--- _id: '10773' abstract: - lang: eng text: The Voronoi tessellation in Rd is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. Discretizing the two piecewise quadratic functions, we get the alpha shapes as sublevel sets of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel sets of the discrete function on the Voronoi tessellation. For the same non-critical value, the corresponding shapes are disjoint, separated by a narrow channel that contains no critical points but the entire level set of the piecewise linear function. acknowledgement: Open access funding provided by the Institute of Science and Technology (IST Austria). 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 last_name: Saghafian citation: ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. 2022;67:811-842. doi:10.1007/s00454-022-00371-2 apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M. (2022). Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-022-00371-2 chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. “Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics.” Discrete and Computational Geometry. Springer Nature, 2022. https://doi.org/10.1007/s00454-022-00371-2. ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics,” Discrete and Computational Geometry, vol. 67. Springer Nature, pp. 811–842, 2022. ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2022. Continuous and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete and Computational Geometry. 67, 811–842. mla: Biswas, Ranita, et al. “Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics.” Discrete and Computational Geometry, vol. 67, Springer Nature, 2022, pp. 811–42, doi:10.1007/s00454-022-00371-2. short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Discrete and Computational Geometry 67 (2022) 811–842. date_created: 2022-02-20T23:01:34Z date_published: 2022-04-01T00:00:00Z date_updated: 2023-08-02T14:31:25Z day: '01' ddc: - '510' department: - _id: HeEd doi: 10.1007/s00454-022-00371-2 external_id: isi: - '000752175300002' file: - access_level: open_access checksum: 9383d3b70561bacee905e335dc922680 content_type: application/pdf creator: dernst date_created: 2022-08-02T06:07:55Z date_updated: 2022-08-02T06:07:55Z file_id: '11718' file_name: 2022_DiscreteCompGeometry_Biswas.pdf file_size: 2518111 relation: main_file success: 1 file_date_updated: 2022-08-02T06:07:55Z has_accepted_license: '1' intvolume: ' 67' isi: 1 language: - iso: eng month: '04' oa: 1 oa_version: Published Version page: 811-842 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: Continuous and discrete radius functions on Voronoi tessellations and 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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 67 year: '2022' ...