{
  "id": "2310.18960",
  "version": "v1",
  "published": "2023-10-29T10:02:20.000Z",
  "updated": "2023-10-29T10:02:20.000Z",
  "title": "Same average in every direction",
  "authors": [
    "Imre Bárány",
    "Gábor Domokos"
  ],
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "abstract": "Given a polytope $P\\subset R^3$ and a non-zero vector $z \\in R^3$, the plane $\\{x\\in R^3:zx=t\\}$ intersects $P$ in convex polygon $P(z,t)$ for $t \\in [t^-,t^+]$ where $t^-=\\min \\{zx: x \\in P\\}$ and $t^+=\\max \\{xz: x\\in P\\}$, $zx$ is the scalar product of $z,x \\in R^3$. Let $A(P,z)$ denote the average number of vertices of $P(z,t)$ on the interval $[t^-,t^+]$. For what polytopes is $A(P,z)$ a constant independent of $z$?",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-29T10:02:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A15"
    ],
    "keywords": [
      "non-zero vector",
      "convex polygon",
      "scalar product",
      "average number",
      "constant independent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}