[{"month":"04","department":[{"_id":"HeEd"}],"file":[{"relation":"main_file","checksum":"e52a832f1def52a2b23d21bcc09e646f","file_name":"2021_Geometry_Edelsbrunner.pdf","success":1,"content_type":"application/pdf","access_level":"open_access","file_id":"9544","date_created":"2021-06-11T13:16:26Z","file_size":694706,"creator":"kschuh","date_updated":"2021-06-11T13:16:26Z"}],"article_number":"15","oa":1,"language":[{"iso":"eng"}],"issue":"1","citation":{"short":"H. Edelsbrunner, A. Nikitenko, G.F. Osang, Journal of Geometry 112 (2021).","ieee":"H. Edelsbrunner, A. Nikitenko, and G. F. Osang, “A step in the Delaunay mosaic of order k,” <i>Journal of Geometry</i>, vol. 112, no. 1. Springer Nature, 2021.","ama":"Edelsbrunner H, Nikitenko A, Osang GF. A step in the Delaunay mosaic of order k. <i>Journal of Geometry</i>. 2021;112(1). doi:<a href=\"https://doi.org/10.1007/s00022-021-00577-4\">10.1007/s00022-021-00577-4</a>","mla":"Edelsbrunner, Herbert, et al. “A Step in the Delaunay Mosaic of Order K.” <i>Journal of Geometry</i>, vol. 112, no. 1, 15, Springer Nature, 2021, doi:<a href=\"https://doi.org/10.1007/s00022-021-00577-4\">10.1007/s00022-021-00577-4</a>.","apa":"Edelsbrunner, H., Nikitenko, A., &#38; Osang, G. F. (2021). A step in the Delaunay mosaic of order k. <i>Journal of Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00022-021-00577-4\">https://doi.org/10.1007/s00022-021-00577-4</a>","ista":"Edelsbrunner H, Nikitenko A, Osang GF. 2021. A step in the Delaunay mosaic of order k. Journal of Geometry. 112(1), 15.","chicago":"Edelsbrunner, Herbert, Anton Nikitenko, and Georg F Osang. “A Step in the Delaunay Mosaic of Order K.” <i>Journal of Geometry</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00022-021-00577-4\">https://doi.org/10.1007/s00022-021-00577-4</a>."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"last_name":"Nikitenko","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","first_name":"Anton"},{"last_name":"Osang","full_name":"Osang, Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","first_name":"Georg F"}],"day":"01","scopus_import":"1","oa_version":"Published Version","title":"A step in the Delaunay mosaic of order k","volume":112,"article_type":"original","date_created":"2021-06-06T22:01:29Z","has_accepted_license":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"intvolume":"       112","abstract":[{"lang":"eng","text":"Given a locally finite set 𝑋⊆ℝ𝑑 and an integer 𝑘≥0, we consider the function 𝐰𝑘:Del𝑘(𝑋)→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space."}],"publication_identifier":{"eissn":["14208997"],"issn":["00472468"]},"publication_status":"published","file_date_updated":"2021-06-11T13:16:26Z","year":"2021","status":"public","publication":"Journal of Geometry","date_published":"2021-04-01T00:00:00Z","doi":"10.1007/s00022-021-00577-4","article_processing_charge":"Yes (via OA deal)","publisher":"Springer Nature","date_updated":"2022-05-12T11:41:45Z","_id":"9465","type":"journal_article","ddc":["510"],"quality_controlled":"1"},{"arxiv":1,"month":"10","department":[{"_id":"HeEd"}],"file":[{"relation":"main_file","checksum":"3514382e3a1eb87fa6c61ad622874415","success":1,"file_name":"2023_ExperimentalMath_Akopyan.pdf","access_level":"open_access","content_type":"application/pdf","file_id":"14053","file_size":1966019,"date_created":"2023-08-14T11:55:10Z","date_updated":"2023-08-14T11:55:10Z","creator":"dernst"}],"oa":1,"language":[{"iso":"eng"}],"citation":{"short":"A. Akopyan, H. Edelsbrunner, A. Nikitenko, Experimental Mathematics (2021) 1–15.","ieee":"A. Akopyan, H. Edelsbrunner, and A. Nikitenko, “The beauty of random polytopes inscribed in the 2-sphere,” <i>Experimental Mathematics</i>. Taylor and Francis, pp. 1–15, 2021.","ama":"Akopyan A, Edelsbrunner H, Nikitenko A. The beauty of random polytopes inscribed in the 2-sphere. <i>Experimental Mathematics</i>. 2021:1-15. doi:<a href=\"https://doi.org/10.1080/10586458.2021.1980459\">10.1080/10586458.2021.1980459</a>","mla":"Akopyan, Arseniy, et al. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” <i>Experimental Mathematics</i>, Taylor and Francis, 2021, pp. 1–15, doi:<a href=\"https://doi.org/10.1080/10586458.2021.1980459\">10.1080/10586458.2021.1980459</a>.","apa":"Akopyan, A., Edelsbrunner, H., &#38; Nikitenko, A. (2021). The beauty of random polytopes inscribed in the 2-sphere. <i>Experimental Mathematics</i>. Taylor and Francis. <a href=\"https://doi.org/10.1080/10586458.2021.1980459\">https://doi.org/10.1080/10586458.2021.1980459</a>","ista":"Akopyan A, Edelsbrunner H, Nikitenko A. 2021. The beauty of random polytopes inscribed in the 2-sphere. Experimental Mathematics., 1–15.","chicago":"Akopyan, Arseniy, Herbert Edelsbrunner, and Anton Nikitenko. “The Beauty of Random Polytopes Inscribed in the 2-Sphere.” <i>Experimental Mathematics</i>. Taylor and Francis, 2021. <a href=\"https://doi.org/10.1080/10586458.2021.1980459\">https://doi.org/10.1080/10586458.2021.1980459</a>."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","full_name":"Akopyan, Arseniy","last_name":"Akopyan","first_name":"Arseniy","orcid":"0000-0002-2548-617X"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Anton","orcid":"0000-0002-0659-3201","last_name":"Nikitenko","full_name":"Nikitenko, Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"}],"scopus_import":"1","day":"25","title":"The beauty of random polytopes inscribed in the 2-sphere","oa_version":"Published Version","article_type":"original","date_created":"2021-11-07T23:01:25Z","has_accepted_license":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"lang":"eng","text":"Consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density."}],"publication_identifier":{"eissn":["1944-950X"],"issn":["1058-6458"]},"publication_status":"published","file_date_updated":"2023-08-14T11:55:10Z","isi":1,"year":"2021","external_id":{"isi":["000710893500001"],"arxiv":["2007.07783"]},"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"The Wittgenstein Prize","grant_number":"Z00342"},{"name":"Discretization in Geometry and Dynamics","grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"publication":"Experimental Mathematics","status":"public","ec_funded":1,"acknowledgement":"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.\r\nWe are grateful to Dmitry Zaporozhets and Christoph Thäle for valuable comments and for directing us to relevant references. We also thank to Anton Mellit for a useful discussion on Bessel functions.","date_published":"2021-10-25T00:00:00Z","doi":"10.1080/10586458.2021.1980459","article_processing_charge":"Yes (via OA deal)","publisher":"Taylor and Francis","date_updated":"2023-08-14T11:57:07Z","_id":"10222","type":"journal_article","page":"1-15","ddc":["510"],"quality_controlled":"1"},{"year":"2020","date_published":"2020-06-22T00:00:00Z","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).","ec_funded":1,"project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"_id":"2533E772-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"638176","name":"Efficient Simulation of Natural Phenomena at Extremely Large Scales"},{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"publication":"Topological Data Analysis","status":"public","type":"conference","date_updated":"2021-01-12T08:17:06Z","_id":"8135","publisher":"Springer Nature","doi":"10.1007/978-3-030-43408-3_8","alternative_title":["Abel Symposia"],"article_processing_charge":"No","quality_controlled":"1","ddc":["510"],"page":"181-218","file":[{"date_updated":"2020-10-08T08:56:14Z","creator":"dernst","file_size":2207071,"date_created":"2020-10-08T08:56:14Z","file_id":"8628","access_level":"open_access","content_type":"application/pdf","success":1,"file_name":"2020-B-01-PoissonExperimentalSurvey.pdf","checksum":"7b5e0de10675d787a2ddb2091370b8d8","relation":"main_file"}],"department":[{"_id":"HeEd"}],"month":"06","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"short":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, P. Synak, in:, Topological Data Analysis, Springer Nature, 2020, pp. 181–218.","ieee":"H. Edelsbrunner, A. Nikitenko, K. Ölsböck, and P. Synak, “Radius functions on Poisson–Delaunay mosaics and related complexes experimentally,” in <i>Topological Data Analysis</i>, 2020, vol. 15, pp. 181–218.","ama":"Edelsbrunner H, Nikitenko A, Ölsböck K, Synak P. Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In: <i>Topological Data Analysis</i>. Vol 15. Springer Nature; 2020:181-218. doi:<a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">10.1007/978-3-030-43408-3_8</a>","mla":"Edelsbrunner, Herbert, et al. “Radius Functions on Poisson–Delaunay Mosaics and Related Complexes Experimentally.” <i>Topological Data Analysis</i>, vol. 15, Springer Nature, 2020, pp. 181–218, doi:<a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">10.1007/978-3-030-43408-3_8</a>.","apa":"Edelsbrunner, H., Nikitenko, A., Ölsböck, K., &#38; Synak, P. (2020). Radius functions on Poisson–Delaunay mosaics and related complexes experimentally. In <i>Topological Data Analysis</i> (Vol. 15, pp. 181–218). Springer Nature. <a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">https://doi.org/10.1007/978-3-030-43408-3_8</a>","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 <i>Topological Data Analysis</i>, 15:181–218. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/978-3-030-43408-3_8\">https://doi.org/10.1007/978-3-030-43408-3_8</a>."},"language":[{"iso":"eng"}],"oa":1,"date_created":"2020-07-19T22:00:59Z","volume":15,"title":"Radius functions on Poisson–Delaunay mosaics and related complexes experimentally","oa_version":"Submitted Version","author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert"},{"first_name":"Anton","last_name":"Nikitenko","full_name":"Nikitenko, Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"},{"id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","full_name":"Ölsböck, Katharina","last_name":"Ölsböck","first_name":"Katharina"},{"first_name":"Peter","last_name":"Synak","id":"331776E2-F248-11E8-B48F-1D18A9856A87","full_name":"Synak, Peter"}],"scopus_import":"1","day":"22","publication_status":"published","publication_identifier":{"isbn":["9783030434076"],"issn":["21932808"],"eissn":["21978549"]},"file_date_updated":"2020-10-08T08:56:14Z","intvolume":"        15","abstract":[{"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.","lang":"eng"}],"has_accepted_license":"1"},{"external_id":{"arxiv":["1705.08735"],"isi":["000551393100007"]},"year":"2020","isi":1,"date_published":"2020-02-13T00:00:00Z","ec_funded":1,"status":"public","publication":"Theory of Probability and its Applications","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183","name":"Alpha Shape Theory Extended"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","_id":"7554","date_updated":"2023-08-18T06:45:48Z","publisher":"SIAM","article_processing_charge":"No","doi":"10.1137/S0040585X97T989726","quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1705.08735","open_access":"1"}],"page":"595-614","department":[{"_id":"HeEd"}],"month":"02","arxiv":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","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.” <i>Theory of Probability and Its Applications</i>. SIAM, 2020. <a href=\"https://doi.org/10.1137/S0040585X97T989726\">https://doi.org/10.1137/S0040585X97T989726</a>.","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Weighted Poisson–Delaunay Mosaics.” <i>Theory of Probability and Its Applications</i>, vol. 64, no. 4, SIAM, 2020, pp. 595–614, doi:<a href=\"https://doi.org/10.1137/S0040585X97T989726\">10.1137/S0040585X97T989726</a>.","apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2020). Weighted Poisson–Delaunay mosaics. <i>Theory of Probability and Its Applications</i>. SIAM. <a href=\"https://doi.org/10.1137/S0040585X97T989726\">https://doi.org/10.1137/S0040585X97T989726</a>","ama":"Edelsbrunner H, Nikitenko A. Weighted Poisson–Delaunay mosaics. <i>Theory of Probability and its Applications</i>. 2020;64(4):595-614. doi:<a href=\"https://doi.org/10.1137/S0040585X97T989726\">10.1137/S0040585X97T989726</a>","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,” <i>Theory of Probability and its Applications</i>, vol. 64, no. 4. SIAM, pp. 595–614, 2020."},"issue":"4","language":[{"iso":"eng"}],"oa":1,"date_created":"2020-03-01T23:00:39Z","article_type":"original","volume":64,"oa_version":"Preprint","title":"Weighted Poisson–Delaunay mosaics","scopus_import":"1","day":"13","author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","last_name":"Nikitenko","first_name":"Anton","orcid":"0000-0002-0659-3201"}],"publication_status":"published","publication_identifier":{"issn":["0040585X"],"eissn":["10957219"]},"intvolume":"        64","abstract":[{"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 ${R}^n$.","lang":"eng"}]},{"has_accepted_license":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"intvolume":"        62","abstract":[{"text":"The order-k Voronoi tessellation of a locally finite set 𝑋⊆ℝ𝑛 decomposes ℝ𝑛 into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold.","lang":"eng"}],"publication_identifier":{"issn":["01795376"],"eissn":["14320444"]},"publication_status":"published","file_date_updated":"2020-07-14T12:47:10Z","author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"last_name":"Nikitenko","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","first_name":"Anton","orcid":"0000-0002-0659-3201"}],"day":"01","scopus_import":"1","oa_version":"Published Version","title":"Poisson–Delaunay Mosaics of Order k","volume":62,"article_type":"original","date_created":"2018-12-16T22:59:20Z","oa":1,"language":[{"iso":"eng"}],"issue":"4","citation":{"ista":"Edelsbrunner H, Nikitenko A. 2019. Poisson–Delaunay Mosaics of Order k. Discrete and Computational Geometry. 62(4), 865–878.","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Poisson–Delaunay Mosaics of Order K.” <i>Discrete and Computational Geometry</i>. Springer, 2019. <a href=\"https://doi.org/10.1007/s00454-018-0049-2\">https://doi.org/10.1007/s00454-018-0049-2</a>.","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Poisson–Delaunay Mosaics of Order K.” <i>Discrete and Computational Geometry</i>, vol. 62, no. 4, Springer, 2019, pp. 865–878, doi:<a href=\"https://doi.org/10.1007/s00454-018-0049-2\">10.1007/s00454-018-0049-2</a>.","apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2019). Poisson–Delaunay Mosaics of Order k. <i>Discrete and Computational Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/s00454-018-0049-2\">https://doi.org/10.1007/s00454-018-0049-2</a>","ama":"Edelsbrunner H, Nikitenko A. Poisson–Delaunay Mosaics of Order k. <i>Discrete and Computational Geometry</i>. 2019;62(4):865–878. doi:<a href=\"https://doi.org/10.1007/s00454-018-0049-2\">10.1007/s00454-018-0049-2</a>","short":"H. Edelsbrunner, A. Nikitenko, Discrete and Computational Geometry 62 (2019) 865–878.","ieee":"H. Edelsbrunner and A. Nikitenko, “Poisson–Delaunay Mosaics of Order k,” <i>Discrete and Computational Geometry</i>, vol. 62, no. 4. Springer, pp. 865–878, 2019."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","arxiv":1,"month":"12","department":[{"_id":"HeEd"}],"file":[{"relation":"main_file","checksum":"f9d00e166efaccb5a76bbcbb4dcea3b4","file_name":"2018_DiscreteCompGeometry_Edelsbrunner.pdf","content_type":"application/pdf","access_level":"open_access","file_id":"5932","file_size":599339,"date_created":"2019-02-06T10:10:46Z","date_updated":"2020-07-14T12:47:10Z","creator":"dernst"}],"page":"865–878","ddc":["516"],"quality_controlled":"1","doi":"10.1007/s00454-018-0049-2","article_processing_charge":"Yes (via OA deal)","publisher":"Springer","date_updated":"2023-09-07T12:07:12Z","_id":"5678","type":"journal_article","project":[{"call_identifier":"H2020","name":"Alpha Shape Theory Extended","grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"publication":"Discrete and Computational Geometry","status":"public","ec_funded":1,"date_published":"2019-12-01T00:00:00Z","isi":1,"year":"2019","external_id":{"isi":["000494042900008"],"arxiv":["1709.09380"]},"related_material":{"record":[{"id":"6287","status":"public","relation":"dissertation_contains"}]}},{"publication":"Annals of Applied Probability","status":"public","project":[{"name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"date_published":"2018-10-01T00:00:00Z","isi":1,"year":"2018","related_material":{"record":[{"id":"6287","relation":"dissertation_contains","status":"public"}]},"external_id":{"isi":["000442893500018"],"arxiv":["1705.02870"]},"publist_id":"7967","page":"3215 - 3238","main_file_link":[{"url":"https://arxiv.org/abs/1705.02870","open_access":"1"}],"quality_controlled":"1","article_processing_charge":"No","doi":"10.1214/18-AAP1389","publisher":"Institute of Mathematical Statistics","_id":"87","date_updated":"2023-09-15T12:10:35Z","type":"journal_article","oa":1,"language":[{"iso":"eng"}],"citation":{"apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2018). Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/18-AAP1389\">https://doi.org/10.1214/18-AAP1389</a>","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics.” <i>Annals of Applied Probability</i>, vol. 28, no. 5, Institute of Mathematical Statistics, 2018, pp. 3215–38, doi:<a href=\"https://doi.org/10.1214/18-AAP1389\">10.1214/18-AAP1389</a>.","ista":"Edelsbrunner H, Nikitenko A. 2018. Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics. Annals of Applied Probability. 28(5), 3215–3238.","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2018. <a href=\"https://doi.org/10.1214/18-AAP1389\">https://doi.org/10.1214/18-AAP1389</a>.","ieee":"H. Edelsbrunner and A. Nikitenko, “Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics,” <i>Annals of Applied Probability</i>, vol. 28, no. 5. Institute of Mathematical Statistics, pp. 3215–3238, 2018.","short":"H. Edelsbrunner, A. Nikitenko, Annals of Applied Probability 28 (2018) 3215–3238.","ama":"Edelsbrunner H, Nikitenko A. Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics. <i>Annals of Applied Probability</i>. 2018;28(5):3215-3238. doi:<a href=\"https://doi.org/10.1214/18-AAP1389\">10.1214/18-AAP1389</a>"},"issue":"5","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","month":"10","arxiv":1,"department":[{"_id":"HeEd"}],"abstract":[{"text":"Using the geodesic distance on the n-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. We find that the expectations are essentially the same as for the Poisson–Delaunay mosaic in n-dimensional Euclidean space. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in Rn+1, so we also get the expected number of faces of a random inscribed polytope. As proved in Antonelli et al. [Adv. in Appl. Probab. 9–12 (1977–1980)], an orthant section of the n-sphere is isometric to the standard n-simplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the n-dimensional Euclidean space. Our results are therefore relevant in information geometry and in population genetics.","lang":"eng"}],"intvolume":"        28","publication_status":"published","day":"01","scopus_import":"1","author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"last_name":"Nikitenko","full_name":"Nikitenko, Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0659-3201","first_name":"Anton"}],"title":"Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics","oa_version":"Preprint","volume":28,"date_created":"2018-12-11T11:44:33Z","article_type":"original"},{"isi":1,"year":"2017","external_id":{"isi":["000418056000005"]},"publist_id":"6182","publication":"Combinatorica","status":"public","project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","grant_number":"318493","call_identifier":"FP7"}],"ec_funded":1,"date_published":"2017-10-01T00:00:00Z","acknowledgement":"This research is partially supported by the Russian Government under the Mega Project 11.G34.31.0053, by the Toposys project FP7-ICT-318493-STREP, by ESF under the ACAT Research Network Programme, by RFBR grant 11-01-00735, and by NSF grants DMS-1101688, DMS-1400876.","article_processing_charge":"No","doi":"10.1007/s00493-016-3308-y","publisher":"Springer","_id":"1173","date_updated":"2023-09-20T11:23:53Z","type":"journal_article","page":"887 - 910","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1411.6337"}],"quality_controlled":"1","month":"10","department":[{"_id":"HeEd"}],"oa":1,"language":[{"iso":"eng"}],"citation":{"chicago":"Edelsbrunner, Herbert, Alexey Glazyrin, Oleg Musin, and Anton Nikitenko. “The Voronoi Functional Is Maximized by the Delaunay Triangulation in the Plane.” <i>Combinatorica</i>. Springer, 2017. <a href=\"https://doi.org/10.1007/s00493-016-3308-y\">https://doi.org/10.1007/s00493-016-3308-y</a>.","ista":"Edelsbrunner H, Glazyrin A, Musin O, Nikitenko A. 2017. The Voronoi functional is maximized by the Delaunay triangulation in the plane. Combinatorica. 37(5), 887–910.","apa":"Edelsbrunner, H., Glazyrin, A., Musin, O., &#38; Nikitenko, A. (2017). The Voronoi functional is maximized by the Delaunay triangulation in the plane. <i>Combinatorica</i>. Springer. <a href=\"https://doi.org/10.1007/s00493-016-3308-y\">https://doi.org/10.1007/s00493-016-3308-y</a>","mla":"Edelsbrunner, Herbert, et al. “The Voronoi Functional Is Maximized by the Delaunay Triangulation in the Plane.” <i>Combinatorica</i>, vol. 37, no. 5, Springer, 2017, pp. 887–910, doi:<a href=\"https://doi.org/10.1007/s00493-016-3308-y\">10.1007/s00493-016-3308-y</a>.","ama":"Edelsbrunner H, Glazyrin A, Musin O, Nikitenko A. The Voronoi functional is maximized by the Delaunay triangulation in the plane. <i>Combinatorica</i>. 2017;37(5):887-910. doi:<a href=\"https://doi.org/10.1007/s00493-016-3308-y\">10.1007/s00493-016-3308-y</a>","ieee":"H. Edelsbrunner, A. Glazyrin, O. Musin, and A. Nikitenko, “The Voronoi functional is maximized by the Delaunay triangulation in the plane,” <i>Combinatorica</i>, vol. 37, no. 5. Springer, pp. 887–910, 2017.","short":"H. Edelsbrunner, A. Glazyrin, O. Musin, A. Nikitenko, Combinatorica 37 (2017) 887–910."},"issue":"5","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","day":"01","scopus_import":"1","author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"last_name":"Glazyrin","full_name":"Glazyrin, Alexey","first_name":"Alexey"},{"first_name":"Oleg","full_name":"Musin, Oleg","last_name":"Musin"},{"id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","last_name":"Nikitenko","orcid":"0000-0002-0659-3201","first_name":"Anton"}],"title":"The Voronoi functional is maximized by the Delaunay triangulation in the plane","oa_version":"Submitted Version","volume":37,"date_created":"2018-12-11T11:50:32Z","intvolume":"        37","abstract":[{"text":"We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.","lang":"eng"}],"publication_identifier":{"issn":["02099683"]},"publication_status":"published"},{"publist_id":"6962","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"6287"}]},"external_id":{"arxiv":["1607.05915"]},"year":"2017","date_published":"2017-09-01T00:00:00Z","ec_funded":1,"publication":"Advances in Applied Probability","status":"public","project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Topological Complex Systems","grant_number":"318493"},{"call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","_id":"718","date_updated":"2023-09-07T12:07:12Z","publisher":"Cambridge University Press","doi":"10.1017/apr.2017.20","quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1607.05915","open_access":"1"}],"page":"745 - 767","department":[{"_id":"HeEd"}],"arxiv":1,"month":"09","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"ieee":"H. Edelsbrunner, A. Nikitenko, and M. Reitzner, “Expected sizes of poisson Delaunay mosaics and their discrete Morse functions,” <i>Advances in Applied Probability</i>, vol. 49, no. 3. Cambridge University Press, pp. 745–767, 2017.","short":"H. Edelsbrunner, A. Nikitenko, M. Reitzner, Advances in Applied Probability 49 (2017) 745–767.","ama":"Edelsbrunner H, Nikitenko A, Reitzner M. Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. <i>Advances in Applied Probability</i>. 2017;49(3):745-767. doi:<a href=\"https://doi.org/10.1017/apr.2017.20\">10.1017/apr.2017.20</a>","apa":"Edelsbrunner, H., Nikitenko, A., &#38; Reitzner, M. (2017). Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. <i>Advances in Applied Probability</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/apr.2017.20\">https://doi.org/10.1017/apr.2017.20</a>","mla":"Edelsbrunner, Herbert, et al. “Expected Sizes of Poisson Delaunay Mosaics and Their Discrete Morse Functions.” <i>Advances in Applied Probability</i>, vol. 49, no. 3, Cambridge University Press, 2017, pp. 745–67, doi:<a href=\"https://doi.org/10.1017/apr.2017.20\">10.1017/apr.2017.20</a>.","chicago":"Edelsbrunner, Herbert, Anton Nikitenko, and Matthias Reitzner. “Expected Sizes of Poisson Delaunay Mosaics and Their Discrete Morse Functions.” <i>Advances in Applied Probability</i>. Cambridge University Press, 2017. <a href=\"https://doi.org/10.1017/apr.2017.20\">https://doi.org/10.1017/apr.2017.20</a>.","ista":"Edelsbrunner H, Nikitenko A, Reitzner M. 2017. Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. Advances in Applied Probability. 49(3), 745–767."},"issue":"3","language":[{"iso":"eng"}],"oa":1,"date_created":"2018-12-11T11:48:07Z","volume":49,"title":"Expected sizes of poisson Delaunay mosaics and their discrete Morse functions","oa_version":"Preprint","day":"01","scopus_import":1,"author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"first_name":"Anton","orcid":"0000-0002-0659-3201","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","last_name":"Nikitenko"},{"first_name":"Matthias","full_name":"Reitzner, Matthias","last_name":"Reitzner"}],"publication_identifier":{"issn":["00018678"]},"publication_status":"published","abstract":[{"lang":"eng","text":"Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝ n , we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4."}],"intvolume":"        49"},{"publication_status":"published","publication_identifier":{"issn":["2663-337X"]},"file_date_updated":"2020-07-14T12:47:26Z","has_accepted_license":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"lang":"eng","text":"The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the Čech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's."}],"date_created":"2019-04-09T15:04:32Z","author":[{"orcid":"0000-0002-0659-3201","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","full_name":"Nikitenko, Anton","last_name":"Nikitenko"}],"day":"27","oa_version":"Published Version","title":"Discrete Morse theory for random complexes ","citation":{"chicago":"Nikitenko, Anton. “Discrete Morse Theory for Random Complexes .” Institute of Science and Technology Austria, 2017. <a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">https://doi.org/10.15479/AT:ISTA:th_873</a>.","ista":"Nikitenko A. 2017. Discrete Morse theory for random complexes . Institute of Science and Technology Austria.","apa":"Nikitenko, A. (2017). <i>Discrete Morse theory for random complexes </i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">https://doi.org/10.15479/AT:ISTA:th_873</a>","mla":"Nikitenko, Anton. <i>Discrete Morse Theory for Random Complexes </i>. Institute of Science and Technology Austria, 2017, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">10.15479/AT:ISTA:th_873</a>.","ama":"Nikitenko A. Discrete Morse theory for random complexes . 2017. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_873\">10.15479/AT:ISTA:th_873</a>","ieee":"A. Nikitenko, “Discrete Morse theory for random complexes ,” Institute of Science and Technology Austria, 2017.","short":"A. Nikitenko, Discrete Morse Theory for Random Complexes , Institute of Science and Technology Austria, 2017."},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa":1,"language":[{"iso":"eng"}],"pubrep_id":"873","department":[{"_id":"HeEd"}],"file":[{"content_type":"application/pdf","access_level":"open_access","file_name":"2017_Thesis_Nikitenko.pdf","checksum":"ece7e598a2f060b263c2febf7f3fe7f9","relation":"main_file","date_updated":"2020-07-14T12:47:26Z","creator":"dernst","date_created":"2019-04-09T14:54:51Z","file_size":2324870,"file_id":"6289"},{"file_name":"2017_Thesis_Nikitenko_source.zip","content_type":"application/zip","access_level":"closed","relation":"source_file","checksum":"99b7ad76e317efd447af60f91e29b49b","file_size":2863219,"date_created":"2019-04-09T14:54:51Z","creator":"dernst","date_updated":"2020-07-14T12:47:26Z","file_id":"6290"}],"supervisor":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"}],"month":"10","page":"86","ddc":["514","516","519"],"date_updated":"2023-09-15T12:10:34Z","_id":"6287","type":"dissertation","doi":"10.15479/AT:ISTA:th_873","alternative_title":["ISTA Thesis"],"article_processing_charge":"No","publisher":"Institute of Science and Technology Austria","date_published":"2017-10-27T00:00:00Z","degree_awarded":"PhD","status":"public","year":"2017","related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"718"},{"id":"5678","relation":"part_of_dissertation","status":"public"},{"id":"87","status":"public","relation":"part_of_dissertation"}]}},{"publist_id":"6111","year":"2016","acknowledgement":"We wish to thank Alexey Tarasov, Vladislav Volkov and Brittany Fasy for some useful comments and remarks, and especially Thom Sulanke for modifying surftri to suit our purposes. Oleg R. Musin was partially supported by the NSF Grant DMS-1400876 and by the RFBR Grant 15-01-99563. Anton V. Nikitenko was supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government Grant 11.G34.31.0026.","date_published":"2016-01-01T00:00:00Z","status":"public","publication":"Discrete & Computational Geometry","_id":"1222","date_updated":"2021-01-12T06:49:11Z","type":"journal_article","doi":"10.1007/s00454-015-9742-6","publisher":"Springer","main_file_link":[{"url":"https://arxiv.org/abs/1212.0649","open_access":"1"}],"quality_controlled":"1","page":"1 - 20","department":[{"_id":"HeEd"}],"month":"01","citation":{"short":"O. Musin, A. Nikitenko, Discrete &#38; Computational Geometry 55 (2016) 1–20.","ieee":"O. Musin and A. Nikitenko, “Optimal packings of congruent circles on a square flat torus,” <i>Discrete &#38; Computational Geometry</i>, vol. 55, no. 1. Springer, pp. 1–20, 2016.","ama":"Musin O, Nikitenko A. Optimal packings of congruent circles on a square flat torus. <i>Discrete &#38; Computational Geometry</i>. 2016;55(1):1-20. doi:<a href=\"https://doi.org/10.1007/s00454-015-9742-6\">10.1007/s00454-015-9742-6</a>","mla":"Musin, Oleg, and Anton Nikitenko. “Optimal Packings of Congruent Circles on a Square Flat Torus.” <i>Discrete &#38; Computational Geometry</i>, vol. 55, no. 1, Springer, 2016, pp. 1–20, doi:<a href=\"https://doi.org/10.1007/s00454-015-9742-6\">10.1007/s00454-015-9742-6</a>.","apa":"Musin, O., &#38; Nikitenko, A. (2016). Optimal packings of congruent circles on a square flat torus. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/s00454-015-9742-6\">https://doi.org/10.1007/s00454-015-9742-6</a>","ista":"Musin O, Nikitenko A. 2016. Optimal packings of congruent circles on a square flat torus. Discrete &#38; Computational Geometry. 55(1), 1–20.","chicago":"Musin, Oleg, and Anton Nikitenko. “Optimal Packings of Congruent Circles on a Square Flat Torus.” <i>Discrete &#38; Computational Geometry</i>. Springer, 2016. <a href=\"https://doi.org/10.1007/s00454-015-9742-6\">https://doi.org/10.1007/s00454-015-9742-6</a>."},"issue":"1","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa":1,"language":[{"iso":"eng"}],"volume":55,"date_created":"2018-12-11T11:50:48Z","scopus_import":1,"day":"01","author":[{"full_name":"Musin, Oleg","last_name":"Musin","first_name":"Oleg"},{"first_name":"Anton","full_name":"Nikitenko, Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","last_name":"Nikitenko"}],"title":"Optimal packings of congruent circles on a square flat torus","oa_version":"Preprint","publication_status":"published","abstract":[{"lang":"eng","text":"We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason—the problem of “super resolution of images.” We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs."}],"intvolume":"        55"}]