{"id":"https://openalex.org/W2895110763","doi":"https://doi.org/10.3390/sym10100447","title":"A Note on the Minimum Size of a Point Set Containing Three Nonintersecting Empty Convex Polygons","display_name":"A Note on the Minimum Size of a Point Set Containing Three Nonintersecting Empty Convex Polygons","publication_year":2018,"publication_date":"2018-09-29","ids":{"openalex":"https://openalex.org/W2895110763","doi":"https://doi.org/10.3390/sym10100447","mag":"2895110763"},"language":"en","primary_location":{"id":"doi:10.3390/sym10100447","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10100447","pdf_url":"https://www.mdpi.com/2073-8994/10/10/447/pdf?version=1538191017","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/10/10/447/pdf?version=1538191017","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5007428030","display_name":"Qing Yang","orcid":"https://orcid.org/0000-0002-0683-5848"},"institutions":[{"id":"https://openalex.org/I170751988","display_name":"Shanxi University of Finance and Economics","ror":"https://ror.org/04nte7y58","country_code":"CN","type":"education","lineage":["https://openalex.org/I170751988"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Qing Yang","raw_affiliation_strings":["Faculty of Accounting, Shanxi University of Finance and Economics, Taiyuan 030006, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Accounting, Shanxi University of Finance and Economics, Taiyuan 030006, China","institution_ids":["https://openalex.org/I170751988"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087629036","display_name":"Zengtai You","orcid":null},"institutions":[{"id":"https://openalex.org/I61565387","display_name":"Dalian Minzu University","ror":"https://ror.org/02hxfx521","country_code":"CN","type":"education","lineage":["https://openalex.org/I61565387"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zengtai You","raw_affiliation_strings":["College of Computer Science and Engineering, Dalian Minzu University, Dalian 116600, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Computer Science and Engineering, Dalian Minzu University, Dalian 116600, China","institution_ids":["https://openalex.org/I61565387"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041496400","display_name":"Xinshang You","orcid":null},"institutions":[{"id":"https://openalex.org/I80143920","display_name":"Shandong University of Science and Technology","ror":"https://ror.org/04gtjhw98","country_code":"CN","type":"education","lineage":["https://openalex.org/I80143920"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Xinshang You","raw_affiliation_strings":["College of Economics and Management, Shandong University of Science and Technology, Qingdao 266590, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Economics and Management, Shandong University of Science and Technology, Qingdao 266590, China","institution_ids":["https://openalex.org/I80143920"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5041496400"],"corresponding_institution_ids":["https://openalex.org/I80143920"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.15336345,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"10","issue":"10","first_page":"447","last_page":"447"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12923","display_name":"Digital Image Processing Techniques","score":0.988099992275238,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10719","display_name":"3D Shape Modeling and Analysis","score":0.9843000173568726,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8128677606582642},{"id":"https://openalex.org/keywords/convex-polygon","display_name":"Convex polygon","score":0.7280452847480774},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.7005454301834106},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6787192821502686},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.6417105197906494},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.637752890586853},{"id":"https://openalex.org/keywords/polygon","display_name":"Polygon (computer graphics)","score":0.5508570075035095},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.5441447496414185},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4921674132347107},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.4667789041996002},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4492841064929962},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.3408935070037842},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.11760947108268738},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10266679525375366}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8128677606582642},{"id":"https://openalex.org/C2780034980","wikidata":"https://www.wikidata.org/wiki/Q3259505","display_name":"Convex polygon","level":3,"score":0.7280452847480774},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.7005454301834106},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6787192821502686},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.6417105197906494},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.637752890586853},{"id":"https://openalex.org/C190694206","wikidata":"https://www.wikidata.org/wiki/Q3276654","display_name":"Polygon (computer graphics)","level":3,"score":0.5508570075035095},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.5441447496414185},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4921674132347107},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.4667789041996002},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4492841064929962},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3408935070037842},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.11760947108268738},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10266679525375366},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0},{"id":"https://openalex.org/C126042441","wikidata":"https://www.wikidata.org/wiki/Q1324888","display_name":"Frame (networking)","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym10100447","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10100447","pdf_url":"https://www.mdpi.com/2073-8994/10/10/447/pdf?version=1538191017","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:c357b0e6518a4c82a55a658c83251bad","is_oa":true,"landing_page_url":"https://doaj.org/article/c357b0e6518a4c82a55a658c83251bad","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 10, Iss 10, p 447 (2018)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym10100447","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10100447","pdf_url":"https://www.mdpi.com/2073-8994/10/10/447/pdf?version=1538191017","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4048898866","display_name":null,"funder_award_id":"18CGL018","funder_id":"https://openalex.org/F4320327557","funder_display_name":"National Office for Philosophy and Social Sciences"}],"funders":[{"id":"https://openalex.org/F4320327557","display_name":"National Office for Philosophy and Social Sciences","ror":"https://ror.org/04m0ms912"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2895110763.pdf","grobid_xml":"https://content.openalex.org/works/W2895110763.grobid-xml"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W1487872891","https://openalex.org/W1488784459","https://openalex.org/W1561962265","https://openalex.org/W1798441839","https://openalex.org/W1977266879","https://openalex.org/W1981051536","https://openalex.org/W2007033927","https://openalex.org/W2018024429","https://openalex.org/W2032581171","https://openalex.org/W2054769584","https://openalex.org/W2072795940","https://openalex.org/W2131974367","https://openalex.org/W2152525037","https://openalex.org/W2164902489","https://openalex.org/W2299568790","https://openalex.org/W2402697736","https://openalex.org/W2889282505","https://openalex.org/W2895100662","https://openalex.org/W6679616306","https://openalex.org/W6713239059"],"related_works":["https://openalex.org/W2072510267","https://openalex.org/W2314484939","https://openalex.org/W4240339429","https://openalex.org/W2154141401","https://openalex.org/W338234595","https://openalex.org/W2123516473","https://openalex.org/W1819380083","https://openalex.org/W4300013361","https://openalex.org/W2950477787","https://openalex.org/W1963890352"],"abstract_inverted_index":{"Let":[0],"P":[1,15,20],"be":[2,16],"a":[3,17,29,61,63,66],"planar":[4,47],"point":[5],"set":[6,49],"with":[7,55],"no":[8,56],"three":[9,57],"points":[10,13,24,33,48,54,58],"collinear,":[11,59],"k":[12,23],"of":[14,19,28,34],"k-hole":[18],"if":[21],"the":[22,26,41],"are":[25,69],"vertices":[27],"convex":[30],"polygon":[31],"without":[32],"P.":[35],"This":[36],"article":[37],"proves":[38],"13":[39,53],"is":[40],"smallest":[42],"integer":[43],"such":[44],"that":[45],"any":[46],"containing":[50],"at":[51],"least":[52],"contains":[60],"3-hole,":[62],"4-hole":[64],"and":[65],"5-hole":[67],"which":[68],"pairwise":[70],"disjoint.":[71]},"counts_by_year":[],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}
