{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T12:10:19Z","timestamp":1778242219549,"version":"3.51.4"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We establish a lower bound theorem for the number of $k$-faces ($1\\le k\\le d-2$) in a $d$-dimensional polytope $P$ (abbreviated as a $d$-polytope) with $2d+2$ vertices, building on the known case for $k=1$. There are two distinct lower bounds depending on the number of facets in the $d$-polytope. We identify all minimisers for $d\\le 5$. If $P$ has $d+2$ facets, the lower bound is tight when $d$ is odd. For $d\\ge 5$ and $P$ with at least $d+3$ facets, the lower bound is always tight. Moreover, for $1\\le k\\le \\lceil d\/3\\rceil-2$, minimisers among $d$-polytopes with $2d+2$ vertices are those with at least $d+3$ facets, while for $\\lfloor 0.4d\\rfloor\\le k\\le d-1$, the lower bound arises from $d$-polytopes with $d+2$ facets. These results support Pineda-Villavicencio's lower bound conjecture for $d$-polytopes with at most $3d-1$ vertices.<\/jats:p>","DOI":"10.37236\/13433","type":"journal-article","created":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T11:31:06Z","timestamp":1778239866000},"source":"Crossref","is-referenced-by-count":0,"title":["A Lower Bound Theorem for $d$-Polytopes with $2d+2$ Vertices"],"prefix":"10.37236","volume":"33","author":[{"given":"Guillermo","family":"Pineda-Villavicencio","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aholiab","family":"Tritama","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jie","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Yost","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2026,5,8]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i2p26\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i2p26\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T11:31:06Z","timestamp":1778239866000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i2p26"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,8]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,4,14]]}},"URL":"https:\/\/doi.org\/10.37236\/13433","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,8]]},"article-number":"P2.26"}}