{"id":"https://openalex.org/W1976668598","doi":"https://doi.org/10.1016/s0195-6698(81)80037-6","title":"The Numbers of Faces of Polytope Pairs and Unbounded Polyhedra","display_name":"The Numbers of Faces of Polytope Pairs and Unbounded Polyhedra","publication_year":1981,"publication_date":"1981-12-01","ids":{"openalex":"https://openalex.org/W1976668598","doi":"https://doi.org/10.1016/s0195-6698(81)80037-6","mag":"1976668598"},"language":"en","primary_location":{"id":"doi:10.1016/s0195-6698(81)80037-6","is_oa":false,"landing_page_url":"https://doi.org/10.1016/s0195-6698(81)80037-6","pdf_url":null,"source":{"id":"https://openalex.org/S204851967","display_name":"European Journal of Combinatorics","issn_l":"0195-6698","issn":["0195-6698","1095-9971"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"European Journal of Combinatorics","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/A5008530739","display_name":"Louis J. Billera","orcid":null},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Louis J. Billera","raw_affiliation_strings":["School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853, U.S.A"],"affiliations":[{"raw_affiliation_string":"School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853, U.S.A","institution_ids":["https://openalex.org/I205783295"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5017349412","display_name":"Carl W. Lee","orcid":null},"institutions":[{"id":"https://openalex.org/I143302722","display_name":"University of Kentucky","ror":"https://ror.org/02k3smh20","country_code":"US","type":"education","lineage":["https://openalex.org/I143302722"]},{"id":"https://openalex.org/I4210114115","display_name":"IBM Research - Thomas J. Watson Research Center","ror":"https://ror.org/0265w5591","country_code":"US","type":"facility","lineage":["https://openalex.org/I1341412227","https://openalex.org/I4210114115"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Carl W. Lee","raw_affiliation_strings":["Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, U.S.A","IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, U.S.A"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, U.S.A","institution_ids":["https://openalex.org/I143302722"]},{"raw_affiliation_string":"IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, U.S.A","institution_ids":["https://openalex.org/I4210114115"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5008530739"],"corresponding_institution_ids":["https://openalex.org/I205783295"],"apc_list":{"value":2880,"currency":"USD","value_usd":2880},"apc_paid":null,"fwci":2.6654,"has_fulltext":false,"cited_by_count":27,"citation_normalized_percentile":{"value":0.9047619,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"2","issue":"4","first_page":"307","last_page":"322"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9998000264167786,"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/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11476","display_name":"Graph theory and applications","score":0.987500011920929,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8906658887863159},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8672862648963928},{"id":"https://openalex.org/keywords/polytope","display_name":"Polytope","score":0.7933588624000549},{"id":"https://openalex.org/keywords/polyhedron","display_name":"Polyhedron","score":0.7345824241638184},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6495466828346252},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.5973142981529236},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5903952717781067},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5753876566886902},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.4436001777648926},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.19527167081832886}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8906658887863159},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8672862648963928},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.7933588624000549},{"id":"https://openalex.org/C54829058","wikidata":"https://www.wikidata.org/wiki/Q172937","display_name":"Polyhedron","level":2,"score":0.7345824241638184},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6495466828346252},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.5973142981529236},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5903952717781067},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5753876566886902},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4436001777648926},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.19527167081832886},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1016/s0195-6698(81)80037-6","is_oa":false,"landing_page_url":"https://doi.org/10.1016/s0195-6698(81)80037-6","pdf_url":null,"source":{"id":"https://openalex.org/S204851967","display_name":"European Journal of Combinatorics","issn_l":"0195-6698","issn":["0195-6698","1095-9971"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"European Journal of Combinatorics","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":24,"referenced_works":["https://openalex.org/W30254118","https://openalex.org/W569838548","https://openalex.org/W1546959834","https://openalex.org/W1579678295","https://openalex.org/W1966068899","https://openalex.org/W1996522633","https://openalex.org/W2018645515","https://openalex.org/W2041186059","https://openalex.org/W2045645047","https://openalex.org/W2057551462","https://openalex.org/W2061450130","https://openalex.org/W2063531543","https://openalex.org/W2076724178","https://openalex.org/W2134654723","https://openalex.org/W2143716978","https://openalex.org/W2144564022","https://openalex.org/W2145201379","https://openalex.org/W2148642357","https://openalex.org/W2218040560","https://openalex.org/W4285719527","https://openalex.org/W6616224346","https://openalex.org/W6634722365","https://openalex.org/W6681417330","https://openalex.org/W6688946424"],"related_works":["https://openalex.org/W2916043681","https://openalex.org/W2896131632","https://openalex.org/W4280529077","https://openalex.org/W2391438543","https://openalex.org/W2008231869","https://openalex.org/W4297950433","https://openalex.org/W2558235681","https://openalex.org/W4299416706","https://openalex.org/W175452553","https://openalex.org/W4312417404"],"abstract_inverted_index":null,"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-03-10T14:07:55.174380","created_date":"2025-10-10T00:00:00"}