{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T18:30:06Z","timestamp":1771353006128,"version":"3.50.1"},"reference-count":21,"publisher":"Elsevier BV","issue":"4","license":[{"start":{"date-parts":[[1981,12,1]],"date-time":"1981-12-01T00:00:00Z","timestamp":376012800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2013,9,13]],"date-time":"2013-09-13T00:00:00Z","timestamp":1379030400000},"content-version":"vor","delay-in-days":11609,"URL":"https:\/\/www.elsevier.com\/open-access\/userlicense\/1.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["European Journal of Combinatorics"],"published-print":{"date-parts":[[1981,12]]},"DOI":"10.1016\/s0195-6698(81)80037-6","type":"journal-article","created":{"date-parts":[[2013,9,12]],"date-time":"2013-09-12T12:16:45Z","timestamp":1378988205000},"page":"307-322","source":"Crossref","is-referenced-by-count":16,"title":["The Numbers of Faces of Polytope Pairs and Unbounded Polyhedra"],"prefix":"10.1016","volume":"2","author":[{"given":"Louis J.","family":"Billera","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carl W.","family":"Lee","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"key":"10.1016\/S0195-6698(81)80037-6_bib1","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/BF02771522","article-title":"The minimum number of vertices of a simple polytope","volume":"10","author":"Barnette","year":"1971","journal-title":"Israel J. Math."},{"key":"10.1016\/S0195-6698(81)80037-6_bib2","doi-asserted-by":"crossref","first-page":"349","DOI":"10.2140\/pjm.1973.46.349","article-title":"A proof of the lower bound conjecture for convex polytopes","volume":"46","author":"Barnette","year":"1973","journal-title":"Pacific J. Math."},{"key":"10.1016\/S0195-6698(81)80037-6_bib3","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1090\/S0273-0979-1980-14712-6","article-title":"Sufficiency of McMullen's conditions for f-vectors of simplicial polytopes","volume":"2","author":"Billera","year":"1980","journal-title":"Bull. (New Series) Amer. Math. Soc."},{"key":"10.1016\/S0195-6698(81)80037-6_bib4","doi-asserted-by":"crossref","unstructured":"L. J. Billera and C. W. Lee, A proof of the sufficiency of McMullen's conditions for f -vectors of simplicial convex polytopes, J. Combin. Theory Ser. A (to appear).","DOI":"10.1016\/0097-3165(81)90058-3"},{"key":"10.1016\/S0195-6698(81)80037-6_bib5","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/S0195-6698(80)80017-5","article-title":"The minimum number of faces of a simple polyhedron","volume":"1","author":"Bj\u00f6rner","year":"1980","journal-title":"Eur. J. Combin."},{"key":"10.1016\/S0195-6698(81)80037-6_bib6","doi-asserted-by":"crossref","first-page":"197","DOI":"10.7146\/math.scand.a-11045","article-title":"Shellable decompositions of cells and spheres","volume":"29","author":"Bruggesser","year":"1971","journal-title":"Math. Scand."},{"key":"10.1016\/S0195-6698(81)80037-6_bib7","series-title":"Convex Polytopes","author":"Gr\u00f6nbaum","year":"1967"},{"key":"10.1016\/S0195-6698(81)80037-6_bib8","doi-asserted-by":"crossref","first-page":"517","DOI":"10.4153\/CJM-1964-053-0","article-title":"A combinatorial analogue of Poincar\u00e9's duality theorem","volume":"16","author":"Klee","year":"1964","journal-title":"Canad. J. Math."},{"key":"10.1016\/S0195-6698(81)80037-6_bib9","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1007\/BF02162086","article-title":"A comparison of primal and dual methods for linear programming","volume":"9","author":"Klee","year":"1966","journal-title":"Numer. Math."},{"key":"10.1016\/S0195-6698(81)80037-6_bib10","first-page":"485","article-title":"Convex polyhedra and mathematical programming","volume":"Vol. 1","author":"Klee","year":"1974"},{"key":"10.1016\/S0195-6698(81)80037-6_bib11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02392139","article-title":"Polytope pairs and their relationship to linear programming","volume":"133","author":"Klee","year":"1974","journal-title":"Acta Math."},{"key":"10.1016\/S0195-6698(81)80037-6_bib12","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/BF02395040","article-title":"The d-step conjecture for polyhedra of dimension d < 6","volume":"117","author":"Klee","year":"1967","journal-title":"Acta Math."},{"key":"10.1016\/S0195-6698(81)80037-6_bib13","series-title":"Ph.D. Thesis","article-title":"Counting the faces of simplicial convex polytopes","author":"Lee","year":"1981"},{"key":"10.1016\/S0195-6698(81)80037-6_bib14","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1112\/S0025579300002850","article-title":"The maximum numbers of faces of a convex polytope","volume":"17","author":"McMullen","year":"1970","journal-title":"Mathematika"},{"key":"10.1016\/S0195-6698(81)80037-6_bib15","series-title":"London Math. Soc. Lecture Note Series 3","article-title":"Convex polytopes and the upper bound conjecture","author":"McMullen","year":"1971"},{"key":"10.1016\/S0195-6698(81)80037-6_bib16","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1112\/S0025579300005520","article-title":"A generalized lower-bound conjecture for simplicial polytopes","volume":"18","author":"McMullen","year":"1971","journal-title":"Mathematika"},{"key":"10.1016\/S0195-6698(81)80037-6_bib17","series-title":"Tech. Rep. No. 402","article-title":"Simplicial complexes associated with convex polyhedra, I: Constructions and combinatorial examples","author":"Provan","year":"1979"},{"key":"10.1016\/S0195-6698(81)80037-6_bib18","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1002\/sapm1975542135","article-title":"The upper bound conjecture and Cohen-Macaulay rings","volume":"54","author":"Stanley","year":"1975","journal-title":"Stud. Appl. Math."},{"key":"10.1016\/S0195-6698(81)80037-6_bib19","series-title":"Higher Combinatorics","article-title":"Cohen-Macaulay complexes","author":"Stanley","year":"1977"},{"key":"10.1016\/S0195-6698(81)80037-6_bib20","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/0001-8708(78)90045-2","article-title":"Hilbert functions of graded algebras","volume":"28","author":"Stanley","year":"1978","journal-title":"Adv. in Math."},{"key":"10.1016\/S0195-6698(81)80037-6_bib21","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/0001-8708(80)90050-X","article-title":"The number of faces of a simplicial convex polytope","volume":"35","author":"Stanley","year":"1980","journal-title":"Adv. in Math."}],"container-title":["European Journal of Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0195669881800376?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0195669881800376?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2018,10,13]],"date-time":"2018-10-13T00:53:34Z","timestamp":1539392014000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0195669881800376"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1981,12]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1981,12]]}},"alternative-id":["S0195669881800376"],"URL":"https:\/\/doi.org\/10.1016\/s0195-6698(81)80037-6","relation":{},"ISSN":["0195-6698"],"issn-type":[{"value":"0195-6698","type":"print"}],"subject":[],"published":{"date-parts":[[1981,12]]}}}