{"id":"https://openalex.org/W135008847","doi":"https://doi.org/10.1007/bf01204726","title":"Note dual polytopes of rational convex polytopes","display_name":"Note dual polytopes of rational convex polytopes","publication_year":1992,"publication_date":"1992-06-01","ids":{"openalex":"https://openalex.org/W135008847","doi":"https://doi.org/10.1007/bf01204726","mag":"135008847"},"language":"en","primary_location":{"id":"doi:10.1007/bf01204726","is_oa":false,"landing_page_url":"https://doi.org/10.1007/bf01204726","pdf_url":null,"source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005668319","display_name":"Takayuki Hibi","orcid":"https://orcid.org/0000-0002-7619-1152"},"institutions":[{"id":"https://openalex.org/I205349734","display_name":"Hokkaido University","ror":"https://ror.org/02e16g702","country_code":"JP","type":"education","lineage":["https://openalex.org/I205349734"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Takayuki Hibi","raw_affiliation_strings":["Department of Mathematics Faculty of Science, Hokkaido University, Kita-ku, Sapporo 060, Japan","Department of Mathematics Faculty of Science Hokkaido University Sapporo 060, Japan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics Faculty of Science, Hokkaido University, Kita-ku, Sapporo 060, Japan","institution_ids":["https://openalex.org/I205349734"]},{"raw_affiliation_string":"Department of Mathematics Faculty of Science Hokkaido University Sapporo 060, Japan","institution_ids":["https://openalex.org/I205349734"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5005668319"],"corresponding_institution_ids":["https://openalex.org/I205349734"],"apc_list":null,"apc_paid":null,"fwci":1.533,"has_fulltext":false,"cited_by_count":139,"citation_normalized_percentile":{"value":0.78986587,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":100},"biblio":{"volume":"12","issue":"2","first_page":"237","last_page":"240"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9955000281333923,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9955000281333923,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9843000173568726,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9452999830245972,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/polytope","display_name":"Polytope","score":0.94115149974823},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.813620388507843},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7557007074356079},{"id":"https://openalex.org/keywords/dual","display_name":"Dual (grammatical number)","score":0.7179731726646423},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.6657332181930542},{"id":"https://openalex.org/keywords/enumeration","display_name":"Enumeration","score":0.5823734402656555},{"id":"https://openalex.org/keywords/convex-polytope","display_name":"Convex polytope","score":0.4161243140697479},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3543124198913574},{"id":"https://openalex.org/keywords/convex-analysis","display_name":"Convex analysis","score":0.2577298581600189},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.13408157229423523},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07428035140037537}],"concepts":[{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.94115149974823},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.813620388507843},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7557007074356079},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.7179731726646423},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.6657332181930542},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.5823734402656555},{"id":"https://openalex.org/C110943637","wikidata":"https://www.wikidata.org/wiki/Q4129097","display_name":"Convex polytope","level":5,"score":0.4161243140697479},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3543124198913574},{"id":"https://openalex.org/C12108790","wikidata":"https://www.wikidata.org/wiki/Q2234833","display_name":"Convex analysis","level":4,"score":0.2577298581600189},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.13408157229423523},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07428035140037537},{"id":"https://openalex.org/C124952713","wikidata":"https://www.wikidata.org/wiki/Q8242","display_name":"Literature","level":1,"score":0.0},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/bf01204726","is_oa":false,"landing_page_url":"https://doi.org/10.1007/bf01204726","pdf_url":null,"source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":2,"referenced_works":["https://openalex.org/W2914659449","https://openalex.org/W4300302225"],"related_works":["https://openalex.org/W4287022798","https://openalex.org/W2901329027","https://openalex.org/W1990911204","https://openalex.org/W2069107464","https://openalex.org/W2768257653","https://openalex.org/W2557288725","https://openalex.org/W3012067785","https://openalex.org/W4243575619","https://openalex.org/W1607687497","https://openalex.org/W2045384787"],"abstract_inverted_index":null,"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":10},{"year":2020,"cited_by_count":15},{"year":2019,"cited_by_count":9},{"year":2018,"cited_by_count":7},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":9},{"year":2015,"cited_by_count":19},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
