--- _id: '7952' abstract: - lang: eng text: "Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f: ℝ^d → ℝ^(d-n). A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation \U0001D4AF of the ambient space ℝ^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation \U0001D4AF. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary. " alternative_title: - LIPIcs article_number: 20:1-20:18 article_processing_charge: No author: - first_name: Jean-Daniel full_name: Boissonnat, Jean-Daniel last_name: Boissonnat - first_name: Mathijs full_name: Wintraecken, Mathijs id: 307CFBC8-F248-11E8-B48F-1D18A9856A87 last_name: Wintraecken orcid: 0000-0002-7472-2220 citation: ama: 'Boissonnat J-D, Wintraecken M. The topological correctness of PL-approximations of isomanifolds. In: 36th International Symposium on Computational Geometry. Vol 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:10.4230/LIPIcs.SoCG.2020.20' apa: 'Boissonnat, J.-D., & Wintraecken, M. (2020). The topological correctness of PL-approximations of isomanifolds. In 36th International Symposium on Computational Geometry (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2020.20' chicago: Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL-Approximations of Isomanifolds.” In 36th International Symposium on Computational Geometry, Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. https://doi.org/10.4230/LIPIcs.SoCG.2020.20. ieee: J.-D. Boissonnat and M. Wintraecken, “The topological correctness of PL-approximations of isomanifolds,” in 36th International Symposium on Computational Geometry, Zürich, Switzerland, 2020, vol. 164. ista: 'Boissonnat J-D, Wintraecken M. 2020. The topological correctness of PL-approximations of isomanifolds. 36th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 164, 20:1-20:18.' mla: Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL-Approximations of Isomanifolds.” 36th International Symposium on Computational Geometry, vol. 164, 20:1-20:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020, doi:10.4230/LIPIcs.SoCG.2020.20. short: J.-D. Boissonnat, M. Wintraecken, in:, 36th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. conference: end_date: 2020-06-26 location: Zürich, Switzerland name: 'SoCG: Symposium on Computational Geometry' start_date: 2020-06-22 date_created: 2020-06-09T07:24:11Z date_published: 2020-06-01T00:00:00Z date_updated: 2023-08-02T06:49:16Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.4230/LIPIcs.SoCG.2020.20 ec_funded: 1 file: - access_level: open_access checksum: 38cbfa4f5d484d267a35d44d210df044 content_type: application/pdf creator: dernst date_created: 2020-06-17T10:13:34Z date_updated: 2020-07-14T12:48:06Z file_id: '7969' file_name: 2020_LIPIcsSoCG_Boissonnat.pdf file_size: 1009739 relation: main_file file_date_updated: 2020-07-14T12:48:06Z has_accepted_license: '1' intvolume: ' 164' language: - iso: eng month: '06' oa: 1 oa_version: Published Version project: - _id: 260C2330-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '754411' name: ISTplus - Postdoctoral Fellowships publication: 36th International Symposium on Computational Geometry publication_identifier: isbn: - 978-3-95977-143-6 issn: - 1868-8969 publication_status: published publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik quality_controlled: '1' related_material: record: - id: '9649' relation: later_version status: public scopus_import: '1' status: public title: The topological correctness of PL-approximations of isomanifolds 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: 164 year: '2020' ... --- _id: '74' abstract: - lang: eng text: "We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about \ the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily \ radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class\r\nof compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2.\r\nWe use a simpler form of Gromov’s pancake argument \ to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian\r\nmeasures." article_processing_charge: No author: - first_name: Arseniy full_name: Akopyan, Arseniy id: 430D2C90-F248-11E8-B48F-1D18A9856A87 last_name: Akopyan orcid: 0000-0002-2548-617X - first_name: Roman full_name: Karasev, Roman last_name: Karasev citation: ama: 'Akopyan A, Karasev R. Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In: Klartag B, Milman E, eds. Geometric Aspects of Functional Analysis. Vol 2256. LNM. Springer Nature; 2020:1-27. doi:10.1007/978-3-030-36020-7_1' apa: Akopyan, A., & Karasev, R. (2020). Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In B. Klartag & E. Milman (Eds.), Geometric Aspects of Functional Analysis (Vol. 2256, pp. 1–27). Springer Nature. https://doi.org/10.1007/978-3-030-36020-7_1 chicago: Akopyan, Arseniy, and Roman Karasev. “Gromov’s Waist of Non-Radial Gaussian Measures and Radial Non-Gaussian Measures.” In Geometric Aspects of Functional Analysis, edited by Bo’az Klartag and Emanuel Milman, 2256:1–27. LNM. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-36020-7_1. ieee: A. Akopyan and R. Karasev, “Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures,” in Geometric Aspects of Functional Analysis, vol. 2256, B. Klartag and E. Milman, Eds. Springer Nature, 2020, pp. 1–27. ista: 'Akopyan A, Karasev R. 2020.Gromov’s waist of non-radial Gaussian measures and radial non-Gaussian measures. In: Geometric Aspects of Functional Analysis. vol. 2256, 1–27.' mla: Akopyan, Arseniy, and Roman Karasev. “Gromov’s Waist of Non-Radial Gaussian Measures and Radial Non-Gaussian Measures.” Geometric Aspects of Functional Analysis, edited by Bo’az Klartag and Emanuel Milman, vol. 2256, Springer Nature, 2020, pp. 1–27, doi:10.1007/978-3-030-36020-7_1. short: A. Akopyan, R. Karasev, in:, B. Klartag, E. Milman (Eds.), Geometric Aspects of Functional Analysis, Springer Nature, 2020, pp. 1–27. date_created: 2018-12-11T11:44:29Z date_published: 2020-06-21T00:00:00Z date_updated: 2023-08-17T13:48:31Z day: '21' department: - _id: HeEd - _id: JaMa doi: 10.1007/978-3-030-36020-7_1 ec_funded: 1 editor: - first_name: Bo'az full_name: Klartag, Bo'az last_name: Klartag - first_name: Emanuel full_name: Milman, Emanuel last_name: Milman external_id: arxiv: - '1808.07350' isi: - '000557689300003' intvolume: ' 2256' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1808.07350 month: '06' oa: 1 oa_version: Preprint page: 1-27 project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics publication: Geometric Aspects of Functional Analysis publication_identifier: eisbn: - '9783030360207' eissn: - '16179692' isbn: - '9783030360191' issn: - '00758434' publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' series_title: LNM status: public title: Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures type: book_chapter user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 2256 year: '2020' ... --- _id: '7554' abstract: - lang: eng text: Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in ${R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in ${R}^n$. article_processing_charge: No 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: 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. Weighted Poisson–Delaunay mosaics. Theory of Probability and its Applications. 2020;64(4):595-614. doi:10.1137/S0040585X97T989726 apa: Edelsbrunner, H., & Nikitenko, A. (2020). Weighted Poisson–Delaunay mosaics. Theory of Probability and Its Applications. SIAM. https://doi.org/10.1137/S0040585X97T989726 chicago: Edelsbrunner, Herbert, and Anton Nikitenko. “Weighted Poisson–Delaunay Mosaics.” Theory of Probability and Its Applications. SIAM, 2020. https://doi.org/10.1137/S0040585X97T989726. ieee: H. Edelsbrunner and A. Nikitenko, “Weighted Poisson–Delaunay mosaics,” Theory of Probability and its Applications, vol. 64, no. 4. SIAM, pp. 595–614, 2020. ista: Edelsbrunner H, Nikitenko A. 2020. Weighted Poisson–Delaunay mosaics. Theory of Probability and its Applications. 64(4), 595–614. mla: Edelsbrunner, Herbert, and Anton Nikitenko. “Weighted Poisson–Delaunay Mosaics.” Theory of Probability and Its Applications, vol. 64, no. 4, SIAM, 2020, pp. 595–614, doi:10.1137/S0040585X97T989726. short: H. Edelsbrunner, A. Nikitenko, Theory of Probability and Its Applications 64 (2020) 595–614. date_created: 2020-03-01T23:00:39Z date_published: 2020-02-13T00:00:00Z date_updated: 2023-08-18T06:45:48Z day: '13' department: - _id: HeEd doi: 10.1137/S0040585X97T989726 ec_funded: 1 external_id: arxiv: - '1705.08735' isi: - '000551393100007' intvolume: ' 64' isi: 1 issue: '4' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1705.08735 month: '02' oa: 1 oa_version: Preprint page: 595-614 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: Theory of Probability and its Applications publication_identifier: eissn: - '10957219' issn: - 0040585X publication_status: published publisher: SIAM quality_controlled: '1' scopus_import: '1' status: public title: Weighted Poisson–Delaunay mosaics type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 64 year: '2020' ... --- _id: '7666' abstract: - lang: eng text: Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups. acknowledgement: This project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). It is also partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF). article_processing_charge: Yes (via OA deal) 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 citation: ama: Edelsbrunner H, Ölsböck K. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 2020;64:759-775. doi:10.1007/s00454-020-00188-x apa: Edelsbrunner, H., & Ölsböck, K. (2020). Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00188-x chicago: Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00188-x. ieee: H. Edelsbrunner and K. Ölsböck, “Tri-partitions and bases of an ordered complex,” Discrete and Computational Geometry, vol. 64. Springer Nature, pp. 759–775, 2020. ista: Edelsbrunner H, Ölsböck K. 2020. Tri-partitions and bases of an ordered complex. Discrete and Computational Geometry. 64, 759–775. mla: Edelsbrunner, Herbert, and Katharina Ölsböck. “Tri-Partitions and Bases of an Ordered Complex.” Discrete and Computational Geometry, vol. 64, Springer Nature, 2020, pp. 759–75, doi:10.1007/s00454-020-00188-x. short: H. Edelsbrunner, K. Ölsböck, Discrete and Computational Geometry 64 (2020) 759–775. date_created: 2020-04-19T22:00:56Z date_published: 2020-03-20T00:00:00Z date_updated: 2023-08-21T06:13:48Z day: '20' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-020-00188-x ec_funded: 1 external_id: isi: - '000520918800001' file: - access_level: open_access checksum: f8cc96e497f00c38340b5dafe0cb91d7 content_type: application/pdf creator: dernst date_created: 2020-11-20T13:22:21Z date_updated: 2020-11-20T13:22:21Z file_id: '8786' file_name: 2020_DiscreteCompGeo_Edelsbrunner.pdf file_size: 701673 relation: main_file success: 1 file_date_updated: 2020-11-20T13:22:21Z has_accepted_license: '1' intvolume: ' 64' isi: 1 language: - iso: eng month: '03' oa: 1 oa_version: Published Version page: 759-775 project: - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund - _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: Discrete and Computational Geometry publication_identifier: eissn: - '14320444' issn: - '01795376' publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Tri-partitions and bases of an ordered complex 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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 64 year: '2020' ... --- _id: '7962' abstract: - lang: eng text: 'A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets.' article_processing_charge: No article_type: original author: - first_name: János full_name: Pach, János id: E62E3130-B088-11EA-B919-BF823C25FEA4 last_name: Pach - first_name: Bruce full_name: Reed, Bruce last_name: Reed - first_name: Yelena full_name: Yuditsky, Yelena last_name: Yuditsky citation: ama: Pach J, Reed B, Yuditsky Y. Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. 2020;63(4):888-917. doi:10.1007/s00454-020-00213-z apa: Pach, J., Reed, B., & Yuditsky, Y. (2020). Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00213-z chicago: Pach, János, Bruce Reed, and Yelena Yuditsky. “Almost All String Graphs Are Intersection Graphs of Plane Convex Sets.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00213-z. ieee: J. Pach, B. Reed, and Y. Yuditsky, “Almost all string graphs are intersection graphs of plane convex sets,” Discrete and Computational Geometry, vol. 63, no. 4. Springer Nature, pp. 888–917, 2020. ista: Pach J, Reed B, Yuditsky Y. 2020. Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. 63(4), 888–917. mla: Pach, János, et al. “Almost All String Graphs Are Intersection Graphs of Plane Convex Sets.” Discrete and Computational Geometry, vol. 63, no. 4, Springer Nature, 2020, pp. 888–917, doi:10.1007/s00454-020-00213-z. short: J. Pach, B. Reed, Y. Yuditsky, Discrete and Computational Geometry 63 (2020) 888–917. date_created: 2020-06-14T22:00:51Z date_published: 2020-06-05T00:00:00Z date_updated: 2023-08-21T08:49:18Z day: '05' department: - _id: HeEd doi: 10.1007/s00454-020-00213-z external_id: arxiv: - '1803.06710' isi: - '000538229000001' intvolume: ' 63' isi: 1 issue: '4' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1803.06710 month: '06' oa: 1 oa_version: Preprint page: 888-917 project: - _id: 268116B8-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: Z00342 name: The Wittgenstein Prize publication: Discrete and Computational Geometry publication_identifier: eissn: - '14320444' issn: - '01795376' publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Almost all string graphs are intersection graphs of plane convex sets type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 63 year: '2020' ...