--- OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '20456' abstract: - lang: eng text: Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with n=#A, and the coloring is random, then the chromatic Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets and Poisson point processes in Rd, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications. acknowledgement: The fourth author thanks Boris Aronov for insightful discussions on the size of the overlay of Voronoi tessellations. Open access funding provided by Institute of Science and Technology (IST Austria). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. 2026;75:24-47. doi:10.1007/s00454-025-00778-7 apa: Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (2026). On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00778-7 chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.” Discrete and Computational Geometry. Springer Nature, 2026. https://doi.org/10.1007/s00454-025-00778-7. ieee: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “On the size of chromatic Delaunay mosaics,” Discrete and Computational Geometry, vol. 75. Springer Nature, pp. 24–47, 2026. ista: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2026. On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry. 75, 24–47. mla: Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” Discrete and Computational Geometry, vol. 75, Springer Nature, 2026, pp. 24–47, doi:10.1007/s00454-025-00778-7. short: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 75 (2026) 24–47. corr_author: '1' date_created: 2025-10-12T22:01:26Z date_published: 2026-01-01T00:00:00Z date_updated: 2026-01-05T13:21:56Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-025-00778-7 ec_funded: 1 external_id: arxiv: - '2212.03121' isi: - '001584166900001' file: - access_level: open_access checksum: 0addb5c1b78142f9fb453bfa04695400 content_type: application/pdf creator: dernst date_created: 2026-01-05T13:21:20Z date_updated: 2026-01-05T13:21:20Z file_id: '20952' file_name: 2026_DiscreteCompGeom_Biswas.pdf file_size: 570922 relation: main_file success: 1 file_date_updated: 2026-01-05T13:21:20Z has_accepted_license: '1' intvolume: ' 75' isi: 1 language: - iso: eng month: '01' oa: 1 oa_version: Published Version page: 24-47 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Discrete and Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '15090' relation: earlier_version status: public scopus_import: '1' status: public title: On the size of chromatic Delaunay mosaics tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 75 year: '2026' ... --- OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '21407' abstract: - lang: eng text: "This note proves that only a linear number of holes in a Cech complex of n points in R^d\r\ncan persist over an interval of constant length. Specifically, for any fixed dimension p <\r\nd and fixed ε > 0, the number of p-dimensional holes in the ˇ Cech complex at radius 1\r\nthat persist to radius 1+ε is bounded above by a constant times n,where n is the number\r\nof points. The proof uses a packing argument supported by relating theCˇ ech complexes\r\nwith corresponding snap complexes over the cells in a partition of space. The argument\r\nis self-contained and elementary, relying on geometric and combinatorial constructions\r\nrather than on the existing theory of sparse approximations or interleavings. The bound\r\nalso applies to Alpha complexes and Vietoris–Rips complexes. While our result can be\r\ninferred from prior work on sparse filtrations, to our knowledge, no explicit statement\r\nor direct proof of this bound appears in the literature." acknowledgement: The authors would like to thank Michael Lesnick and Primoz Skraba for their helpful comments regarding sparse approximations of filtrations. We are also grateful to the anonymous referees for their careful reading and constructive suggestions. The three authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35, the U.S. National Science Foundation (NSF-DMS), grant no. 2005630, and a JSPS Grant-in-Aid for Transformative Research Areas (A) (22H05107, Y.H.), EPSRC Research Grant EP/Y008642/1. article_number: '5' article_processing_charge: Yes (in subscription journal) article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Matthew full_name: Kahle, Matthew last_name: Kahle - first_name: Shu full_name: Kanazawa, Shu last_name: Kanazawa citation: ama: Edelsbrunner H, Kahle M, Kanazawa S. Maximum persistent Betti numbers of Čech complexes. Journal of Applied and Computational Topology. 2026;10. doi:10.1007/s41468-026-00233-3 apa: Edelsbrunner, H., Kahle, M., & Kanazawa, S. (2026). Maximum persistent Betti numbers of Čech complexes. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-026-00233-3 chicago: Edelsbrunner, Herbert, Matthew Kahle, and Shu Kanazawa. “Maximum Persistent Betti Numbers of Čech Complexes.” Journal of Applied and Computational Topology. Springer Nature, 2026. https://doi.org/10.1007/s41468-026-00233-3. ieee: H. Edelsbrunner, M. Kahle, and S. Kanazawa, “Maximum persistent Betti numbers of Čech complexes,” Journal of Applied and Computational Topology, vol. 10. Springer Nature, 2026. ista: Edelsbrunner H, Kahle M, Kanazawa S. 2026. Maximum persistent Betti numbers of Čech complexes. Journal of Applied and Computational Topology. 10, 5. mla: Edelsbrunner, Herbert, et al. “Maximum Persistent Betti Numbers of Čech Complexes.” Journal of Applied and Computational Topology, vol. 10, 5, Springer Nature, 2026, doi:10.1007/s41468-026-00233-3. short: H. Edelsbrunner, M. Kahle, S. Kanazawa, Journal of Applied and Computational Topology 10 (2026). date_created: 2026-03-08T23:01:45Z date_published: 2026-03-01T00:00:00Z date_updated: 2026-03-09T11:31:29Z day: '01' ddc: - '500' department: - _id: HeEd doi: 10.1007/s41468-026-00233-3 external_id: arxiv: - '2409.05241' file: - access_level: open_access checksum: 0bf6dc430cafa40c08f260fe17d54595 content_type: application/pdf creator: dernst date_created: 2026-03-09T11:29:30Z date_updated: 2026-03-09T11:29:30Z file_id: '21416' file_name: 2026_JourAppliedCompTopology_Edelsbrunner.pdf file_size: 323111 relation: main_file success: 1 file_date_updated: 2026-03-09T11:29:30Z has_accepted_license: '1' intvolume: ' 10' language: - iso: eng month: '03' oa: 1 oa_version: Published Version project: - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Journal of Applied and Computational Topology publication_identifier: eissn: - 2367-1734 issn: - 2367-1726 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Maximum persistent Betti numbers of Čech complexes tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 10 year: '2026' ... --- OA_place: repository OA_type: green _id: '18626' abstract: - lang: eng text: "The local angle property of the (order-1) Delaunay triangulations of a generic set in R2\r\n asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well." acknowledgement: Work by the first and third authors is partially supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is partially supported by the Alexander von Humboldt Foundation. article_number: '110055' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Edelsbrunner H, Garber A, Saghafian M. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 2025;461. doi:10.1016/j.aim.2024.110055 apa: Edelsbrunner, H., Garber, A., & Saghafian, M. (2025). Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2024.110055 chicago: Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “Order-2 Delaunay Triangulations Optimize Angles.” Advances in Mathematics. Elsevier, 2025. https://doi.org/10.1016/j.aim.2024.110055. ieee: H. Edelsbrunner, A. Garber, and M. Saghafian, “Order-2 Delaunay triangulations optimize angles,” Advances in Mathematics, vol. 461. Elsevier, 2025. ista: Edelsbrunner H, Garber A, Saghafian M. 2025. Order-2 Delaunay triangulations optimize angles. Advances in Mathematics. 461, 110055. mla: Edelsbrunner, Herbert, et al. “Order-2 Delaunay Triangulations Optimize Angles.” Advances in Mathematics, vol. 461, 110055, Elsevier, 2025, doi:10.1016/j.aim.2024.110055. short: H. Edelsbrunner, A. Garber, M. Saghafian, Advances in Mathematics 461 (2025). corr_author: '1' date_created: 2024-12-08T23:01:54Z date_published: 2025-02-01T00:00:00Z date_updated: 2025-04-15T07:16:53Z day: '01' department: - _id: HeEd doi: 10.1016/j.aim.2024.110055 ec_funded: 1 external_id: arxiv: - '2310.18238' isi: - '001370682500001' intvolume: ' 461' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2310.18238 month: '02' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Advances in Mathematics publication_identifier: eissn: - 1090-2082 issn: - 0001-8708 publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: Order-2 Delaunay triangulations optimize angles type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 461 year: '2025' ... --- OA_place: publisher OA_type: gold _id: '20005' abstract: - lang: eng text: "We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in \U0001D54A^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers." acknowledgement: "Herbert Edelsbrunner: partially supported by the Wittgenstein Prize, Austrian Science\r\nFund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,\r\nAustrian Science Fund (FWF), grant no. I 02979-N35.\r\nAlexey Garber: partially supported by the Simons Foundation.\r\nMorteza Saghafian: partially supported by the Wittgenstein Prize, Austrian Science Fund (FWF),\r\ngrant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund (FWF), grant no. I 02979-N35" alternative_title: - LIPIcs article_number: '43' article_processing_charge: Yes arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In: 41st International Symposium on Computational Geometry. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:10.4230/LIPIcs.SoCG.2025.43' apa: 'Edelsbrunner, H., Garber, A., & Saghafian, M. (2025). On spheres with k points inside. In 41st International Symposium on Computational Geometry (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2025.43' chicago: Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres with k Points Inside.” In 41st International Symposium on Computational Geometry, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. https://doi.org/10.4230/LIPIcs.SoCG.2025.43. ieee: H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,” in 41st International Symposium on Computational Geometry, Kanazawa, Japan, 2025, vol. 332. ista: 'Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 43.' mla: Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” 41st International Symposium on Computational Geometry, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:10.4230/LIPIcs.SoCG.2025.43. short: H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. conference: end_date: 2025-06-27 location: Kanazawa, Japan name: 'SoCG: Symposium on Computational Geometry' start_date: 2025-06-23 corr_author: '1' date_created: 2025-07-13T22:01:22Z date_published: 2025-06-20T00:00:00Z date_updated: 2025-07-14T07:26:14Z day: '20' ddc: - '510' department: - _id: HeEd doi: 10.4230/LIPIcs.SoCG.2025.43 external_id: arxiv: - '2410.21204' file: - access_level: open_access checksum: b5313ed8575ea87913c71a6e3c7513c8 content_type: application/pdf creator: dernst date_created: 2025-07-14T07:24:22Z date_updated: 2025-07-14T07:24:22Z file_id: '20016' file_name: 2025_LIPIcs.SoCG_Edelsbrunner.pdf file_size: 661893 relation: main_file success: 1 file_date_updated: 2025-07-14T07:24:22Z has_accepted_license: '1' intvolume: ' 332' language: - iso: eng month: '06' oa: 1 oa_version: Published Version project: - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: 41st International Symposium on Computational Geometry publication_identifier: eissn: - 1868-8969 isbn: - '9783959773706' publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: On spheres with k points inside tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 332 year: '2025' ... --- OA_place: publisher OA_type: gold _id: '20006' abstract: - lang: eng text: In numerous fields, dynamic time series data require continuous updates, necessitating efficient data processing techniques for accurate analysis. This paper examines the banana tree data structure, specifically designed to efficiently maintain the multi-scale topological descriptor commonly known as persistent homology for dynamically changing time series data. We implement this data structure and conduct an experimental study to assess its properties and runtime for update operations. Our findings indicate that banana trees are highly effective with unbiased random data, outperforming state-of-the-art static algorithms in these scenarios. Additionally, our results show that real-world time series share structural properties with unbiased random walks, suggesting potential practical utility for our implementation. acknowledgement: "Lara Ost: Supported by the Vienna Graduate School on Computational Optimization\r\n(VGSCO), FWF project no. W1260-N35.\r\nSebastiano Cultrera di Montesano: Supported by the Eric and Wendy Schmidt Center at the Broad Institute of MIT and Harvard.\r\nHerbert Edelsbrunner: Partially supported by the Wittgenstein Prize, FWF grant no. Z 342-N31,\r\nand by the DFG Collaborative Research Center TRR 109, FWF grant no. I 02979-N35." alternative_title: - LIPIcs article_number: '71' article_processing_charge: Yes arxiv: 1 author: - first_name: Lara full_name: Ost, Lara last_name: Ost - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 citation: ama: 'Ost L, Cultrera di Montesano S, Edelsbrunner H. Banana trees for the persistence in time series experimentally. In: 41st International Symposium on Computational Geometry. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:10.4230/LIPIcs.SoCG.2025.71' apa: 'Ost, L., Cultrera di Montesano, S., & Edelsbrunner, H. (2025). Banana trees for the persistence in time series experimentally. In 41st International Symposium on Computational Geometry (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2025.71' chicago: Ost, Lara, Sebastiano Cultrera di Montesano, and Herbert Edelsbrunner. “Banana Trees for the Persistence in Time Series Experimentally.” In 41st International Symposium on Computational Geometry, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. https://doi.org/10.4230/LIPIcs.SoCG.2025.71. ieee: L. Ost, S. Cultrera di Montesano, and H. Edelsbrunner, “Banana trees for the persistence in time series experimentally,” in 41st International Symposium on Computational Geometry, Kanazawa, Japan, 2025, vol. 332. ista: 'Ost L, Cultrera di Montesano S, Edelsbrunner H. 2025. Banana trees for the persistence in time series experimentally. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 71.' mla: Ost, Lara, et al. “Banana Trees for the Persistence in Time Series Experimentally.” 41st International Symposium on Computational Geometry, vol. 332, 71, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:10.4230/LIPIcs.SoCG.2025.71. short: L. Ost, S. Cultrera di Montesano, H. Edelsbrunner, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. conference: end_date: 2025-06-27 location: Kanazawa, Japan name: 'SoCG: Symposium on Computational Geometry' start_date: 2025-06-23 corr_author: '1' date_created: 2025-07-13T22:01:22Z date_published: 2025-06-20T00:00:00Z date_updated: 2025-12-30T11:04:33Z day: '20' ddc: - '000' department: - _id: HeEd doi: 10.4230/LIPIcs.SoCG.2025.71 external_id: arxiv: - '2405.17920' file: - access_level: open_access checksum: 3a4a7a707a56e0cfdf51428782dee55a content_type: application/pdf creator: dernst date_created: 2025-07-14T08:23:38Z date_updated: 2025-07-14T08:23:38Z file_id: '20017' file_name: 2025_LIPIcs.SoCG_Ost.pdf file_size: 834623 relation: main_file success: 1 file_date_updated: 2025-07-14T08:23:38Z has_accepted_license: '1' intvolume: ' 332' language: - iso: eng month: '06' oa: 1 oa_version: Published Version project: - _id: 9B9290DE-BA93-11EA-9121-9846C619BF3A grant_number: W1260-N35 name: Vienna Graduate School on Computational Optimization - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication: 41st International Symposium on Computational Geometry publication_identifier: eissn: - 1868-8969 isbn: - '9783959773706' publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' related_material: link: - relation: software url: https://github.com/laraost/BananaPersist scopus_import: '1' status: public title: Banana trees for the persistence in time series experimentally tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 332 year: '2025' ... --- DOAJ_listed: '1' OA_place: publisher OA_type: gold PlanS_conform: '1' _id: '20293' abstract: - lang: eng text: Motivated by questions arising at the intersection of information theory and geometry, we compare two dissimilarity measures between finite categorical distributions. One is the well-known Jensen–Shannon divergence, which is easy to compute and whose square root is a proper metric. The other is what we call the minmax divergence, which is harder to compute. Just like the Jensen–Shannon divergence, it arises naturally from the Kullback–Leibler divergence. The main contribution of this paper is a proof showing that the minmax divergence can be tightly approximated by the Jensen–Shannon divergence. The bounds suggest that the square root of the minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional case. The general case remains open. Finally, we consider analogous questions in the context of another Bregman divergence and the corresponding Burbea–Rao (Jensen–Bregman) divergence. acknowledgement: "This research received partial funding from the European Research Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation programme, grant no. 788183, the\r\nWittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, the DFG Collaborative\r\nResearch Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35, and the 2022 Google Research Scholar Award for project ‘Algorithms for Topological Analysis of Neural Networks’. The APC was waived." article_number: '854' article_processing_charge: Yes article_type: original author: - first_name: Arseniy full_name: Akopyan, Arseniy id: 430D2C90-F248-11E8-B48F-1D18A9856A87 last_name: Akopyan orcid: 0000-0002-2548-617X - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Ziga full_name: Virk, Ziga id: 2E36B656-F248-11E8-B48F-1D18A9856A87 last_name: Virk - first_name: Hubert full_name: Wagner, Hubert id: 379CA8B8-F248-11E8-B48F-1D18A9856A87 last_name: Wagner citation: ama: Akopyan A, Edelsbrunner H, Virk Z, Wagner H. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. Entropy. 2025;27(8). doi:10.3390/e27080854 apa: Akopyan, A., Edelsbrunner, H., Virk, Z., & Wagner, H. (2025). Tight bounds between the Jensen–Shannon divergence and the minmax divergence. Entropy. MDPI. https://doi.org/10.3390/e27080854 chicago: Akopyan, Arseniy, Herbert Edelsbrunner, Ziga Virk, and Hubert Wagner. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” Entropy. MDPI, 2025. https://doi.org/10.3390/e27080854. ieee: A. Akopyan, H. Edelsbrunner, Z. Virk, and H. Wagner, “Tight bounds between the Jensen–Shannon divergence and the minmax divergence,” Entropy, vol. 27, no. 8. MDPI, 2025. ista: Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854. mla: Akopyan, Arseniy, et al. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” Entropy, vol. 27, no. 8, 854, MDPI, 2025, doi:10.3390/e27080854. short: A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025). corr_author: '1' date_created: 2025-09-07T22:01:33Z date_published: 2025-08-01T00:00:00Z date_updated: 2025-09-30T14:32:31Z day: '01' ddc: - '500' department: - _id: HeEd doi: 10.3390/e27080854 ec_funded: 1 external_id: isi: - '001557476000001' pmid: - '40870326' file: - access_level: open_access checksum: 65c5399c4015d9c8abb8c7a96f3d7836 content_type: application/pdf creator: dernst date_created: 2025-09-08T07:55:48Z date_updated: 2025-09-08T07:55:48Z file_id: '20309' file_name: 2025_Entropy_Akopyan.pdf file_size: 379340 relation: main_file success: 1 file_date_updated: 2025-09-08T07:55:48Z has_accepted_license: '1' intvolume: ' 27' isi: 1 issue: '8' language: - iso: eng month: '08' oa: 1 oa_version: Published Version pmid: 1 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Entropy publication_identifier: eissn: - 1099-4300 publication_status: published publisher: MDPI quality_controlled: '1' scopus_import: '1' status: public title: Tight bounds between the Jensen–Shannon divergence and the minmax divergence tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 27 year: '2025' ... --- OA_place: repository OA_type: green _id: '20490' abstract: - lang: eng text: "We study flips in hypertriangulations of planar points sets. Here a level-k hypertriangulation of n\r\n points in the plane is a subdivision induced by the projection of a k-hypersimplex, which is the convex hull of the barycenters of the (k-1)-dimensional faces of the standard (n-1)-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.\r\n" acknowledgement: Work by all authors but the second is supported by the European Research Council (ERC), grant no. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. Work by the second author is partially supported by the Alexander von Humboldt Foundation and by the Simons Foundation . The second author thanks Jesús A. De Loera for useful discussions on flips and non-flips and Pavel Galashin and Alexey Balitskiy for useful discussions on plabic graphs. article_number: '104248' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Mohadese full_name: Ghafari, Mohadese last_name: Ghafari - first_name: Teresa full_name: Heiss, Teresa id: 4879BB4E-F248-11E8-B48F-1D18A9856A87 last_name: Heiss orcid: 0000-0002-1780-2689 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 2025;132. doi:10.1016/j.ejc.2025.104248 apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2025). Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. Elsevier. https://doi.org/10.1016/j.ejc.2025.104248 chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “Flips in Two-Dimensional Hypertriangulations.” European Journal of Combinatorics. Elsevier, 2025. https://doi.org/10.1016/j.ejc.2025.104248. ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “Flips in two-dimensional hypertriangulations,” European Journal of Combinatorics, vol. 132. Elsevier, 2025. ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2025. Flips in two-dimensional hypertriangulations. European Journal of Combinatorics. 132, 104248. mla: Edelsbrunner, Herbert, et al. “Flips in Two-Dimensional Hypertriangulations.” European Journal of Combinatorics, vol. 132, 104248, Elsevier, 2025, doi:10.1016/j.ejc.2025.104248. short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, European Journal of Combinatorics 132 (2025). corr_author: '1' date_created: 2025-10-19T22:01:31Z date_published: 2025-10-10T00:00:00Z date_updated: 2025-12-01T12:57:29Z day: '10' department: - _id: HeEd doi: 10.1016/j.ejc.2025.104248 ec_funded: 1 external_id: arxiv: - '2212.11380' isi: - '001599061500002' intvolume: ' 132' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2212.11380 month: '10' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: European Journal of Combinatorics publication_identifier: issn: - 0195-6698 publication_status: epub_ahead publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: Flips in two-dimensional hypertriangulations type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 132 year: '2025' ... --- OA_place: repository OA_type: green _id: '20585' abstract: - lang: eng text: Motivated by applications in medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided. acknowledgement: "This project has received funding from the European Research\r\nCouncil (ERC) under the European Union’s Horizon 2020 research and innovation\r\nprogramme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund\r\n(FWF), grant no. Z 342-N31, and from the DFG Collaborative Research Center TRR\r\n109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35." article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic alpha complexes. Foundations of Data Science. 2025;8:30-62. doi:10.3934/fods.2025003 apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (2025). Chromatic alpha complexes. Foundations of Data Science. American Institute of Mathematical Sciences. https://doi.org/10.3934/fods.2025003 chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Alpha Complexes.” Foundations of Data Science. American Institute of Mathematical Sciences, 2025. https://doi.org/10.3934/fods.2025003. ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic alpha complexes,” Foundations of Data Science, vol. 8. American Institute of Mathematical Sciences, pp. 30–62, 2025. ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2025. Chromatic alpha complexes. Foundations of Data Science. 8, 30–62. mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Alpha Complexes.” Foundations of Data Science, vol. 8, American Institute of Mathematical Sciences, 2025, pp. 30–62, doi:10.3934/fods.2025003. short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, Foundations of Data Science 8 (2025) 30–62. corr_author: '1' date_created: 2025-11-02T23:01:33Z date_published: 2025-03-01T00:00:00Z date_updated: 2025-11-04T12:25:47Z day: '01' department: - _id: HeEd doi: 10.3934/fods.2025003 ec_funded: 1 external_id: arxiv: - '2212.03128' intvolume: ' 8' language: - iso: eng month: '03' oa_version: Preprint page: 30-62 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Foundations of Data Science publication_identifier: eissn: - 2639-8001 publication_status: epub_ahead publisher: American Institute of Mathematical Sciences quality_controlled: '1' related_material: record: - id: '15091' relation: earlier_version status: public scopus_import: '1' status: public title: Chromatic alpha complexes type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 8 year: '2025' ... --- OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '20657' abstract: - lang: eng text: 'The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². ' acknowledgement: The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. The second author is supported by the European Research Council (ERC), grant “GeoScape” and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: János full_name: Pach, János id: E62E3130-B088-11EA-B919-BF823C25FEA4 last_name: Pach citation: ama: Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. Discrete & Computational Geometry. 2025. doi:10.1007/s00454-025-00796-5 apa: Edelsbrunner, H., & Pach, J. (2025). Maximum Betti numbers of Čech complexes. Discrete & Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00796-5 chicago: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” Discrete & Computational Geometry. Springer Nature, 2025. https://doi.org/10.1007/s00454-025-00796-5. ieee: H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” Discrete & Computational Geometry. Springer Nature, 2025. ista: Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete & Computational Geometry. mla: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” Discrete & Computational Geometry, Springer Nature, 2025, doi:10.1007/s00454-025-00796-5. short: H. Edelsbrunner, J. Pach, Discrete & Computational Geometry (2025). corr_author: '1' date_created: 2025-11-19T09:44:58Z date_published: 2025-11-10T00:00:00Z date_updated: 2025-12-01T15:19:21Z day: '10' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-025-00796-5 ec_funded: 1 external_id: arxiv: - '2310.14801' isi: - '001610592600001' has_accepted_license: '1' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.1007/s00454-025-00796-5 month: '11' oa: 1 oa_version: Published Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication: Discrete & Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: epub_ahead publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '17146' relation: earlier_version status: public scopus_import: '1' status: public title: Maximum Betti numbers of Čech complexes tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2025' ... --- OA_place: repository OA_type: green _id: '20658' abstract: - lang: eng text: The medial axis of a smoothly embedded surface in R^3 consists of all points for which the Euclidean distance function on the surface has at least two global minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided. alternative_title: - LNCS article_processing_charge: No arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Elizabeth R full_name: Stephenson, Elizabeth R id: 2D04F932-F248-11E8-B48F-1D18A9856A87 last_name: Stephenson orcid: 0000-0002-6862-208X - first_name: Martin H full_name: Thoresen, Martin H id: 47CB1472-F248-11E8-B48F-1D18A9856A87 last_name: Thoresen citation: ama: 'Edelsbrunner H, Stephenson ER, Thoresen MH. The mid-sphere cousin of the medial axis transform. In: 4th International Joint Conference on Discrete Geometry and Mathematical Morphology. Vol 16296. Springer Nature; 2025:133-147. doi:10.1007/978-3-032-09544-2_10' apa: 'Edelsbrunner, H., Stephenson, E. R., & Thoresen, M. H. (2025). The mid-sphere cousin of the medial axis transform. In 4th International Joint Conference on Discrete Geometry and Mathematical Morphology (Vol. 16296, pp. 133–147). Groningen, The Netherlands: Springer Nature. https://doi.org/10.1007/978-3-032-09544-2_10' chicago: Edelsbrunner, Herbert, Elizabeth R Stephenson, and Martin H Thoresen. “The Mid-Sphere Cousin of the Medial Axis Transform.” In 4th International Joint Conference on Discrete Geometry and Mathematical Morphology, 16296:133–47. Springer Nature, 2025. https://doi.org/10.1007/978-3-032-09544-2_10. ieee: H. Edelsbrunner, E. R. Stephenson, and M. H. Thoresen, “The mid-sphere cousin of the medial axis transform,” in 4th International Joint Conference on Discrete Geometry and Mathematical Morphology, Groningen, The Netherlands, 2025, vol. 16296, pp. 133–147. ista: 'Edelsbrunner H, Stephenson ER, Thoresen MH. 2025. The mid-sphere cousin of the medial axis transform. 4th International Joint Conference on Discrete Geometry and Mathematical Morphology. DGMM: Discrete Geometry and Mathematical Morphology, LNCS, vol. 16296, 133–147.' mla: Edelsbrunner, Herbert, et al. “The Mid-Sphere Cousin of the Medial Axis Transform.” 4th International Joint Conference on Discrete Geometry and Mathematical Morphology, vol. 16296, Springer Nature, 2025, pp. 133–47, doi:10.1007/978-3-032-09544-2_10. short: H. Edelsbrunner, E.R. Stephenson, M.H. Thoresen, in:, 4th International Joint Conference on Discrete Geometry and Mathematical Morphology, Springer Nature, 2025, pp. 133–147. conference: end_date: 2025-11-06 location: Groningen, The Netherlands name: 'DGMM: Discrete Geometry and Mathematical Morphology' start_date: 2025-11-03 date_created: 2025-11-23T23:01:37Z date_published: 2025-11-01T00:00:00Z date_updated: 2025-11-24T10:05:11Z day: '01' department: - _id: HeEd doi: 10.1007/978-3-032-09544-2_10 external_id: arxiv: - '2504.14743' intvolume: ' 16296' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2504.14743 month: '11' oa: 1 oa_version: Preprint page: 133-147 publication: 4th International Joint Conference on Discrete Geometry and Mathematical Morphology publication_identifier: eissn: - 1611-3349 isbn: - '9783032095435' issn: - 0302-9743 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: The mid-sphere cousin of the medial axis transform type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 16296 year: '2025' ... --- OA_place: publisher OA_type: hybrid _id: '17149' abstract: - lang: eng text: The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about 4/Pi. We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes. acknowledgement: "The authors thank Ranita Biswas and Tatiana Ezubova for the collaboration on computational experiments that motivated the work reported in this paper. The authors also thank Daniel Bonnema for proofreading and noticing an issue with the original proof of Lemma 4.3.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).\r\nThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35." article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Anton full_name: Nikitenko, Anton id: 3E4FF1BA-F248-11E8-B48F-1D18A9856A87 last_name: Nikitenko orcid: 0000-0002-0659-3201 citation: ama: Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths and scapes. Discrete & Computational Geometry. 2025;73:490-499. doi:10.1007/s00454-024-00660-y apa: Edelsbrunner, H., & Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. Discrete & Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-024-00660-y chicago: Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” Discrete & Computational Geometry. Springer Nature, 2025. https://doi.org/10.1007/s00454-024-00660-y. ieee: H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi paths and scapes,” Discrete & Computational Geometry, vol. 73. Springer Nature, pp. 490–499, 2025. ista: Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete & Computational Geometry. 73, 490–499. mla: Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” Discrete & Computational Geometry, vol. 73, Springer Nature, 2025, pp. 490–99, doi:10.1007/s00454-024-00660-y. short: H. Edelsbrunner, A. Nikitenko, Discrete & Computational Geometry 73 (2025) 490–499. corr_author: '1' date_created: 2024-06-16T22:01:07Z date_published: 2025-03-01T00:00:00Z date_updated: 2026-02-16T12:18:50Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-024-00660-y ec_funded: 1 external_id: arxiv: - '2012.03350' isi: - '001238566200004' pmid: - '39974750' file: - access_level: open_access checksum: ffb0c818222138f9f113f4bbea41e834 content_type: application/pdf creator: dernst date_created: 2025-04-23T07:31:32Z date_updated: 2025-04-23T07:31:32Z file_id: '19610' file_name: 2025_DiscreteComputGeom_EdelsbrunnerHe.pdf file_size: 283443 relation: main_file success: 1 file_date_updated: 2025-04-23T07:31:32Z has_accepted_license: '1' intvolume: ' 73' isi: 1 language: - iso: eng month: '03' oa: 1 oa_version: Published Version page: 490-499 pmid: 1 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Discrete & Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Average and expected distortion of Voronoi paths and scapes tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 73 year: '2025' ... --- OA_place: repository _id: '21050' abstract: - lang: eng text: "In 1873, James C. Maxwell conjectured that the electric field generated by $n$ point charges in generic position has at most $(n-1)^2$ isolated zeroes. The first (non-optimal) upper bound was only obtained in 2007 by Gabrielov, Novikov and Shapiro, who also posed two additional interesting conjectures.\r\n In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to a conjecture of Gabrielov, Novikov and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges.\r\n Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find smaller bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day." article_processing_charge: No arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Christopher D full_name: Fillmore, Christopher D id: 35638A5C-AAC7-11E9-B0BF-5503E6697425 last_name: Fillmore - first_name: Gonçalo full_name: Olivera, Gonçalo last_name: Olivera citation: ama: Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. arXiv. doi:10.48550/ARXIV.2501.05315 apa: Edelsbrunner, H., Fillmore, C. D., & Olivera, G. (n.d.). Counting equilibria of the electrostatic potential. arXiv. https://doi.org/10.48550/ARXIV.2501.05315 chicago: Edelsbrunner, Herbert, Christopher D Fillmore, and Gonçalo Olivera. “Counting Equilibria of the Electrostatic Potential.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2501.05315. ieee: H. Edelsbrunner, C. D. Fillmore, and G. Olivera, “Counting equilibria of the electrostatic potential,” arXiv. . ista: Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. arXiv, 10.48550/ARXIV.2501.05315. mla: Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” ArXiv, doi:10.48550/ARXIV.2501.05315. short: H. Edelsbrunner, C.D. Fillmore, G. Olivera, ArXiv (n.d.). corr_author: '1' date_created: 2026-01-27T14:29:27Z date_published: 2025-03-20T00:00:00Z date_updated: 2026-04-07T11:42:48Z day: '20' department: - _id: HeEd doi: 10.48550/ARXIV.2501.05315 external_id: arxiv: - '2501.05315' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2501.05315 month: '03' oa: 1 oa_version: Preprint publication: arXiv publication_status: draft related_material: record: - id: '21021' relation: dissertation_contains status: public status: public title: Counting equilibria of the electrostatic potential tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: preprint user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2025' ... --- _id: '14345' abstract: - lang: eng text: For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970)). acknowledgement: Work by all authors but A. Garber is supported by the European Research Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially supported by the Alexander von Humboldt Foundation. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Mohadese full_name: Ghafari, Mohadese last_name: Ghafari - first_name: Teresa full_name: Heiss, Teresa id: 4879BB4E-F248-11E8-B48F-1D18A9856A87 last_name: Heiss orcid: 0000-0002-1780-2689 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. 2024;72:29-48. doi:10.1007/s00454-023-00566-1 apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2024). On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-023-00566-1 chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” Discrete and Computational Geometry. Springer Nature, 2024. https://doi.org/10.1007/s00454-023-00566-1. ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles in higher order Brillouin tessellations and related tilings in the plane,” Discrete and Computational Geometry, vol. 72. Springer Nature, pp. 29–48, 2024. ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. 72, 29–48. mla: Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” Discrete and Computational Geometry, vol. 72, Springer Nature, 2024, pp. 29–48, doi:10.1007/s00454-023-00566-1. short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry 72 (2024) 29–48. corr_author: '1' date_created: 2023-09-17T22:01:10Z date_published: 2024-07-01T00:00:00Z date_updated: 2025-04-23T08:41:59Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-023-00566-1 ec_funded: 1 external_id: arxiv: - '2204.01076' isi: - '001060727600004' pmid: - '39610762' file: - access_level: open_access checksum: b207b4e00f904e8ea8a30e24f0251f79 content_type: application/pdf creator: dernst date_created: 2024-07-22T09:43:19Z date_updated: 2024-07-22T09:43:19Z file_id: '17301' file_name: 2024_DiscreteComputGeom_Edelsbrunner.pdf file_size: 892019 relation: main_file success: 1 file_date_updated: 2024-07-22T09:43:19Z has_accepted_license: '1' intvolume: ' 72' isi: 1 language: - iso: eng month: '07' oa: 1 oa_version: Published Version page: 29-48 pmid: 1 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Discrete and Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: On angles in higher order Brillouin tessellations and related tilings in the plane tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 72 year: '2024' ... --- _id: '17891' abstract: - lang: eng text: "Abstract\r\nMethods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback–Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics." acknowledgement: We thank Anton Nikitenko for first observing that the Wrap complex can be characterized as stated in Claim (ii) of the Wrap Complex Lemma, and Ondrej Draganov for correcting a critical mistake in one of our formulas in Section 2. article_number: '637' article_processing_charge: Yes article_type: original author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Katharina full_name: Ölsböck, Katharina id: 4D4AA390-F248-11E8-B48F-1D18A9856A87 last_name: Ölsböck orcid: 0000-0002-4672-8297 - first_name: Hubert full_name: Wagner, Hubert id: 379CA8B8-F248-11E8-B48F-1D18A9856A87 last_name: Wagner citation: ama: Edelsbrunner H, Ölsböck K, Wagner H. Understanding higher-order interactions in information space. Entropy. 2024;26(8). doi:10.3390/e26080637 apa: Edelsbrunner, H., Ölsböck, K., & Wagner, H. (2024). Understanding higher-order interactions in information space. Entropy. MDPI. https://doi.org/10.3390/e26080637 chicago: Edelsbrunner, Herbert, Katharina Ölsböck, and Hubert Wagner. “Understanding Higher-Order Interactions in Information Space.” Entropy. MDPI, 2024. https://doi.org/10.3390/e26080637. ieee: H. Edelsbrunner, K. Ölsböck, and H. Wagner, “Understanding higher-order interactions in information space,” Entropy, vol. 26, no. 8. MDPI, 2024. ista: Edelsbrunner H, Ölsböck K, Wagner H. 2024. Understanding higher-order interactions in information space. Entropy. 26(8), 637. mla: Edelsbrunner, Herbert, et al. “Understanding Higher-Order Interactions in Information Space.” Entropy, vol. 26, no. 8, 637, MDPI, 2024, doi:10.3390/e26080637. short: H. Edelsbrunner, K. Ölsböck, H. Wagner, Entropy 26 (2024). date_created: 2024-09-08T22:01:11Z date_published: 2024-08-01T00:00:00Z date_updated: 2025-09-08T09:13:44Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.3390/e26080637 external_id: isi: - '001305543500001' pmid: - '39202107' file: - access_level: open_access checksum: 624a9e2c5b49d6c38b88b0f675467ba3 content_type: application/pdf creator: dernst date_created: 2024-09-09T09:01:12Z date_updated: 2024-09-09T09:01:12Z file_id: '17948' file_name: 2024_Entropy_Edelsbrunner.pdf file_size: 8025139 relation: main_file success: 1 file_date_updated: 2024-09-09T09:01:12Z has_accepted_license: '1' intvolume: ' 26' isi: 1 issue: '8' language: - iso: eng month: '08' oa: 1 oa_version: Published Version pmid: 1 publication: Entropy publication_identifier: eissn: - 1099-4300 publication_status: published publisher: MDPI quality_controlled: '1' related_material: link: - relation: software url: https://git.ista.ac.at/katharina.oelsboeck/wrap_2_3-public/ scopus_import: '1' status: public title: Understanding higher-order interactions in information space tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 26 year: '2024' ... --- OA_place: publisher OA_type: gold _id: '18556' abstract: - lang: eng text: Given a finite set, A ⊆ ℝ², and a subset, B ⊆ A, the MST-ratio is the combined length of the minimum spanning trees of B and A⧵B divided by the length of the minimum spanning tree of A. The question of the supremum, over all sets A, of the maximum, over all subsets B, is related to the Steiner ratio, and we prove this sup-max is between 2.154 and 2.427. Restricting ourselves to 2-dimensional lattices, we prove that the sup-max is 2, while the inf-max is 1.25. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than 1.25. 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. alternative_title: - LIPIcs article_number: '3' article_processing_charge: Yes arxiv: 1 author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. The Euclidean MST-ratio for bi-colored lattices. In: 32nd International Symposium on Graph Drawing and Network Visualization. Vol 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.GD.2024.3' apa: 'Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (2024). The Euclidean MST-ratio for bi-colored lattices. In 32nd International Symposium on Graph Drawing and Network Visualization (Vol. 320). Vienna, Austria: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.GD.2024.3' chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “The Euclidean MST-Ratio for Bi-Colored Lattices.” In 32nd International Symposium on Graph Drawing and Network Visualization, Vol. 320. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.GD.2024.3. ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “The Euclidean MST-ratio for bi-colored lattices,” in 32nd International Symposium on Graph Drawing and Network Visualization, Vienna, Austria, 2024, vol. 320. ista: 'Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. 2024. The Euclidean MST-ratio for bi-colored lattices. 32nd International Symposium on Graph Drawing and Network Visualization. GD: Graph Drawing and Network Visualization, LIPIcs, vol. 320, 3.' mla: Cultrera di Montesano, Sebastiano, et al. “The Euclidean MST-Ratio for Bi-Colored Lattices.” 32nd International Symposium on Graph Drawing and Network Visualization, vol. 320, 3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.GD.2024.3. short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, in:, 32nd International Symposium on Graph Drawing and Network Visualization, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. conference: end_date: 2024-09-20 location: Vienna, Austria name: 'GD: Graph Drawing and Network Visualization' start_date: 2024-09-18 corr_author: '1' date_created: 2024-11-17T23:01:47Z date_published: 2024-10-28T00:00:00Z date_updated: 2025-12-02T13:50:50Z day: '28' ddc: - '510' department: - _id: HeEd doi: 10.4230/LIPIcs.GD.2024.3 ec_funded: 1 external_id: arxiv: - '2403.10204' isi: - '001540278400001' file: - access_level: open_access checksum: 5f9b35e115c3d375e99be78da9054cb4 content_type: application/pdf creator: dernst date_created: 2024-11-18T07:49:25Z date_updated: 2024-11-18T07:49:25Z file_id: '18560' file_name: 2024_LIPIcs_CultreradiMontesano.pdf file_size: 908541 relation: main_file success: 1 file_date_updated: 2024-11-18T07:49:25Z has_accepted_license: '1' intvolume: ' 320' isi: 1 language: - iso: eng month: '10' oa: 1 oa_version: Published Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: 32nd International Symposium on Graph Drawing and Network Visualization publication_identifier: isbn: - '9783959773430' issn: - 1868-8969 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' scopus_import: '1' status: public title: The Euclidean MST-ratio for bi-colored lattices tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 320 year: '2024' ... --- OA_place: repository OA_type: green _id: '18999' abstract: - lang: eng text: Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial interactions among multiple configurations, each distinguished by a color. It describes advances in this area and prepares the ground for further exploration by mentioning unresolved questions and promising research avenues while focusing on the overlap with discrete geometry. article_number: '2406.04102' article_processing_charge: No arxiv: 1 author: - first_name: Sebastiano full_name: Cultrera di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera di Montesano orcid: 0000-0001-6249-0832 - first_name: Ondrej full_name: Draganov, Ondrej id: 2B23F01E-F248-11E8-B48F-1D18A9856A87 last_name: Draganov orcid: 0000-0003-0464-3823 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. arXiv. doi:10.48550/ARXIV.2406.04102 apa: Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., & Saghafian, M. (n.d.). Chromatic topological data analysis. arXiv. https://doi.org/10.48550/ARXIV.2406.04102 chicago: Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Topological Data Analysis.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2406.04102. ieee: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic topological data analysis,” arXiv. . ista: Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. arXiv, 2406.04102. mla: Cultrera di Montesano, Sebastiano, et al. “Chromatic Topological Data Analysis.” ArXiv, 2406.04102, doi:10.48550/ARXIV.2406.04102. short: S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.). corr_author: '1' date_created: 2025-02-04T16:21:21Z date_published: 2024-06-06T00:00:00Z date_updated: 2025-02-10T08:14:27Z day: '06' ddc: - '510' department: - _id: GradSch - _id: HeEd doi: 10.48550/ARXIV.2406.04102 external_id: arxiv: - '2406.04102' has_accepted_license: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2406.04102 month: '06' oa: 1 oa_version: Preprint publication: arXiv publication_status: submitted status: public title: Chromatic topological data analysis tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ... --- OA_place: publisher OA_type: hybrid _id: '15380' abstract: - lang: eng text: The depth of a cell in an arrangement of n (non-vertical) great-spheres in Sd is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn–Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements. acknowledgement: "The authors thank Uli Wagner and Emo Welzl for comments on an earlier version of this paper, and for pointing out related work in the prior literature.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35." article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Ranita full_name: Biswas, Ranita id: 3C2B033E-F248-11E8-B48F-1D18A9856A87 last_name: Biswas orcid: 0000-0002-5372-7890 - first_name: Sebastiano full_name: Cultrera Di Montesano, Sebastiano id: 34D2A09C-F248-11E8-B48F-1D18A9856A87 last_name: Cultrera Di Montesano orcid: 0000-0001-6249-0832 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Morteza full_name: Saghafian, Morteza id: f86f7148-b140-11ec-9577-95435b8df824 last_name: Saghafian citation: ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. 2024;8:557-578. doi:10.1007/s41468-024-00173-w' apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2024). Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-024-00173-w' chicago: 'Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Journal of Applied and Computational Topology. Springer Nature, 2024. https://doi.org/10.1007/s41468-024-00173-w.' ieee: 'R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth in arrangements: Dehn–Sommerville–Euler relations with applications,” Journal of Applied and Computational Topology, vol. 8. Springer Nature, pp. 557–578, 2024.' ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2024. Depth in arrangements: Dehn–Sommerville–Euler relations with applications. Journal of Applied and Computational Topology. 8, 557–578.' mla: 'Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations with Applications.” Journal of Applied and Computational Topology, vol. 8, Springer Nature, 2024, pp. 557–78, doi:10.1007/s41468-024-00173-w.' short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal of Applied and Computational Topology 8 (2024) 557–578. corr_author: '1' date_created: 2024-05-12T22:01:03Z date_published: 2024-09-01T00:00:00Z date_updated: 2025-05-14T09:27:57Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s41468-024-00173-w ec_funded: 1 external_id: pmid: - '39308789' file: - access_level: open_access checksum: 0ee15c1493a6413cf356ab2f32c81a9e content_type: application/pdf creator: dernst date_created: 2025-04-23T08:01:36Z date_updated: 2025-04-23T08:01:36Z file_id: '19612' file_name: 2024_JourApplCompTopo_BiswasRa.pdf file_size: 522831 relation: main_file success: 1 file_date_updated: 2025-04-23T08:01:36Z has_accepted_license: '1' intvolume: ' 8' language: - iso: eng month: '09' oa: 1 oa_version: Published Version page: 557-578 pmid: 1 project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: Journal of Applied and Computational Topology publication_identifier: eissn: - 2367-1734 issn: - 2367-1726 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '11658' relation: earlier_version status: public scopus_import: '1' status: public title: 'Depth in arrangements: Dehn–Sommerville–Euler relations with applications' tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 8 year: '2024' ... --- _id: '17146' abstract: - lang: eng text: The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry. acknowledgement: "The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. {I 02979-N35.} The second author is supported by the European Research Council (ERC), grant \"GeoScape\" and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.\r\nThe authors thank Matt Kahle for communicating the question about extremal Čech complexes, Ben Schweinhart for early discussions on the linked circles construction in three dimensions, and Gábor Tardos for helpful remarks and suggestions." alternative_title: - LIPIcs article_number: '53' article_processing_charge: No arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: János full_name: Pach, János id: E62E3130-B088-11EA-B919-BF823C25FEA4 last_name: Pach citation: ama: 'Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: 40th International Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.53' apa: 'Edelsbrunner, H., & Pach, J. (2024). Maximum Betti numbers of Čech complexes. In 40th International Symposium on Computational Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.53' chicago: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” In 40th International Symposium on Computational Geometry, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.53. ieee: H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” in 40th International Symposium on Computational Geometry, Athens, Greece, 2024, vol. 293. ista: 'Edelsbrunner H, Pach J. 2024. Maximum Betti numbers of Čech complexes. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 53.' mla: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” 40th International Symposium on Computational Geometry, vol. 293, 53, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.53. short: H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. conference: end_date: 2024-06-14 location: Athens, Greece name: 'SoCG: Symposium on Computational Geometry' start_date: 2024-06-11 date_created: 2024-06-16T22:01:06Z date_published: 2024-06-01T00:00:00Z date_updated: 2025-12-01T15:19:20Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.4230/LIPIcs.SoCG.2024.53 ec_funded: 1 external_id: arxiv: - '2310.14801' file: - access_level: open_access checksum: 5442d44fb89d77477a87668d6e61aac9 content_type: application/pdf creator: dernst date_created: 2024-06-17T08:46:33Z date_updated: 2024-06-17T08:46:33Z file_id: '17152' file_name: 2024_LIPICS_Edelsbrunner.pdf file_size: 766562 relation: main_file success: 1 file_date_updated: 2024-06-17T08:46:33Z has_accepted_license: '1' intvolume: ' 293' language: - iso: eng month: '06' oa: 1 oa_version: Published Version project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication: 40th International Symposium on Computational Geometry publication_identifier: isbn: - '9783959773164' issn: - 1868-8969 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' related_material: record: - id: '20657' relation: later_version status: public scopus_import: '1' status: public title: Maximum Betti numbers of Čech complexes tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 293 year: '2024' ... --- _id: '17190' abstract: - lang: eng text: "For a locally finite set, \U0001D434⊆ℝ\U0001D451\r\n, the \U0001D458\r\nth Brillouin zone of \U0001D44E∈\U0001D434\r\n is the region of points \U0001D465∈ℝ\U0001D451\r\n for which ‖\U0001D465−\U0001D44E‖\r\n is the \U0001D458\r\nth smallest among the Euclidean distances between \U0001D465\r\n and the points in \U0001D434\r\n. If \U0001D434\r\n is a lattice, the \U0001D458\r\nth Brillouin zones of the points in \U0001D434\r\n are translates of each other, and together they tile space. Depending on the value of \U0001D458\r\n, they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in ℝ2\r\n, and the convergence of the maximum volume of a chamber to zero for the integer lattice." acknowledgement: The second author is partially supported by the Alexander von Humboldt Foundation. The sixth author is supported by the European Union's Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement 754411, and by Austrian Science Fund(FWF) grant M-3073. All other authors are supported by European Research Council (ERC) grant 788183, by the Wittgenstein Prize, by Austrian Science Fund (FWF) grant Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF) grant I 02979-N35. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Garber, Alexey last_name: Garber - first_name: Mohadese full_name: Ghafaris, Mohadese last_name: Ghafaris - first_name: Teresa full_name: Heiss, Teresa id: 4879BB4E-F248-11E8-B48F-1D18A9856A87 last_name: Heiss orcid: 0000-0002-1780-2689 - first_name: Morteza full_name: Saghafiant, Morteza last_name: Saghafiant - first_name: Mathijs full_name: Wintraecken, Mathijs id: 307CFBC8-F248-11E8-B48F-1D18A9856A87 last_name: Wintraecken orcid: 0000-0002-7472-2220 citation: ama: Edelsbrunner H, Garber A, Ghafaris M, Heiss T, Saghafiant M, Wintraecken M. Brillouin zones of integer lattices and their perturbations. SIAM Journal on Discrete Mathematics. 2024;38(2):1784-1807. doi:10.1137/22M1489071 apa: Edelsbrunner, H., Garber, A., Ghafaris, M., Heiss, T., Saghafiant, M., & Wintraecken, M. (2024). Brillouin zones of integer lattices and their perturbations. SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/22M1489071 chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafaris, Teresa Heiss, Morteza Saghafiant, and Mathijs Wintraecken. “Brillouin Zones of Integer Lattices and Their Perturbations.” SIAM Journal on Discrete Mathematics. Society for Industrial and Applied Mathematics, 2024. https://doi.org/10.1137/22M1489071. ieee: H. Edelsbrunner, A. Garber, M. Ghafaris, T. Heiss, M. Saghafiant, and M. Wintraecken, “Brillouin zones of integer lattices and their perturbations,” SIAM Journal on Discrete Mathematics, vol. 38, no. 2. Society for Industrial and Applied Mathematics, pp. 1784–1807, 2024. ista: Edelsbrunner H, Garber A, Ghafaris M, Heiss T, Saghafiant M, Wintraecken M. 2024. Brillouin zones of integer lattices and their perturbations. SIAM Journal on Discrete Mathematics. 38(2), 1784–1807. mla: Edelsbrunner, Herbert, et al. “Brillouin Zones of Integer Lattices and Their Perturbations.” SIAM Journal on Discrete Mathematics, vol. 38, no. 2, Society for Industrial and Applied Mathematics, 2024, pp. 1784–807, doi:10.1137/22M1489071. short: H. Edelsbrunner, A. Garber, M. Ghafaris, T. Heiss, M. Saghafiant, M. Wintraecken, SIAM Journal on Discrete Mathematics 38 (2024) 1784–1807. corr_author: '1' date_created: 2024-06-30T22:01:05Z date_published: 2024-06-07T00:00:00Z date_updated: 2025-09-08T08:06:04Z day: '07' department: - _id: HeEd doi: 10.1137/22M1489071 ec_funded: 1 external_id: arxiv: - '2204.01077' isi: - '001292728600001' intvolume: ' 38' isi: 1 issue: '2' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2204.01077 month: '06' oa: 1 oa_version: Preprint page: 1784-1807 project: - _id: 260C2330-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '754411' name: ISTplus - Postdoctoral Fellowships - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: fc390959-9c52-11eb-aca3-afa58bd282b2 grant_number: M03073 name: Learning and triangulating manifolds via collapses - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: Mathematics, Computer Science publication: SIAM Journal on Discrete Mathematics publication_identifier: issn: - 0895-4801 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: Brillouin zones of integer lattices and their perturbations type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 38 year: '2024' ... --- OA_place: repository _id: '18673' abstract: - lang: eng text: "Motivated by applications to crystalline materials, we generalize the merge tree and the related barcode of a filtered complex to the periodic setting in Euclidean space. They are invariant under isometries, changing bases, and indeed changing lattices. In addition, we prove stability under perturbations and provide an algorithm that under mild geometric conditions typically satisfied by crystalline materials takes O((n+m)logn) time, in which n and m are the numbers of vertices and edges in the quotient complex, respectively.\r\n" acknowledgement: "Both authors are partially supported by the European Research Council (ERC) Horizon 2020 project\r\n‘Alpha Shape Theory Extended’, grant no. 788183. The first author is also partially supported by the DFG\r\nCollaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund\r\n(FWF), grant no. I 02979-N35." article_processing_charge: No arxiv: 1 author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Teresa full_name: Heiss, Teresa id: 4879BB4E-F248-11E8-B48F-1D18A9856A87 last_name: Heiss orcid: 0000-0002-1780-2689 citation: ama: Edelsbrunner H, Heiss T. Merge trees of periodic filtrations. arXiv. doi:10.48550/arXiv.2408.16575 apa: Edelsbrunner, H., & Heiss, T. (n.d.). Merge trees of periodic filtrations. arXiv. https://doi.org/10.48550/arXiv.2408.16575 chicago: Edelsbrunner, Herbert, and Teresa Heiss. “Merge Trees of Periodic Filtrations.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2408.16575. ieee: H. Edelsbrunner and T. Heiss, “Merge trees of periodic filtrations,” arXiv. . ista: Edelsbrunner H, Heiss T. Merge trees of periodic filtrations. arXiv, 10.48550/arXiv.2408.16575. mla: Edelsbrunner, Herbert, and Teresa Heiss. “Merge Trees of Periodic Filtrations.” ArXiv, doi:10.48550/arXiv.2408.16575. short: H. Edelsbrunner, T. Heiss, ArXiv (n.d.). corr_author: '1' date_created: 2024-12-18T14:06:57Z date_published: 2024-08-29T00:00:00Z date_updated: 2026-04-07T12:54:09Z day: '29' department: - _id: HeEd doi: 10.48550/arXiv.2408.16575 ec_funded: 1 external_id: arxiv: - '2408.16575' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2408.16575 month: '08' oa: 1 oa_version: Preprint project: - _id: 266A2E9E-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '788183' name: Alpha Shape Theory Extended - _id: 2561EBF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: I02979-N35 name: Persistence and stability of geometric complexes publication: arXiv publication_status: draft related_material: record: - id: '18667' relation: dissertation_contains status: public status: public title: Merge trees of periodic filtrations tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ...