[{"publisher":"Springer","_id":"5678","author":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"orcid":"0000-0002-0659-3201","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","last_name":"Nikitenko","full_name":"Nikitenko, Anton"}],"article_processing_charge":"Yes (via OA deal)","department":[{"_id":"HeEd"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","year":"2019","day":"01","date_published":"2019-12-01T00:00:00Z","arxiv":1,"file":[{"content_type":"application/pdf","checksum":"f9d00e166efaccb5a76bbcbb4dcea3b4","file_name":"2018_DiscreteCompGeometry_Edelsbrunner.pdf","creator":"dernst","file_id":"5932","relation":"main_file","access_level":"open_access","date_updated":"2020-07-14T12:47:10Z","date_created":"2019-02-06T10:10:46Z","file_size":599339}],"license":"https://creativecommons.org/licenses/by/4.0/","status":"public","ec_funded":1,"file_date_updated":"2020-07-14T12:47:10Z","volume":62,"issue":"4","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"6287"}]},"publication_identifier":{"issn":["01795376"],"eissn":["14320444"]},"has_accepted_license":"1","type":"journal_article","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"quality_controlled":"1","citation":{"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>.","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>","ista":"Edelsbrunner H, Nikitenko A. 2019. Poisson–Delaunay Mosaics of Order k. Discrete and Computational Geometry. 62(4), 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.","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>","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>.","short":"H. Edelsbrunner, A. Nikitenko, Discrete and Computational Geometry 62 (2019) 865–878."},"article_type":"original","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)"},"scopus_import":"1","publication_status":"published","date_updated":"2026-04-08T14:19:30Z","oa_version":"Published Version","language":[{"iso":"eng"}],"doi":"10.1007/s00454-018-0049-2","title":"Poisson–Delaunay Mosaics of Order k","abstract":[{"lang":"eng","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."}],"external_id":{"isi":["000494042900008"],"arxiv":["1709.09380"]},"isi":1,"ddc":["516"],"month":"12","corr_author":"1","publication":"Discrete and Computational Geometry","page":"865–878","date_created":"2018-12-16T22:59:20Z","oa":1,"intvolume":"        62"},{"citation":{"ama":"Aichholzer O, Akitaya HA, Cheung KC, et al. Folding polyominoes with holes into a cube. In: <i>Proceedings of the 31st Canadian Conference on Computational Geometry</i>. Canadian Conference on Computational Geometry; 2019:164-170.","ista":"Aichholzer O, Akitaya HA, Cheung KC, Demaine ED, Demaine ML, Fekete SP, Kleist L, Kostitsyna I, Löffler M, Masárová Z, Mundilova K, Schmidt C. 2019. Folding polyominoes with holes into a cube. Proceedings of the 31st Canadian Conference on Computational Geometry. CCCG: Canadian Conference in Computational Geometry, 164–170.","chicago":"Aichholzer, Oswin, Hugo A Akitaya, Kenneth C Cheung, Erik D Demaine, Martin L Demaine, Sandor P Fekete, Linda Kleist, et al. “Folding Polyominoes with Holes into a Cube.” In <i>Proceedings of the 31st Canadian Conference on Computational Geometry</i>, 164–70. Canadian Conference on Computational Geometry, 2019.","short":"O. Aichholzer, H.A. Akitaya, K.C. Cheung, E.D. Demaine, M.L. Demaine, S.P. Fekete, L. Kleist, I. Kostitsyna, M. Löffler, Z. Masárová, K. Mundilova, C. Schmidt, in:, Proceedings of the 31st Canadian Conference on Computational Geometry, Canadian Conference on Computational Geometry, 2019, pp. 164–170.","apa":"Aichholzer, O., Akitaya, H. A., Cheung, K. C., Demaine, E. D., Demaine, M. L., Fekete, S. P., … Schmidt, C. (2019). Folding polyominoes with holes into a cube. In <i>Proceedings of the 31st Canadian Conference on Computational Geometry</i> (pp. 164–170). Edmonton, Canada: Canadian Conference on Computational Geometry.","ieee":"O. Aichholzer <i>et al.</i>, “Folding polyominoes with holes into a cube,” in <i>Proceedings of the 31st Canadian Conference on Computational Geometry</i>, Edmonton, Canada, 2019, pp. 164–170.","mla":"Aichholzer, Oswin, et al. “Folding Polyominoes with Holes into a Cube.” <i>Proceedings of the 31st Canadian Conference on Computational Geometry</i>, Canadian Conference on Computational Geometry, 2019, pp. 164–70."},"type":"conference","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://sites.ualberta.ca/~cccg2019/cccg2019_proceedings.pdf"}],"related_material":{"record":[{"relation":"extended_version","status":"public","id":"8317"}]},"status":"public","arxiv":1,"acknowledgement":"This research was performed in part at the 33rd Bellairs Winter Workshop on Computational  Geometry. We thank all other participants for a fruitful atmosphere.","conference":{"location":"Edmonton, Canada","end_date":"2019-08-10","start_date":"2019-08-08","name":"CCCG: Canadian Conference in Computational Geometry"},"department":[{"_id":"HeEd"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"01","year":"2019","date_published":"2019-08-01T00:00:00Z","publisher":"Canadian Conference on Computational Geometry","_id":"6989","author":[{"full_name":"Aichholzer, Oswin","last_name":"Aichholzer","first_name":"Oswin"},{"last_name":"Akitaya","full_name":"Akitaya, Hugo A","first_name":"Hugo A"},{"first_name":"Kenneth C","full_name":"Cheung, Kenneth C","last_name":"Cheung"},{"first_name":"Erik D","full_name":"Demaine, Erik D","last_name":"Demaine"},{"full_name":"Demaine, Martin L","last_name":"Demaine","first_name":"Martin L"},{"full_name":"Fekete, Sandor P","last_name":"Fekete","first_name":"Sandor P"},{"full_name":"Kleist, Linda","last_name":"Kleist","first_name":"Linda"},{"first_name":"Irina","last_name":"Kostitsyna","full_name":"Kostitsyna, Irina"},{"full_name":"Löffler, Maarten","last_name":"Löffler","first_name":"Maarten"},{"orcid":"0000-0002-6660-1322","id":"45CFE238-F248-11E8-B48F-1D18A9856A87","first_name":"Zuzana","last_name":"Masárová","full_name":"Masárová, Zuzana"},{"first_name":"Klara","last_name":"Mundilova","full_name":"Mundilova, Klara"},{"first_name":"Christiane","full_name":"Schmidt, Christiane","last_name":"Schmidt"}],"article_processing_charge":"No","oa":1,"publication":"Proceedings of the 31st Canadian Conference on Computational Geometry","page":"164-170","date_created":"2019-11-04T16:46:11Z","month":"08","abstract":[{"text":"When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with hole(s) to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special simple holes guarantee foldability. ","lang":"eng"}],"title":"Folding polyominoes with holes into a cube","external_id":{"arxiv":["1910.09917"]},"oa_version":"Published Version","language":[{"iso":"eng"}],"scopus_import":"1","date_updated":"2026-04-16T09:14:30Z","publication_status":"published"},{"intvolume":"        99","oa":1,"page":"35:1 - 35:13","date_created":"2018-12-11T11:45:05Z","ddc":["000"],"month":"06","title":"Smallest enclosing spheres and Chernoff points in Bregman geometry","abstract":[{"lang":"eng","text":"Smallest enclosing spheres of finite point sets are central to methods in topological data analysis. Focusing on Bregman divergences to measure dissimilarity, we prove bounds on the location of the center of a smallest enclosing sphere. These bounds depend on the range of radii for which Bregman balls are convex."}],"doi":"10.4230/LIPIcs.SoCG.2018.35","oa_version":"Published Version","language":[{"iso":"eng"}],"scopus_import":1,"date_updated":"2021-01-12T06:53:48Z","publication_status":"published","citation":{"mla":"Edelsbrunner, Herbert, et al. <i>Smallest Enclosing Spheres and Chernoff Points in Bregman Geometry</i>. Vol. 99, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, p. 35:1-35:13, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.35\">10.4230/LIPIcs.SoCG.2018.35</a>.","ama":"Edelsbrunner H, Virk Z, Wagner H. Smallest enclosing spheres and Chernoff points in Bregman geometry. In: Vol 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2018:35:1-35:13. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.35\">10.4230/LIPIcs.SoCG.2018.35</a>","ista":"Edelsbrunner H, Virk Z, Wagner H. 2018. Smallest enclosing spheres and Chernoff points in Bregman geometry. SoCG: Symposium on Computational Geometry, Leibniz International Proceedings in Information, LIPIcs, vol. 99, 35:1-35:13.","short":"H. Edelsbrunner, Z. Virk, H. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, p. 35:1-35:13.","chicago":"Edelsbrunner, Herbert, Ziga Virk, and Hubert Wagner. “Smallest Enclosing Spheres and Chernoff Points in Bregman Geometry,” 99:35:1-35:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.35\">https://doi.org/10.4230/LIPIcs.SoCG.2018.35</a>.","ieee":"H. Edelsbrunner, Z. Virk, and H. Wagner, “Smallest enclosing spheres and Chernoff points in Bregman geometry,” presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary, 2018, vol. 99, p. 35:1-35:13.","apa":"Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2018). Smallest enclosing spheres and Chernoff points in Bregman geometry (Vol. 99, p. 35:1-35:13). Presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.35\">https://doi.org/10.4230/LIPIcs.SoCG.2018.35</a>"},"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","quality_controlled":"1","project":[{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"has_accepted_license":"1","file":[{"access_level":"open_access","date_updated":"2020-07-14T12:45:20Z","date_created":"2018-12-17T16:31:31Z","file_size":489080,"checksum":"7509403803b3ac1aee94bbc2ad293d21","content_type":"application/pdf","file_name":"2018_LIPIcs_Edelsbrunner.pdf","creator":"dernst","file_id":"5724","relation":"main_file"}],"status":"public","volume":99,"file_date_updated":"2020-07-14T12:45:20Z","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","conference":{"end_date":"2018-06-14","location":"Budapest, Hungary","name":"SoCG: Symposium on Computational Geometry","start_date":"2018-06-11"},"publist_id":"7733","department":[{"_id":"HeEd"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"11","year":"2018","date_published":"2018-06-11T00:00:00Z","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","alternative_title":["Leibniz International Proceedings in Information, LIPIcs"],"_id":"188","author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Virk, Ziga","last_name":"Virk","first_name":"Ziga"},{"last_name":"Wagner","full_name":"Wagner, Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert"}]},{"oa":1,"isi":1,"month":"06","page":"51 - 65","publication":"Proceedings of the 2018 on Asia Conference on Computer and Communication Security","date_created":"2018-12-11T11:45:07Z","doi":"10.1145/3196494.3196534","external_id":{"isi":["000516620100005"]},"abstract":[{"lang":"eng","text":"We show attacks on five data-independent memory-hard functions (iMHF) that were submitted to the password hashing competition (PHC). Informally, an MHF is a function which cannot be evaluated on dedicated hardware, like ASICs, at significantly lower hardware and/or energy cost than evaluating a single instance on a standard single-core architecture. Data-independent means the memory access pattern of the function is independent of the input; this makes iMHFs harder to construct than data-dependent ones, but the latter can be attacked by various side-channel attacks. Following [Alwen-Blocki'16], we capture the evaluation of an iMHF as a directed acyclic graph (DAG). The cumulative parallel pebbling complexity of this DAG is a measure for the hardware cost of evaluating the iMHF on an ASIC. Ideally, one would like the complexity of a DAG underlying an iMHF to be as close to quadratic in the number of nodes of the graph as possible. Instead, we show that (the DAGs underlying) the following iMHFs are far from this bound: Rig.v2, TwoCats and Gambit each having an exponent no more than 1.75. Moreover, we show that the complexity of the iMHF modes of the PHC finalists Pomelo and Lyra2 have exponents at most 1.83 and 1.67 respectively. To show this we investigate a combinatorial property of each underlying DAG (called its depth-robustness. By establishing upper bounds on this property we are then able to apply the general technique of [Alwen-Block'16] for analyzing the hardware costs of an iMHF."}],"title":"On the memory hardness of data independent password hashing functions","date_updated":"2024-11-04T13:52:29Z","publication_status":"published","scopus_import":"1","oa_version":"Submitted Version","language":[{"iso":"eng"}],"type":"conference","project":[{"grant_number":"616160","call_identifier":"FP7","name":"Discrete Optimization in Computer Vision: Theory and Practice","_id":"25FBA906-B435-11E9-9278-68D0E5697425"},{"grant_number":"682815","call_identifier":"H2020","name":"Teaching Old Crypto New Tricks","_id":"258AA5B2-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","citation":{"chicago":"Alwen, Joel F, Peter Gazi, Chethan Kamath Hosdurg, Karen Klein, Georg F Osang, Krzysztof Z Pietrzak, Lenoid Reyzin, Michal Rolinek, and Michal Rybar. “On the Memory Hardness of Data Independent Password Hashing Functions.” In <i>Proceedings of the 2018 on Asia Conference on Computer and Communication Security</i>, 51–65. ACM, 2018. <a href=\"https://doi.org/10.1145/3196494.3196534\">https://doi.org/10.1145/3196494.3196534</a>.","short":"J.F. Alwen, P. Gazi, C. Kamath Hosdurg, K. Klein, G.F. Osang, K.Z. Pietrzak, L. Reyzin, M. Rolinek, M. Rybar, in:, Proceedings of the 2018 on Asia Conference on Computer and Communication Security, ACM, 2018, pp. 51–65.","apa":"Alwen, J. F., Gazi, P., Kamath Hosdurg, C., Klein, K., Osang, G. F., Pietrzak, K. Z., … Rybar, M. (2018). On the memory hardness of data independent password hashing functions. In <i>Proceedings of the 2018 on Asia Conference on Computer and Communication Security</i> (pp. 51–65). Incheon, Republic of Korea: ACM. <a href=\"https://doi.org/10.1145/3196494.3196534\">https://doi.org/10.1145/3196494.3196534</a>","ieee":"J. F. Alwen <i>et al.</i>, “On the memory hardness of data independent password hashing functions,” in <i>Proceedings of the 2018 on Asia Conference on Computer and Communication Security</i>, Incheon, Republic of Korea, 2018, pp. 51–65.","ama":"Alwen JF, Gazi P, Kamath Hosdurg C, et al. On the memory hardness of data independent password hashing functions. In: <i>Proceedings of the 2018 on Asia Conference on Computer and Communication Security</i>. ACM; 2018:51-65. doi:<a href=\"https://doi.org/10.1145/3196494.3196534\">10.1145/3196494.3196534</a>","ista":"Alwen JF, Gazi P, Kamath Hosdurg C, Klein K, Osang GF, Pietrzak KZ, Reyzin L, Rolinek M, Rybar M. 2018. On the memory hardness of data independent password hashing functions. Proceedings of the 2018 on Asia Conference on Computer and Communication Security. ASIACCS: Asia Conference on Computer and Communications Security , 51–65.","mla":"Alwen, Joel F., et al. “On the Memory Hardness of Data Independent Password Hashing Functions.” <i>Proceedings of the 2018 on Asia Conference on Computer and Communication Security</i>, ACM, 2018, pp. 51–65, doi:<a href=\"https://doi.org/10.1145/3196494.3196534\">10.1145/3196494.3196534</a>."},"main_file_link":[{"url":"https://eprint.iacr.org/2016/783","open_access":"1"}],"publist_id":"7723","conference":{"end_date":"2018-06-08","location":"Incheon, Republic of Korea","name":"ASIACCS: Asia Conference on Computer and Communications Security ","start_date":"2018-06-04"},"acknowledgement":"Leonid Reyzin was supported in part by IST Austria and by US NSF grants 1012910, 1012798, and 1422965; this research was performed while he was visiting IST Austria.","status":"public","ec_funded":1,"_id":"193","author":[{"id":"2A8DFA8C-F248-11E8-B48F-1D18A9856A87","first_name":"Joel F","full_name":"Alwen, Joel F","last_name":"Alwen"},{"first_name":"Peter","last_name":"Gazi","full_name":"Gazi, Peter"},{"first_name":"Chethan","id":"4BD3F30E-F248-11E8-B48F-1D18A9856A87","full_name":"Kamath Hosdurg, Chethan","last_name":"Kamath Hosdurg"},{"full_name":"Klein, Karen","last_name":"Klein","first_name":"Karen","id":"3E83A2F8-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-8882-5116","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","first_name":"Georg F","last_name":"Osang","full_name":"Osang, Georg F"},{"first_name":"Krzysztof Z","id":"3E04A7AA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9139-1654","full_name":"Pietrzak, Krzysztof Z","last_name":"Pietrzak"},{"first_name":"Lenoid","full_name":"Reyzin, Lenoid","last_name":"Reyzin"},{"id":"3CB3BC06-F248-11E8-B48F-1D18A9856A87","first_name":"Michal","full_name":"Rolinek, Michal","last_name":"Rolinek"},{"first_name":"Michal","id":"2B3E3DE8-F248-11E8-B48F-1D18A9856A87","last_name":"Rybar","full_name":"Rybar, Michal"}],"article_processing_charge":"No","publisher":"ACM","year":"2018","day":"01","date_published":"2018-06-01T00:00:00Z","department":[{"_id":"KrPi"},{"_id":"HeEd"},{"_id":"VlKo"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1"},{"title":"Long geodesics on convex surfaces","abstract":[{"lang":"eng","text":"The goal of this article is to introduce the reader to the theory of intrinsic geometry of convex surfaces. We illustrate the power of the tools by proving a theorem on convex surfaces containing an arbitrarily long closed simple geodesic. Let us remind ourselves that a curve in a surface is called geodesic if every sufficiently short arc of the curve is length minimizing; if, in addition, it has no self-intersections, we call it simple geodesic. A tetrahedron with equal opposite edges is called isosceles. The axiomatic method of Alexandrov geometry allows us to work with the metrics of convex surfaces directly, without approximating it first by a smooth or polyhedral metric. Such approximations destroy the closed geodesics on the surface; therefore it is difficult (if at all possible) to apply approximations in the proof of our theorem. On the other hand, a proof in the smooth or polyhedral case usually admits a translation into Alexandrov’s language; such translation makes the result more general. In fact, our proof resembles a translation of the proof given by Protasov. Note that the main theorem implies in particular that a smooth convex surface does not have arbitrarily long simple closed geodesics. However we do not know a proof of this corollary that is essentially simpler than the one presented below."}],"external_id":{"arxiv":["1702.05172"],"isi":["000444141200005"]},"doi":"10.1007/s00283-018-9795-5","language":[{"iso":"eng"}],"oa_version":"Preprint","scopus_import":"1","publication_status":"published","date_updated":"2023-09-13T08:49:16Z","intvolume":"        40","oa":1,"date_created":"2018-12-11T11:44:40Z","page":"26 - 31","publication":"Mathematical Intelligencer","month":"09","isi":1,"status":"public","volume":40,"arxiv":1,"publist_id":"7948","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","department":[{"_id":"HeEd"}],"date_published":"2018-09-01T00:00:00Z","year":"2018","day":"01","publisher":"Springer","article_processing_charge":"No","_id":"106","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan"},{"full_name":"Petrunin, Anton","last_name":"Petrunin","first_name":"Anton"}],"citation":{"ista":"Akopyan A, Petrunin A. 2018. Long geodesics on convex surfaces. Mathematical Intelligencer. 40(3), 26–31.","ama":"Akopyan A, Petrunin A. Long geodesics on convex surfaces. <i>Mathematical Intelligencer</i>. 2018;40(3):26-31. doi:<a href=\"https://doi.org/10.1007/s00283-018-9795-5\">10.1007/s00283-018-9795-5</a>","chicago":"Akopyan, Arseniy, and Anton Petrunin. “Long Geodesics on Convex Surfaces.” <i>Mathematical Intelligencer</i>. Springer, 2018. <a href=\"https://doi.org/10.1007/s00283-018-9795-5\">https://doi.org/10.1007/s00283-018-9795-5</a>.","short":"A. Akopyan, A. Petrunin, Mathematical Intelligencer 40 (2018) 26–31.","apa":"Akopyan, A., &#38; Petrunin, A. (2018). Long geodesics on convex surfaces. <i>Mathematical Intelligencer</i>. Springer. <a href=\"https://doi.org/10.1007/s00283-018-9795-5\">https://doi.org/10.1007/s00283-018-9795-5</a>","ieee":"A. Akopyan and A. Petrunin, “Long geodesics on convex surfaces,” <i>Mathematical Intelligencer</i>, vol. 40, no. 3. Springer, pp. 26–31, 2018.","mla":"Akopyan, Arseniy, and Anton Petrunin. “Long Geodesics on Convex Surfaces.” <i>Mathematical Intelligencer</i>, vol. 40, no. 3, Springer, 2018, pp. 26–31, doi:<a href=\"https://doi.org/10.1007/s00283-018-9795-5\">10.1007/s00283-018-9795-5</a>."},"quality_controlled":"1","type":"journal_article","main_file_link":[{"url":"https://arxiv.org/abs/1702.05172","open_access":"1"}],"issue":"3"},{"ec_funded":1,"status":"public","file":[{"file_size":708357,"date_created":"2019-02-12T06:47:52Z","date_updated":"2020-07-14T12:46:38Z","access_level":"open_access","relation":"main_file","file_id":"5953","creator":"dernst","file_name":"2018_Edelsbrunner.pdf","content_type":"application/pdf","checksum":"1c8d58cd489a66cd3e2064c1141c8c5e"}],"file_date_updated":"2020-07-14T12:46:38Z","volume":68,"publist_id":"7289","department":[{"_id":"HeEd"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_published":"2018-03-01T00:00:00Z","day":"01","year":"2018","publisher":"Elsevier","_id":"530","article_processing_charge":"No","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"first_name":"Mabel","id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","last_name":"Iglesias Ham","full_name":"Iglesias Ham, Mabel"}],"citation":{"mla":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls I: Inclusion–Exclusion.” <i>Computational Geometry: Theory and Applications</i>, vol. 68, Elsevier, 2018, pp. 119–33, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">10.1016/j.comgeo.2017.06.014</a>.","short":"H. Edelsbrunner, M. Iglesias Ham, Computational Geometry: Theory and Applications 68 (2018) 119–133.","chicago":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls I: Inclusion–Exclusion.” <i>Computational Geometry: Theory and Applications</i>. Elsevier, 2018. <a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">https://doi.org/10.1016/j.comgeo.2017.06.014</a>.","apa":"Edelsbrunner, H., &#38; Iglesias Ham, M. (2018). Multiple covers with balls I: Inclusion–exclusion. <i>Computational Geometry: Theory and Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">https://doi.org/10.1016/j.comgeo.2017.06.014</a>","ieee":"H. Edelsbrunner and M. Iglesias Ham, “Multiple covers with balls I: Inclusion–exclusion,” <i>Computational Geometry: Theory and Applications</i>, vol. 68. Elsevier, pp. 119–133, 2018.","ama":"Edelsbrunner H, Iglesias Ham M. Multiple covers with balls I: Inclusion–exclusion. <i>Computational Geometry: Theory and Applications</i>. 2018;68:119-133. doi:<a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">10.1016/j.comgeo.2017.06.014</a>","ista":"Edelsbrunner H, Iglesias Ham M. 2018. Multiple covers with balls I: Inclusion–exclusion. Computational Geometry: Theory and Applications. 68, 119–133."},"quality_controlled":"1","project":[{"call_identifier":"FP7","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493"}],"type":"journal_article","has_accepted_license":"1","title":"Multiple covers with balls I: Inclusion–exclusion","abstract":[{"lang":"eng","text":"Inclusion–exclusion is an effective method for computing the volume of a union of measurable sets. We extend it to multiple coverings, proving short inclusion–exclusion formulas for the subset of Rn covered by at least k balls in a finite set. We implement two of the formulas in dimension n=3 and report on results obtained with our software."}],"external_id":{"isi":["000415778300010"]},"doi":"10.1016/j.comgeo.2017.06.014","language":[{"iso":"eng"}],"oa_version":"Preprint","scopus_import":"1","date_updated":"2025-04-15T08:37:54Z","publication_status":"published","intvolume":"        68","oa":1,"date_created":"2018-12-11T11:46:59Z","page":"119 - 133","publication":"Computational Geometry: Theory and Applications","month":"03","ddc":["000"],"isi":1,"corr_author":"1"},{"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1604.00960"}],"issue":"3","project":[{"grant_number":"291734","call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme"}],"quality_controlled":"1","type":"journal_article","citation":{"mla":"Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal Arrangements.” <i>SIAM Journal on Discrete Mathematics</i>, vol. 32, no. 3, Society for Industrial and Applied Mathematics , 2018, pp. 2242–57, doi:<a href=\"https://doi.org/10.1137/16M110407X\">10.1137/16M110407X</a>.","ama":"Akopyan A, Segal Halevi E. Counting blanks in polygonal arrangements. <i>SIAM Journal on Discrete Mathematics</i>. 2018;32(3):2242-2257. doi:<a href=\"https://doi.org/10.1137/16M110407X\">10.1137/16M110407X</a>","ista":"Akopyan A, Segal Halevi E. 2018. Counting blanks in polygonal arrangements. SIAM Journal on Discrete Mathematics. 32(3), 2242–2257.","ieee":"A. Akopyan and E. Segal Halevi, “Counting blanks in polygonal arrangements,” <i>SIAM Journal on Discrete Mathematics</i>, vol. 32, no. 3. Society for Industrial and Applied Mathematics , pp. 2242–2257, 2018.","apa":"Akopyan, A., &#38; Segal Halevi, E. (2018). Counting blanks in polygonal arrangements. <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied Mathematics . <a href=\"https://doi.org/10.1137/16M110407X\">https://doi.org/10.1137/16M110407X</a>","short":"A. Akopyan, E. Segal Halevi, SIAM Journal on Discrete Mathematics 32 (2018) 2242–2257.","chicago":"Akopyan, Arseniy, and Erel Segal Halevi. “Counting Blanks in Polygonal Arrangements.” <i>SIAM Journal on Discrete Mathematics</i>. Society for Industrial and Applied Mathematics , 2018. <a href=\"https://doi.org/10.1137/16M110407X\">https://doi.org/10.1137/16M110407X</a>."},"publisher":"Society for Industrial and Applied Mathematics ","_id":"58","article_processing_charge":"No","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan"},{"full_name":"Segal Halevi, Erel","last_name":"Segal Halevi","first_name":"Erel"}],"department":[{"_id":"HeEd"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_published":"2018-09-06T00:00:00Z","day":"06","year":"2018","arxiv":1,"publist_id":"7996","ec_funded":1,"status":"public","volume":32,"month":"09","isi":1,"date_created":"2018-12-11T11:44:24Z","page":"2242 - 2257","publication":"SIAM Journal on Discrete Mathematics","oa":1,"intvolume":"        32","scopus_import":"1","publication_status":"published","date_updated":"2025-04-15T06:50:24Z","language":[{"iso":"eng"}],"oa_version":"Preprint","doi":"10.1137/16M110407X","abstract":[{"lang":"eng","text":"Inside a two-dimensional region (``cake&quot;&quot;), there are m nonoverlapping tiles of a certain kind (``toppings&quot;&quot;). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,&quot;&quot; such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks."}],"title":"Counting blanks in polygonal arrangements","external_id":{"isi":["000450810500036"],"arxiv":["1604.00960"]}},{"publication":"Geometriae Dedicata","page":"55 - 64","date_created":"2018-12-11T11:47:57Z","isi":1,"ddc":["510"],"month":"06","corr_author":"1","intvolume":"       194","oa":1,"oa_version":"Published Version","language":[{"iso":"eng"}],"scopus_import":"1","date_updated":"2025-04-15T06:50:29Z","publication_status":"published","title":"3-Webs generated by confocal conics and circles","abstract":[{"text":"We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them.","lang":"eng"}],"external_id":{"isi":["000431418800004"]},"doi":"10.1007/s10711-017-0265-6","has_accepted_license":"1","issue":"1","citation":{"ama":"Akopyan A. 3-Webs generated by confocal conics and circles. <i>Geometriae Dedicata</i>. 2018;194(1):55-64. doi:<a href=\"https://doi.org/10.1007/s10711-017-0265-6\">10.1007/s10711-017-0265-6</a>","ista":"Akopyan A. 2018. 3-Webs generated by confocal conics and circles. Geometriae Dedicata. 194(1), 55–64.","ieee":"A. Akopyan, “3-Webs generated by confocal conics and circles,” <i>Geometriae Dedicata</i>, vol. 194, no. 1. Springer, pp. 55–64, 2018.","apa":"Akopyan, A. (2018). 3-Webs generated by confocal conics and circles. <i>Geometriae Dedicata</i>. Springer. <a href=\"https://doi.org/10.1007/s10711-017-0265-6\">https://doi.org/10.1007/s10711-017-0265-6</a>","chicago":"Akopyan, Arseniy. “3-Webs Generated by Confocal Conics and Circles.” <i>Geometriae Dedicata</i>. Springer, 2018. <a href=\"https://doi.org/10.1007/s10711-017-0265-6\">https://doi.org/10.1007/s10711-017-0265-6</a>.","short":"A. Akopyan, Geometriae Dedicata 194 (2018) 55–64.","mla":"Akopyan, Arseniy. “3-Webs Generated by Confocal Conics and Circles.” <i>Geometriae Dedicata</i>, vol. 194, no. 1, Springer, 2018, pp. 55–64, doi:<a href=\"https://doi.org/10.1007/s10711-017-0265-6\">10.1007/s10711-017-0265-6</a>."},"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","project":[{"grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7"}],"quality_controlled":"1","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","department":[{"_id":"HeEd"}],"day":"01","year":"2018","date_published":"2018-06-01T00:00:00Z","publisher":"Springer","article_processing_charge":"Yes (via OA deal)","_id":"692","author":[{"orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","last_name":"Akopyan","full_name":"Akopyan, Arseniy"}],"file":[{"date_updated":"2020-07-14T12:47:44Z","access_level":"open_access","date_created":"2020-01-03T11:35:08Z","file_size":1140860,"creator":"kschuh","file_name":"2018_Springer_Akopyan.pdf","content_type":"application/pdf","checksum":"1febcfc1266486053a069e3425ea3713","relation":"main_file","file_id":"7222"}],"ec_funded":1,"status":"public","file_date_updated":"2020-07-14T12:47:44Z","volume":194,"publist_id":"7014"},{"month":"04","isi":1,"corr_author":"1","date_created":"2018-12-11T11:46:19Z","publication":"Comptes Rendus Mathematique","page":"412-414","oa":1,"intvolume":"       356","scopus_import":"1","publication_status":"published","date_updated":"2025-07-10T11:52:35Z","language":[{"iso":"eng"}],"oa_version":"Preprint","doi":"10.1016/j.crma.2018.03.005","title":"On the number of non-hexagons in a planar tiling","abstract":[{"text":"We give a simple proof of T. Stehling's result [4], whereby in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except a finite number are hexagons.","lang":"eng"}],"external_id":{"isi":["000430402700009"],"arxiv":["1805.01652"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1805.01652"}],"issue":"4","publication_identifier":{"issn":["1631-073X"]},"quality_controlled":"1","type":"journal_article","citation":{"apa":"Akopyan, A. (2018). On the number of non-hexagons in a planar tiling. <i>Comptes Rendus Mathematique</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.crma.2018.03.005\">https://doi.org/10.1016/j.crma.2018.03.005</a>","ieee":"A. Akopyan, “On the number of non-hexagons in a planar tiling,” <i>Comptes Rendus Mathematique</i>, vol. 356, no. 4. Elsevier, pp. 412–414, 2018.","chicago":"Akopyan, Arseniy. “On the Number of Non-Hexagons in a Planar Tiling.” <i>Comptes Rendus Mathematique</i>. Elsevier, 2018. <a href=\"https://doi.org/10.1016/j.crma.2018.03.005\">https://doi.org/10.1016/j.crma.2018.03.005</a>.","short":"A. Akopyan, Comptes Rendus Mathematique 356 (2018) 412–414.","ama":"Akopyan A. On the number of non-hexagons in a planar tiling. <i>Comptes Rendus Mathematique</i>. 2018;356(4):412-414. doi:<a href=\"https://doi.org/10.1016/j.crma.2018.03.005\">10.1016/j.crma.2018.03.005</a>","ista":"Akopyan A. 2018. On the number of non-hexagons in a planar tiling. Comptes Rendus Mathematique. 356(4), 412–414.","mla":"Akopyan, Arseniy. “On the Number of Non-Hexagons in a Planar Tiling.” <i>Comptes Rendus Mathematique</i>, vol. 356, no. 4, Elsevier, 2018, pp. 412–14, doi:<a href=\"https://doi.org/10.1016/j.crma.2018.03.005\">10.1016/j.crma.2018.03.005</a>."},"article_type":"original","publisher":"Elsevier","author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan"}],"_id":"409","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"HeEd"}],"date_published":"2018-04-01T00:00:00Z","day":"01","year":"2018","arxiv":1,"publist_id":"7420","status":"public","volume":356},{"status":"public","ec_funded":1,"volume":370,"arxiv":1,"acknowledgement":"DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”; People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) REA grant agreement n◦[291734]","publist_id":"7363","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"HeEd"}],"day":"01","year":"2018","date_published":"2018-04-01T00:00:00Z","publisher":"American Mathematical Society","author":[{"full_name":"Akopyan, Arseniy","last_name":"Akopyan","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X"},{"last_name":"Bobenko","full_name":"Bobenko, Alexander","first_name":"Alexander"}],"_id":"458","article_processing_charge":"No","citation":{"mla":"Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” <i>Transactions of the American Mathematical Society</i>, vol. 370, no. 4, American Mathematical Society, 2018, pp. 2825–54, doi:<a href=\"https://doi.org/10.1090/tran/7292\">10.1090/tran/7292</a>.","ista":"Akopyan A, Bobenko A. 2018. Incircular nets and confocal conics. Transactions of the American Mathematical Society. 370(4), 2825–2854.","ama":"Akopyan A, Bobenko A. Incircular nets and confocal conics. <i>Transactions of the American Mathematical Society</i>. 2018;370(4):2825-2854. doi:<a href=\"https://doi.org/10.1090/tran/7292\">10.1090/tran/7292</a>","ieee":"A. Akopyan and A. Bobenko, “Incircular nets and confocal conics,” <i>Transactions of the American Mathematical Society</i>, vol. 370, no. 4. American Mathematical Society, pp. 2825–2854, 2018.","apa":"Akopyan, A., &#38; Bobenko, A. (2018). Incircular nets and confocal conics. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/7292\">https://doi.org/10.1090/tran/7292</a>","chicago":"Akopyan, Arseniy, and Alexander Bobenko. “Incircular Nets and Confocal Conics.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2018. <a href=\"https://doi.org/10.1090/tran/7292\">https://doi.org/10.1090/tran/7292</a>.","short":"A. Akopyan, A. Bobenko, Transactions of the American Mathematical Society 370 (2018) 2825–2854."},"type":"journal_article","quality_controlled":"1","project":[{"name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"291734"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1602.04637"}],"issue":"4","abstract":[{"lang":"eng","text":"We consider congruences of straight lines in a plane with the combinatorics of the square grid, with all elementary quadrilaterals possessing an incircle. It is shown that all the vertices of such nets (we call them incircular or IC-nets) lie on confocal conics. Our main new results are on checkerboard IC-nets in the plane. These are congruences of straight lines in the plane with the combinatorics of the square grid, combinatorially colored as a checkerboard, such that all black coordinate quadrilaterals possess inscribed circles. We show how this larger class of IC-nets appears quite naturally in Laguerre geometry of oriented planes and spheres and leads to new remarkable incidence theorems. Most of our results are valid in hyperbolic and spherical geometries as well. We present also generalizations in spaces of higher dimension, called checkerboard IS-nets. The construction of these nets is based on a new 9 inspheres incidence theorem."}],"title":"Incircular nets and confocal conics","external_id":{"isi":["000423197800019"],"arxiv":["1602.04637"]},"doi":"10.1090/tran/7292","oa_version":"Preprint","language":[{"iso":"eng"}],"scopus_import":"1","date_updated":"2025-06-04T08:06:10Z","publication_status":"published","intvolume":"       370","oa":1,"page":"2825 - 2854","publication":"Transactions of the American Mathematical Society","date_created":"2018-12-11T11:46:35Z","isi":1,"month":"04","corr_author":"1"},{"citation":{"mla":"Edelsbrunner, Herbert, and Georg F. Osang. <i>The Multi-Cover Persistence of Euclidean Balls</i>. Vol. 99, 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.34\">10.4230/LIPIcs.SoCG.2018.34</a>.","ista":"Edelsbrunner H, Osang GF. 2018. The multi-cover persistence of Euclidean balls. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 99, 34.","ama":"Edelsbrunner H, Osang GF. The multi-cover persistence of Euclidean balls. In: Vol 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2018. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.34\">10.4230/LIPIcs.SoCG.2018.34</a>","chicago":"Edelsbrunner, Herbert, and Georg F Osang. “The Multi-Cover Persistence of Euclidean Balls,” Vol. 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.34\">https://doi.org/10.4230/LIPIcs.SoCG.2018.34</a>.","short":"H. Edelsbrunner, G.F. Osang, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.","apa":"Edelsbrunner, H., &#38; Osang, G. F. (2018). The multi-cover persistence of Euclidean balls (Vol. 99). Presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.34\">https://doi.org/10.4230/LIPIcs.SoCG.2018.34</a>","ieee":"H. Edelsbrunner and G. F. Osang, “The multi-cover persistence of Euclidean balls,” presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary, 2018, vol. 99."},"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)"},"project":[{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"quality_controlled":"1","type":"conference","has_accepted_license":"1","article_number":"34","related_material":{"record":[{"id":"9317","status":"public","relation":"later_version"},{"id":"9056","status":"public","relation":"dissertation_contains"}]},"status":"public","file":[{"date_created":"2018-12-18T09:27:22Z","file_size":528018,"access_level":"open_access","date_updated":"2020-07-14T12:45:19Z","file_id":"5738","relation":"main_file","checksum":"d8c0533ad0018eb4ed1077475eb8fc18","file_name":"2018_LIPIcs_Edelsbrunner_Osang.pdf","content_type":"application/pdf","creator":"dernst"}],"file_date_updated":"2020-07-14T12:45:19Z","volume":99,"acknowledgement":"This work is 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).","conference":{"start_date":"2018-06-11","name":"SoCG: Symposium on Computational Geometry","end_date":"2018-06-14","location":"Budapest, Hungary"},"publist_id":"7732","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"HeEd"}],"date_published":"2018-06-11T00:00:00Z","year":"2018","day":"11","alternative_title":["LIPIcs"],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"first_name":"Georg F","id":"464B40D6-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8882-5116","full_name":"Osang, Georg F","last_name":"Osang"}],"_id":"187","intvolume":"        99","oa":1,"date_created":"2018-12-11T11:45:05Z","month":"06","ddc":["516"],"title":"The multi-cover persistence of Euclidean balls","abstract":[{"text":"Given a locally finite X ⊆ ℝd and a radius r ≥ 0, the k-fold cover of X and r consists of all points in ℝd that have k or more points of X within distance r. We consider two filtrations - one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k - and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in ℝd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers. ","lang":"eng"}],"doi":"10.4230/LIPIcs.SoCG.2018.34","language":[{"iso":"eng"}],"oa_version":"Published Version","scopus_import":1,"publication_status":"published","date_updated":"2026-04-08T07:01:29Z"},{"publication":"Forum of Mathematics, Sigma","date_created":"2019-04-30T06:09:57Z","corr_author":"1","isi":1,"ddc":["510"],"month":"05","intvolume":"         6","oa":1,"oa_version":"Published Version","language":[{"iso":"eng"}],"publication_status":"published","date_updated":"2026-04-08T07:25:54Z","external_id":{"arxiv":["1712.10205"],"isi":["000433915500001"]},"title":"Any cyclic quadrilateral can be inscribed in any closed convex smooth curve","abstract":[{"text":"We  prove  that  any  cyclic  quadrilateral  can  be  inscribed  in  any  closed  convex C1-curve.  The smoothness condition is not required if the quadrilateral is a rectangle.","lang":"eng"}],"doi":"10.1017/fms.2018.7","publication_identifier":{"issn":["2050-5094"]},"has_accepted_license":"1","related_material":{"record":[{"id":"8156","status":"public","relation":"dissertation_contains"}]},"article_number":"e7","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)"},"citation":{"chicago":"Akopyan, Arseniy, and Sergey Avvakumov. “Any Cyclic Quadrilateral Can Be Inscribed in Any Closed Convex Smooth Curve.” <i>Forum of Mathematics, Sigma</i>. Cambridge University Press, 2018. <a href=\"https://doi.org/10.1017/fms.2018.7\">https://doi.org/10.1017/fms.2018.7</a>.","short":"A. Akopyan, S. Avvakumov, Forum of Mathematics, Sigma 6 (2018).","apa":"Akopyan, A., &#38; Avvakumov, S. (2018). Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. <i>Forum of Mathematics, Sigma</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/fms.2018.7\">https://doi.org/10.1017/fms.2018.7</a>","ieee":"A. Akopyan and S. Avvakumov, “Any cyclic quadrilateral can be inscribed in any closed convex smooth curve,” <i>Forum of Mathematics, Sigma</i>, vol. 6. Cambridge University Press, 2018.","ista":"Akopyan A, Avvakumov S. 2018. Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. Forum of Mathematics, Sigma. 6, e7.","ama":"Akopyan A, Avvakumov S. Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. <i>Forum of Mathematics, Sigma</i>. 2018;6. doi:<a href=\"https://doi.org/10.1017/fms.2018.7\">10.1017/fms.2018.7</a>","mla":"Akopyan, Arseniy, and Sergey Avvakumov. “Any Cyclic Quadrilateral Can Be Inscribed in Any Closed Convex Smooth Curve.” <i>Forum of Mathematics, Sigma</i>, vol. 6, e7, Cambridge University Press, 2018, doi:<a href=\"https://doi.org/10.1017/fms.2018.7\">10.1017/fms.2018.7</a>."},"type":"journal_article","project":[{"call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117"}],"quality_controlled":"1","year":"2018","day":"31","date_published":"2018-05-31T00:00:00Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","department":[{"_id":"UlWa"},{"_id":"HeEd"},{"_id":"JaMa"}],"_id":"6355","author":[{"full_name":"Akopyan, Arseniy","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy","orcid":"0000-0002-2548-617X"},{"full_name":"Avvakumov, Sergey","last_name":"Avvakumov","first_name":"Sergey","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7840-5062"}],"article_processing_charge":"No","publisher":"Cambridge University Press","file_date_updated":"2020-07-14T12:47:28Z","volume":6,"file":[{"access_level":"open_access","date_updated":"2020-07-14T12:47:28Z","date_created":"2019-04-30T06:14:58Z","file_size":249246,"checksum":"5a71b24ba712a3eb2e46165a38fbc30a","file_name":"2018_ForumMahtematics_Akopyan.pdf","content_type":"application/pdf","creator":"dernst","file_id":"6356","relation":"main_file"}],"status":"public","ec_funded":1,"arxiv":1},{"publication_status":"published","_id":"75","date_updated":"2026-04-08T07:25:54Z","article_processing_charge":"No","author":[{"full_name":"Akopyan, Arseniy","last_name":"Akopyan","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X"},{"full_name":"Avvakumov, Sergey","last_name":"Avvakumov","first_name":"Sergey","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7840-5062"},{"last_name":"Karasev","full_name":"Karasev, Roman","first_name":"Roman"}],"publisher":"arXiv","date_published":"2018-09-13T00:00:00Z","language":[{"iso":"eng"}],"year":"2018","oa_version":"Preprint","day":"13","department":[{"_id":"HeEd"},{"_id":"JaMa"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","arxiv":1,"doi":"10.48550/arXiv.1804.03057","external_id":{"arxiv":["1804.03057"]},"abstract":[{"text":"We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer m≥2; this result was previously known for prime powers m=pk. We also give a higher-dimensional generalization.","lang":"eng"}],"ec_funded":1,"status":"public","title":"Convex fair partitions into arbitrary number of pieces","corr_author":"1","related_material":{"record":[{"id":"8156","relation":"dissertation_contains","status":"public"}]},"article_number":"1804.03057","month":"09","main_file_link":[{"url":"https://arxiv.org/abs/1804.03057","open_access":"1"}],"date_created":"2018-12-11T11:44:30Z","project":[{"call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117"}],"type":"preprint","oa":1,"citation":{"ista":"Akopyan A, Avvakumov S, Karasev R. 2018. Convex fair partitions into arbitrary number of pieces. 1804.03057.","ama":"Akopyan A, Avvakumov S, Karasev R. Convex fair partitions into arbitrary number of pieces. 2018. doi:<a href=\"https://doi.org/10.48550/arXiv.1804.03057\">10.48550/arXiv.1804.03057</a>","chicago":"Akopyan, Arseniy, Sergey Avvakumov, and Roman Karasev. “Convex Fair Partitions into Arbitrary Number of Pieces.” arXiv, 2018. <a href=\"https://doi.org/10.48550/arXiv.1804.03057\">https://doi.org/10.48550/arXiv.1804.03057</a>.","short":"A. Akopyan, S. Avvakumov, R. Karasev, (2018).","apa":"Akopyan, A., Avvakumov, S., &#38; Karasev, R. (2018). Convex fair partitions into arbitrary number of pieces. arXiv. <a href=\"https://doi.org/10.48550/arXiv.1804.03057\">https://doi.org/10.48550/arXiv.1804.03057</a>","ieee":"A. Akopyan, S. Avvakumov, and R. Karasev, “Convex fair partitions into arbitrary number of pieces.” arXiv, 2018.","mla":"Akopyan, Arseniy, et al. <i>Convex Fair Partitions into Arbitrary Number of Pieces</i>. 1804.03057, arXiv, 2018, doi:<a href=\"https://doi.org/10.48550/arXiv.1804.03057\">10.48550/arXiv.1804.03057</a>."}},{"year":"2018","day":"11","date_published":"2018-06-11T00:00:00Z","pubrep_id":"1026","department":[{"_id":"HeEd"}],"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","article_processing_charge":"No","_id":"201","author":[{"id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","first_name":"Mabel","last_name":"Iglesias Ham","full_name":"Iglesias Ham, Mabel"}],"publisher":"Institute of Science and Technology Austria","alternative_title":["ISTA Thesis"],"file_date_updated":"2020-07-14T12:45:24Z","file":[{"file_size":11827713,"date_created":"2019-02-05T07:43:31Z","access_level":"closed","date_updated":"2020-07-14T12:45:24Z","file_id":"5918","relation":"source_file","file_name":"IST-2018-1025-v2+5_ist-thesis-iglesias-11June2018(1).zip","content_type":"application/zip","checksum":"dd699303623e96d1478a6ae07210dd05","creator":"kschuh"},{"file_id":"5919","relation":"main_file","checksum":"ba163849a190d2b41d66fef0e4983294","content_type":"application/pdf","file_name":"IST-2018-1025-v2+4_ThesisIglesiasFinal11June2018.pdf","creator":"kschuh","file_size":4783846,"date_created":"2019-02-05T07:43:45Z","access_level":"open_access","date_updated":"2020-07-14T12:45:24Z"}],"status":"public","publist_id":"7712","publication_identifier":{"issn":["2663-337X"]},"has_accepted_license":"1","citation":{"ista":"Iglesias Ham M. 2018. Multiple covers with balls. Institute of Science and Technology Austria.","ama":"Iglesias Ham M. Multiple covers with balls. 2018. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">10.15479/AT:ISTA:th_1026</a>","ieee":"M. Iglesias Ham, “Multiple covers with balls,” Institute of Science and Technology Austria, 2018.","apa":"Iglesias Ham, M. (2018). <i>Multiple covers with balls</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">https://doi.org/10.15479/AT:ISTA:th_1026</a>","chicago":"Iglesias Ham, Mabel. “Multiple Covers with Balls.” Institute of Science and Technology Austria, 2018. <a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">https://doi.org/10.15479/AT:ISTA:th_1026</a>.","short":"M. Iglesias Ham, Multiple Covers with Balls, Institute of Science and Technology Austria, 2018.","mla":"Iglesias Ham, Mabel. <i>Multiple Covers with Balls</i>. Institute of Science and Technology Austria, 2018, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th_1026\">10.15479/AT:ISTA:th_1026</a>."},"type":"dissertation","oa_version":"Published Version","language":[{"iso":"eng"}],"date_updated":"2026-04-08T14:04:03Z","publication_status":"published","degree_awarded":"PhD","abstract":[{"text":"We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications.","lang":"eng"}],"title":"Multiple covers with balls","doi":"10.15479/AT:ISTA:th_1026","OA_place":"publisher","page":"171","date_created":"2018-12-11T11:45:10Z","corr_author":"1","supervisor":[{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"}],"ddc":["514","516"],"month":"06","oa":1},{"issue":"5","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1705.02870"}],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"6287"}]},"type":"journal_article","quality_controlled":"1","project":[{"grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","call_identifier":"FWF"}],"citation":{"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>.","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>","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.","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.","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>","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>.","short":"H. Edelsbrunner, A. Nikitenko, Annals of Applied Probability 28 (2018) 3215–3238."},"article_type":"original","publisher":"Institute of Mathematical Statistics","_id":"87","article_processing_charge":"No","author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"last_name":"Nikitenko","full_name":"Nikitenko, Anton","orcid":"0000-0002-0659-3201","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","department":[{"_id":"HeEd"}],"year":"2018","day":"01","date_published":"2018-10-01T00:00:00Z","arxiv":1,"publist_id":"7967","status":"public","volume":28,"isi":1,"month":"10","page":"3215 - 3238","publication":"Annals of Applied Probability","date_created":"2018-12-11T11:44:33Z","oa":1,"intvolume":"        28","scopus_import":"1","publication_status":"published","date_updated":"2026-04-08T14:19:30Z","oa_version":"Preprint","language":[{"iso":"eng"}],"doi":"10.1214/18-AAP1389","abstract":[{"lang":"eng","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."}],"title":"Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics","external_id":{"isi":["000442893500018"],"arxiv":["1705.02870"]}},{"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","department":[{"_id":"HeEd"}],"year":"2018","day":"29","date_published":"2018-03-29T00:00:00Z","publisher":"Society for Industrial and Applied Mathematics ","_id":"312","author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Iglesias Ham, Mabel","last_name":"Iglesias Ham","id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","first_name":"Mabel"}],"article_processing_charge":"No","status":"public","volume":32,"acknowledgement":"This work was partially supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics,” through grant I02979-N35 of the Austrian Science Fund (FWF).","publist_id":"7553","publication_identifier":{"issn":["0895-4801"]},"issue":"1","main_file_link":[{"url":"http://pdfs.semanticscholar.org/d2d5/6da00fbc674e6a8b1bb9d857167e54200dc6.pdf","open_access":"1"}],"citation":{"ama":"Edelsbrunner H, Iglesias Ham M. On the optimality of the FCC lattice for soft sphere packing. <i>SIAM J Discrete Math</i>. 2018;32(1):750-782. doi:<a href=\"https://doi.org/10.1137/16M1097201\">10.1137/16M1097201</a>","ista":"Edelsbrunner H, Iglesias Ham M. 2018. On the optimality of the FCC lattice for soft sphere packing. SIAM J Discrete Math. 32(1), 750–782.","apa":"Edelsbrunner, H., &#38; Iglesias Ham, M. (2018). On the optimality of the FCC lattice for soft sphere packing. <i>SIAM J Discrete Math</i>. Society for Industrial and Applied Mathematics . <a href=\"https://doi.org/10.1137/16M1097201\">https://doi.org/10.1137/16M1097201</a>","ieee":"H. Edelsbrunner and M. Iglesias Ham, “On the optimality of the FCC lattice for soft sphere packing,” <i>SIAM J Discrete Math</i>, vol. 32, no. 1. Society for Industrial and Applied Mathematics , pp. 750–782, 2018.","short":"H. Edelsbrunner, M. Iglesias Ham, SIAM J Discrete Math 32 (2018) 750–782.","chicago":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “On the Optimality of the FCC Lattice for Soft Sphere Packing.” <i>SIAM J Discrete Math</i>. Society for Industrial and Applied Mathematics , 2018. <a href=\"https://doi.org/10.1137/16M1097201\">https://doi.org/10.1137/16M1097201</a>.","mla":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “On the Optimality of the FCC Lattice for Soft Sphere Packing.” <i>SIAM J Discrete Math</i>, vol. 32, no. 1, Society for Industrial and Applied Mathematics , 2018, pp. 750–82, doi:<a href=\"https://doi.org/10.1137/16M1097201\">10.1137/16M1097201</a>."},"article_type":"original","type":"journal_article","project":[{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"quality_controlled":"1","oa_version":"Submitted Version","language":[{"iso":"eng"}],"scopus_import":"1","date_updated":"2026-04-16T09:53:02Z","publication_status":"published","abstract":[{"lang":"eng","text":"Motivated by biological questions, we study configurations of equal spheres that neither pack nor cover. Placing their centers on a lattice, we define the soft density of the configuration by penalizing multiple overlaps. Considering the 1-parameter family of diagonally distorted 3-dimensional integer lattices, we show that the soft density is maximized at the FCC lattice."}],"title":"On the optimality of the FCC lattice for soft sphere packing","external_id":{"isi":["000428958900038"]},"doi":"10.1137/16M1097201","publication":"SIAM J Discrete Math","page":"750 - 782","date_created":"2018-12-11T11:45:46Z","isi":1,"month":"03","intvolume":"        32","oa":1},{"month":"06","ddc":["516","000"],"isi":1,"corr_author":"1","date_created":"2018-12-11T11:49:57Z","page":"1001-1009","publication":"Discrete & Computational Geometry","oa":1,"intvolume":"        59","scopus_import":"1","publication_status":"published","date_updated":"2026-05-20T10:19:33Z","language":[{"iso":"eng"}],"oa_version":"Published Version","doi":"10.1007/s00454-017-9883-x","abstract":[{"text":"In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii r1, … , rn in the plane, it is always possible to cover them by a disk of radius R= ∑ ri, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body K⊂ Rd with homothety coefficients τ1, … , τn> 0 , it is always possible to cover them by a translate of d+12(∑τi)K, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.","lang":"eng"}],"title":"On the circle covering theorem by A.W. Goodman and R.E. Goodman","external_id":{"isi":["000432205500011"]},"issue":"4","has_accepted_license":"1","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"quality_controlled":"1","project":[{"call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734"}],"type":"journal_article","citation":{"mla":"Akopyan, Arseniy, et al. “On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman.” <i>Discrete &#38; Computational Geometry</i>, vol. 59, no. 4, Springer, 2018, pp. 1001–09, doi:<a href=\"https://doi.org/10.1007/s00454-017-9883-x\">10.1007/s00454-017-9883-x</a>.","ama":"Akopyan A, Balitskiy A, Grigorev M. On the circle covering theorem by A.W. Goodman and R.E. Goodman. <i>Discrete &#38; Computational Geometry</i>. 2018;59(4):1001-1009. doi:<a href=\"https://doi.org/10.1007/s00454-017-9883-x\">10.1007/s00454-017-9883-x</a>","ista":"Akopyan A, Balitskiy A, Grigorev M. 2018. On the circle covering theorem by A.W. Goodman and R.E. Goodman. Discrete &#38; Computational Geometry. 59(4), 1001–1009.","chicago":"Akopyan, Arseniy, Alexey Balitskiy, and Mikhail Grigorev. “On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman.” <i>Discrete &#38; Computational Geometry</i>. Springer, 2018. <a href=\"https://doi.org/10.1007/s00454-017-9883-x\">https://doi.org/10.1007/s00454-017-9883-x</a>.","short":"A. Akopyan, A. Balitskiy, M. Grigorev, Discrete &#38; Computational Geometry 59 (2018) 1001–1009.","ieee":"A. Akopyan, A. Balitskiy, and M. Grigorev, “On the circle covering theorem by A.W. Goodman and R.E. Goodman,” <i>Discrete &#38; Computational Geometry</i>, vol. 59, no. 4. Springer, pp. 1001–1009, 2018.","apa":"Akopyan, A., Balitskiy, A., &#38; Grigorev, M. (2018). On the circle covering theorem by A.W. Goodman and R.E. Goodman. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/s00454-017-9883-x\">https://doi.org/10.1007/s00454-017-9883-x</a>"},"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","publisher":"Springer","article_processing_charge":"Yes (via OA deal)","_id":"1064","author":[{"last_name":"Akopyan","full_name":"Akopyan, Arseniy","orcid":"0000-0002-2548-617X","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","first_name":"Arseniy"},{"last_name":"Balitskiy","full_name":"Balitskiy, Alexey","first_name":"Alexey"},{"first_name":"Mikhail","full_name":"Grigorev, Mikhail","last_name":"Grigorev"}],"department":[{"_id":"HeEd"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2018-06-01T00:00:00Z","day":"01","year":"2018","publist_id":"6324","status":"public","ec_funded":1,"file":[{"date_updated":"2019-01-18T09:27:36Z","access_level":"open_access","file_size":482518,"date_created":"2019-01-18T09:27:36Z","success":1,"creator":"dernst","file_name":"2018_DiscreteComp_Akopyan.pdf","content_type":"application/pdf","relation":"main_file","file_id":"5844"}],"volume":59,"file_date_updated":"2019-01-18T09:27:36Z"},{"doi":"10.1007/s00493-016-3308-y","external_id":{"arxiv":["1411.6337"],"isi":["000418056000005"]},"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"}],"title":"The Voronoi functional is maximized by the Delaunay triangulation in the plane","date_updated":"2025-06-04T08:44:44Z","publication_status":"published","scopus_import":"1","oa_version":"Submitted Version","language":[{"iso":"eng"}],"oa":1,"intvolume":"        37","isi":1,"month":"10","publication":"Combinatorica","page":"887 - 910","date_created":"2018-12-11T11:50:32Z","publist_id":"6182","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.","arxiv":1,"volume":37,"ec_funded":1,"status":"public","_id":"1173","article_processing_charge":"No","author":[{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"full_name":"Glazyrin, Alexey","last_name":"Glazyrin","first_name":"Alexey"},{"last_name":"Musin","full_name":"Musin, Oleg","first_name":"Oleg"},{"last_name":"Nikitenko","full_name":"Nikitenko, Anton","orcid":"0000-0002-0659-3201","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"}],"publisher":"Springer","year":"2017","day":"01","date_published":"2017-10-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"HeEd"}],"type":"journal_article","quality_controlled":"1","project":[{"call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","grant_number":"318493"}],"citation":{"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>","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>","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.","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>.","short":"H. Edelsbrunner, A. Glazyrin, O. Musin, A. Nikitenko, Combinatorica 37 (2017) 887–910.","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>."},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1411.6337"}],"issue":"5","publication_identifier":{"issn":["0209-9683"]}},{"oa":1,"intvolume":"       308","month":"02","isi":1,"date_created":"2018-12-11T11:50:34Z","publication":"Advances in Mathematics","page":"627 - 644","doi":"10.1016/j.aim.2016.12.026","external_id":{"arxiv":["1508.07594"],"isi":["000409292900015"]},"title":"Algebraic vertices of non-convex polyhedra","abstract":[{"lang":"eng","text":"In this article we define an algebraic vertex of a generalized polyhedron and show that the set of algebraic vertices is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope P is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of P. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier–Laplace transform. We show that a point v is an algebraic vertex of a generalized polyhedron P if and only if the tangent cone of P, at v, has non-zero Fourier–Laplace transform."}],"date_updated":"2025-06-04T08:45:48Z","publication_status":"published","scopus_import":"1","language":[{"iso":"eng"}],"oa_version":"Submitted Version","quality_controlled":"1","project":[{"grant_number":"291734","name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"type":"journal_article","citation":{"mla":"Akopyan, Arseniy, et al. “Algebraic Vertices of Non-Convex Polyhedra.” <i>Advances in Mathematics</i>, vol. 308, Academic Press, 2017, pp. 627–44, doi:<a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">10.1016/j.aim.2016.12.026</a>.","chicago":"Akopyan, Arseniy, Imre Bárány, and Sinai Robins. “Algebraic Vertices of Non-Convex Polyhedra.” <i>Advances in Mathematics</i>. Academic Press, 2017. <a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">https://doi.org/10.1016/j.aim.2016.12.026</a>.","short":"A. Akopyan, I. Bárány, S. Robins, Advances in Mathematics 308 (2017) 627–644.","ieee":"A. Akopyan, I. Bárány, and S. Robins, “Algebraic vertices of non-convex polyhedra,” <i>Advances in Mathematics</i>, vol. 308. Academic Press, pp. 627–644, 2017.","apa":"Akopyan, A., Bárány, I., &#38; Robins, S. (2017). Algebraic vertices of non-convex polyhedra. <i>Advances in Mathematics</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">https://doi.org/10.1016/j.aim.2016.12.026</a>","ama":"Akopyan A, Bárány I, Robins S. Algebraic vertices of non-convex polyhedra. <i>Advances in Mathematics</i>. 2017;308:627-644. doi:<a href=\"https://doi.org/10.1016/j.aim.2016.12.026\">10.1016/j.aim.2016.12.026</a>","ista":"Akopyan A, Bárány I, Robins S. 2017. Algebraic vertices of non-convex polyhedra. Advances in Mathematics. 308, 627–644."},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1508.07594"}],"publication_identifier":{"issn":["0001-8708"]},"publist_id":"6173","arxiv":1,"volume":308,"ec_funded":1,"status":"public","_id":"1180","article_processing_charge":"No","author":[{"orcid":"0000-0002-2548-617X","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","full_name":"Akopyan, Arseniy"},{"first_name":"Imre","full_name":"Bárány, Imre","last_name":"Bárány"},{"full_name":"Robins, Sinai","last_name":"Robins","first_name":"Sinai"}],"publisher":"Academic Press","date_published":"2017-02-21T00:00:00Z","day":"21","year":"2017","department":[{"_id":"HeEd"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1016/j.jsc.2016.03.008"}],"related_material":{"record":[{"status":"public","relation":"earlier_version","id":"10894"}]},"publication_identifier":{"issn":[" 0747-7171"]},"quality_controlled":"1","project":[{"grant_number":"318493","call_identifier":"FP7","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","citation":{"mla":"Bauer, Ulrich, et al. “Phat - Persistent Homology Algorithms Toolbox.” <i>Journal of Symbolic Computation</i>, vol. 78, Academic Press, 2017, pp. 76–90, doi:<a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">10.1016/j.jsc.2016.03.008</a>.","ama":"Bauer U, Kerber M, Reininghaus J, Wagner H. Phat - Persistent homology algorithms toolbox. <i>Journal of Symbolic Computation</i>. 2017;78:76-90. doi:<a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">10.1016/j.jsc.2016.03.008</a>","ista":"Bauer U, Kerber M, Reininghaus J, Wagner H. 2017. Phat - Persistent homology algorithms toolbox. Journal of Symbolic Computation. 78, 76–90.","ieee":"U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner, “Phat - Persistent homology algorithms toolbox,” <i>Journal of Symbolic Computation</i>, vol. 78. Academic Press, pp. 76–90, 2017.","apa":"Bauer, U., Kerber, M., Reininghaus, J., &#38; Wagner, H. (2017). Phat - Persistent homology algorithms toolbox. <i>Journal of Symbolic Computation</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">https://doi.org/10.1016/j.jsc.2016.03.008</a>","chicago":"Bauer, Ulrich, Michael Kerber, Jan Reininghaus, and Hubert Wagner. “Phat - Persistent Homology Algorithms Toolbox.” <i>Journal of Symbolic Computation</i>. Academic Press, 2017. <a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">https://doi.org/10.1016/j.jsc.2016.03.008</a>.","short":"U. Bauer, M. Kerber, J. Reininghaus, H. Wagner, Journal of Symbolic Computation 78 (2017) 76–90."},"OA_type":"free access","article_type":"original","publisher":"Academic Press","_id":"1433","author":[{"last_name":"Bauer","full_name":"Bauer, Ulrich","first_name":"Ulrich"},{"full_name":"Kerber, Michael","last_name":"Kerber","first_name":"Michael"},{"full_name":"Reininghaus, Jan","last_name":"Reininghaus","first_name":"Jan"},{"first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Hubert","last_name":"Wagner"}],"article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"HeEd"}],"date_published":"2017-01-01T00:00:00Z","year":"2017","day":"01","acknowledgement":"Michael Kerber acknowledges support by the Max Planck Center for Visual Computing and Communications (FKZ-01IMC01 and FKZ-01IM10001). Ulrich Bauer, Jan Reininghaus, and Hubert Wagner acknowledge support by the EU Project TOPOSYS (FP7-ICT-318493-STREP).","publist_id":"5765","ec_funded":1,"status":"public","volume":78,"month":"01","isi":1,"corr_author":"1","date_created":"2018-12-11T11:51:59Z","page":"76 - 90","publication":"Journal of Symbolic Computation","oa":1,"intvolume":"        78","scopus_import":"1","publication_status":"published","date_updated":"2025-10-01T07:39:51Z","language":[{"iso":"eng"}],"oa_version":"Published Version","doi":"10.1016/j.jsc.2016.03.008","abstract":[{"text":"Phat is an open-source C. ++ library for the computation of persistent homology by matrix reduction, targeted towards developers of software for topological data analysis. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. We provide numerous different reduction strategies as well as data types to store and manipulate the boundary matrix. We compare the different combinations through extensive experimental evaluation and identify optimization techniques that work well in practical situations. We also compare our software with various other publicly available libraries for persistent homology.","lang":"eng"}],"title":"Phat - Persistent homology algorithms toolbox","external_id":{"isi":["000384396000005"]}}]