{
  "id": "1912.08017",
  "version": "v1",
  "published": "2019-12-17T13:51:07.000Z",
  "updated": "2019-12-17T13:51:07.000Z",
  "title": "Coefficients of the solid angle and Ehrhart quasi-polynomials",
  "authors": [
    "Fabrício Caluza Machado",
    "Sinai Robins"
  ],
  "comment": "35 pages, 5 figures",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "abstract": "Given a rational polytope $P$, we give a local formula for the codimension two quasi-coefficient of the solid angle sum of $P$, which is one version of a discretized volume of $P$. This local formula is valid for all positive real dilates of a rational polytope in $\\mathbb R^d$. As a consequence, we show that the classical Ehrhart polynomial, which is another discretized volume of $P$, also has a similar local formula for the codimension one and codimension two quasi-coefficients. The formula for the codimension one quasi-coefficient extends the classical interpretation of half the sum of the relative volumes of the facets, valid for integer dilations, to a formula valid for real dilations. Some of the present methods offer a further development for the initial approach of Diaz, Le, and Robins.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-17T13:51:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C07",
      "26B20",
      "52B20",
      "52C22"
    ],
    "keywords": [
      "ehrhart quasi-polynomials",
      "coefficients",
      "codimension",
      "rational polytope",
      "similar local formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}