{
  "id": "2409.14294",
  "version": "v1",
  "published": "2024-09-22T02:06:40.000Z",
  "updated": "2024-09-22T02:06:40.000Z",
  "title": "A lower bound theorem for $d$-polytopes with $2d+2$ vertices",
  "authors": [
    "Guillermo Pineda-Villavicencio",
    "Aholiab Tritama",
    "David Yost"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "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 $1\\le k\\le {d/3}-2$, minimisers among $d$-polytopes with $2d+2$ vertices are those with at least $d+3$ facets, while for ${0.4d}\\le k\\le d-1$, 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-22T02:06:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B05"
    ],
    "keywords": [
      "lower bound theorem",
      "support pineda-villavicencios lower bound conjecture",
      "results support pineda-villavicencios lower bound",
      "distinct lower bounds",
      "lower bound arises"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}