publication_status":"published","language":[{"iso":"eng"}],"issue":"4","volume":64,"ec_funded":1,"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"citation":{"ista":"Edelsbrunner H, Nikitenko A. 2020. Weighted Poisson–Delaunay mosaics. Theory of Probability and its Applications. 64(4), 595–614.","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Weighted Poisson–Delaunay Mosaics.” Theory of Probability and Its Applications. SIAM, 2020. https://doi.org/10.1137/S0040585X97T989726.","short":"H. Edelsbrunner, A. Nikitenko, Theory of Probability and Its Applications 64 (2020) 595–614.","ieee":"H. Edelsbrunner and A. Nikitenko, “Weighted Poisson–Delaunay mosaics,” Theory of Probability and its Applications, vol. 64, no. 4. SIAM, pp. 595–614, 2020.","apa":"Edelsbrunner, H., & Nikitenko, A. (2020). Weighted Poisson–Delaunay mosaics. Theory of Probability and Its Applications. SIAM. https://doi.org/10.1137/S0040585X97T989726","ama":"Edelsbrunner H, Nikitenko A. Weighted Poisson–Delaunay mosaics. Theory of Probability and its Applications. 2020;64(4):595-614. doi:10.1137/S0040585X97T989726","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Weighted Poisson–Delaunay Mosaics.” Theory of Probability and Its Applications, vol. 64, no. 4, SIAM, 2020, pp. 595–614, doi:10.1137/S0040585X97T989726."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","author":[{"full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Nikitenko","orcid":"0000-0002-0659-3201","full_name":"Nikitenko, Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","first_name":"Anton"}],"external_id":{"isi":["000551393100007"],"arxiv":["1705.08735"]},"article_processing_charge":"No","title":"Weighted Poisson–Delaunay mosaics","quality_controlled":"1","publisher":"SIAM","oa":1,"isi":1,"year":"2020","day":"13","publication":"Theory of Probability and its Applications","page":"595-614","date_published":"2020-02-13T00:00:00Z","doi":"10.1137/S0040585X97T989726","date_created":"2020-03-01T23:00:39Z"},{"oa":1,"quality_controlled":"1","publisher":"Springer Nature","acknowledgement":"This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF).","date_created":"2020-04-19T22:00:56Z","doi":"10.1007/s00454-020-00188-x","date_published":"2020-03-20T00:00:00Z","page":"759-775","publication":"Discrete and Computational Geometry","day":"20","year":"2020","isi":1,"has_accepted_license":"1","project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"},{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"title":"Tri-partitions and bases of an ordered complex","external_id":{"isi":["000520918800001"]},"article_processing_charge":"Yes (via OA deal)","author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"last_name":"Ölsböck","orcid":"0000-0002-4672-8297","full_name":"Ölsböck, Katharina","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","first_name":"Katharina"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"ieee":"H. Edelsbrunner and K. Ölsböck, “Tri-partitions and bases of an ordered complex,” Discrete and Computational Geometry, vol. 64. Springer Nature, pp. 759–775, 2020.","short":"H. Edelsbrunner, K. Ölsböck, Discrete and Computational Geometry 64 (2020) 759–775.","ama":"Edelsbrunner H, Ölsböck K. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 2020;64:759-775. doi:10.1007/s00454-020-00188-x","apa":"Edelsbrunner, H., & Ölsböck, K. (2020). Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00188-x","mla":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” Discrete and Computational Geometry, vol. 64, Springer Nature, 2020, pp. 759–75, doi:10.1007/s00454-020-00188-x.","ista":"Edelsbrunner H, Ölsböck K. 2020. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 64, 759–775.","chicago":"Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00188-x."},"intvolume":" 64","month":"03","scopus_import":"1","oa_version":"Published Version","abstract":[{"text":"Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.","lang":"eng"}],"ec_funded":1,"volume":64,"language":[{"iso":"eng"}],"file":[{"file_size":701673,"date_updated":"2020-11-20T13:22:21Z","creator":"dernst","file_name":"2020_DiscreteCompGeo_Edelsbrunner.pdf","date_created":"2020-11-20T13:22:21Z","content_type":"application/pdf","relation":"main_file","access_level":"open_access","success":1,"file_id":"8786","checksum":"f8cc96e497f00c38340b5dafe0cb91d7"}],"publication_status":"published","publication_identifier":{"eissn":["14320444"],"issn":["01795376"]},"status":"public","tmp":{"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)","short":"CC BY (4.0)"},"article_type":"original","type":"journal_article","_id":"7666","file_date_updated":"2020-11-20T13:22:21Z","department":[{"_id":"HeEd"}],"ddc":["510"],"date_updated":"2023-08-21T06:13:48Z"},{"issue":"4","volume":63,"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"eissn":["14320444"],"issn":["01795376"]},"intvolume":" 63","month":"06","main_file_link":[{"url":"https://arxiv.org/abs/1803.06710","open_access":"1"}],"scopus_import":"1","oa_version":"Preprint","abstract":[{"text":"A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets.","lang":"eng"}],"department":[{"_id":"HeEd"}],"date_updated":"2023-08-21T08:49:18Z","status":"public","article_type":"original","type":"journal_article","_id":"7962","date_created":"2020-06-14T22:00:51Z","doi":"10.1007/s00454-020-00213-z","date_published":"2020-06-05T00:00:00Z","page":"888-917","publication":"Discrete and Computational Geometry","day":"05","year":"2020","isi":1,"oa":1,"quality_controlled":"1","publisher":"Springer Nature","title":"Almost all string graphs are intersection graphs of plane convex sets","article_processing_charge":"No","external_id":{"isi":["000538229000001"],"arxiv":["1803.06710"]},"author":[{"last_name":"Pach","full_name":"Pach, János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","first_name":"János"},{"last_name":"Reed","full_name":"Reed, Bruce","first_name":"Bruce"},{"last_name":"Yuditsky","full_name":"Yuditsky, Yelena","first_name":"Yelena"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"mla":"Pach, János, et al. “Almost All String Graphs Are Intersection Graphs of Plane Convex Sets.” Discrete and Computational Geometry, vol. 63, no. 4, Springer Nature, 2020, pp. 888–917, doi:10.1007/s00454-020-00213-z.","ieee":"J. Pach, B. Reed, and Y. Yuditsky, “Almost all string graphs are intersection graphs of plane convex sets,” Discrete and Computational Geometry, vol. 63, no. 4. Springer Nature, pp. 888–917, 2020.","short":"J. Pach, B. Reed, Y. Yuditsky, Discrete and Computational Geometry 63 (2020) 888–917.","apa":"Pach, J., Reed, B., & Yuditsky, Y. (2020). Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00213-z","ama":"Pach J, Reed B, Yuditsky Y. Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. 2020;63(4):888-917. doi:10.1007/s00454-020-00213-z","chicago":"Pach, János, Bruce Reed, and Yelena Yuditsky. “Almost All String Graphs Are Intersection Graphs of Plane Convex Sets.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00213-z.","ista":"Pach J, Reed B, Yuditsky Y. 2020. Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. 63(4), 888–917."},"project":[{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342","name":"The Wittgenstein Prize"}]}]