{
  "id": "2002.04191",
  "version": "v1",
  "published": "2020-02-11T03:52:05.000Z",
  "updated": "2020-02-11T03:52:05.000Z",
  "title": "On the Area Bounded by the Curve $\\prod_{k = 1}^n |x\\sin(kπ/n) - y\\cos(kπ/n)| = 1$",
  "authors": [
    "Anton Mosunov"
  ],
  "comment": "4 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In 2000 Bean and Laugesen proved that for every integer $n \\geq 3$ the area bounded by the curve $$\\prod\\limits_{k = 1}^n\\left|x\\sin\\left(\\frac{k\\pi}{n}\\right) -y\\cos\\left(\\frac{k\\pi}{n}\\right)\\right| = 1 $$ is equal to $4^{1 - 1/n}B\\left(\\frac{1}{2} - \\frac{1}{n}, \\frac{1}{2}\\right)$, where $B(x, y)$ is the beta function. We provide an elementary proof of this fact based on the polar formula for the area calculation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-11T03:52:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "beta function",
      "elementary proof",
      "polar formula",
      "area calculation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}