{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T22:35:37Z","timestamp":1773268537058,"version":"3.50.1"},"reference-count":19,"publisher":"Elsevier BV","issue":"3","license":[{"start":{"date-parts":[[1981,11,1]],"date-time":"1981-11-01T00:00:00Z","timestamp":373420800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2013,7,17]],"date-time":"2013-07-17T00:00:00Z","timestamp":1374019200000},"content-version":"vor","delay-in-days":11581,"URL":"https:\/\/www.elsevier.com\/open-access\/userlicense\/1.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Combinatorial Theory, Series A"],"published-print":{"date-parts":[[1981,11]]},"DOI":"10.1016\/0097-3165(81)90058-3","type":"journal-article","created":{"date-parts":[[2005,2,10]],"date-time":"2005-02-10T04:49:18Z","timestamp":1108010958000},"page":"237-255","source":"Crossref","is-referenced-by-count":140,"title":["A proof of the sufficiency of McMullen's conditions for f-vectors of simplicial convex polytopes"],"prefix":"10.1016","volume":"31","author":[{"given":"Louis J","family":"Billera","sequence":"first","affiliation":[]},{"given":"Carl W","family":"Lee","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/0097-3165(81)90058-3_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\/0097-3165(81)90058-3_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\/0097-3165(81)90058-3_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. Amer. Math. Soc. (New Series)"},{"key":"10.1016\/0097-3165(81)90058-3_BIB4","doi-asserted-by":"crossref","unstructured":"L. J. Billera and C. W. Lee, The numbers of faces of polytope pairs and unbounded polyhedra, European J. Combinatorics, in press.","DOI":"10.1016\/S0195-6698(81)80037-6"},{"key":"10.1016\/0097-3165(81)90058-3_BIB5","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1090\/S0273-0979-1981-14877-1","article-title":"The unimodality conjecture for convex polytopes","volume":"4","author":"Bj\u00f6rner","year":"1981","journal-title":"Bull. Amer. Math. Soc. (New Series)"},{"key":"10.1016\/0097-3165(81)90058-3_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\/0097-3165(81)90058-3_BIB7","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/S0167-5060(08)70320-0","article-title":"Which spheres are shellable?","volume":"2","author":"Danaraj","year":"1978","journal-title":"Ann. Discrete Math."},{"key":"10.1016\/0097-3165(81)90058-3_BIB8","series-title":"Convex Polytopes","author":"Gr\u00fcnbaum","year":"1967"},{"key":"10.1016\/0097-3165(81)90058-3_BIB9","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\/0097-3165(81)90058-3_BIB10","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1016\/0097-3165(72)90068-4","article-title":"Spheres with few vertices","volume":"13","author":"Mani","year":"1972","journal-title":"J. Combinatorial Theory Ser. A"},{"key":"10.1016\/0097-3165(81)90058-3_BIB11","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1112\/S0025579300002850","article-title":"The maximum number of faces of a convex polytope","volume":"17","author":"McMullen","year":"1970","journal-title":"Mathematika"},{"key":"10.1016\/0097-3165(81)90058-3_BIB12","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1007\/BF02771471","article-title":"The numbers of faces of simplicial polytopes","volume":"9","author":"McMullen","year":"1971","journal-title":"Israel J. Math."},{"key":"10.1016\/0097-3165(81)90058-3_BIB13","doi-asserted-by":"crossref","DOI":"10.1016\/0095-8956(71)90042-6","article-title":"Convex Polytopes and the Upper Bound Conjecture","author":"McMullen","year":"1971"},{"key":"10.1016\/0097-3165(81)90058-3_BIB14","doi-asserted-by":"crossref","first-page":"246","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\/0097-3165(81)90058-3_BIB15","series-title":"Algebraic Topology","author":"Spanier","year":"1966"},{"key":"10.1016\/0097-3165(81)90058-3_BIB16","series-title":"Higher Combinatorics","first-page":"51","article-title":"Cohen-Macaulay complexes","author":"Stanley","year":"1977"},{"key":"10.1016\/0097-3165(81)90058-3_BIB17","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":"Advan. Math."},{"key":"10.1016\/0097-3165(81)90058-3_BIB18","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":"Advan. Math."},{"key":"10.1016\/0097-3165(81)90058-3_BIB19","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/BF02392331","article-title":"The lower bound conjecture for 3- and 4-manifolds","volume":"125","author":"Walkup","year":"1970","journal-title":"Acta Math."}],"container-title":["Journal of Combinatorial Theory, Series A"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:0097316581900583?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:0097316581900583?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2019,1,29]],"date-time":"2019-01-29T08:55:35Z","timestamp":1548752135000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/0097316581900583"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1981,11]]},"references-count":19,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1981,11]]}},"alternative-id":["0097316581900583"],"URL":"https:\/\/doi.org\/10.1016\/0097-3165(81)90058-3","relation":{},"ISSN":["0097-3165"],"issn-type":[{"value":"0097-3165","type":"print"}],"subject":[],"published":{"date-parts":[[1981,11]]}}}