{
  "id": "math/9612218",
  "version": "v1",
  "published": "1996-12-11T00:00:00.000Z",
  "updated": "1996-12-11T00:00:00.000Z",
  "title": "The number of faces of a simple polytope",
  "authors": [
    "Anders Björner",
    "Svante Linusson"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Consider the question: Given integers $k<d<n$, does there exist a simple $d$-polytope with $n$ faces of dimension $k$? We show that there exist numbers $G(d,k)$ and $N(d,k)$ such that for $n> N(d,k)$ the answer is yes if and only if $n\\equiv 0\\quad \\pmod {G(d,k)}$. Furthermore, a formula for $G(d,k)$ is given, showing that e.g. $G(d,k)=1$ if $k\\ge \\left\\lfloor\\frac{d+1}{2}\\right\\rfloor$ or if both $d$ and $k$ are even, and also in some other cases (meaning that all numbers beyond $N(d,k)$ occur as the number of $k$-faces of some simple $d$-polytope). This question has previously been studied only for the case of vertices ($k=0$), where Lee \\cite{Le} proved the existence of $N(d,0)$ (with $G(d,0)=1$ or $2$ depending on whether $d$ is even or odd), and Prabhu \\cite{P2} showed that $N(d,0) \\le cd\\sqrt {d}$. We show here that asymptotically the true value of Prabhu's constant is $c=\\sqrt2$ if $d$ is even, and $c=1$ if $d$ is odd.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1996-12-11T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple polytope",
      "true value",
      "prabhus constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1996math.....12218B"
    }
  }
}