{
  "id": "math-ph/0703020",
  "version": "v1",
  "published": "2007-03-05T20:11:05.000Z",
  "updated": "2007-03-05T20:11:05.000Z",
  "title": "On the critical exponent in an isoperimetric inequality for chords",
  "authors": [
    "Pavel Exner",
    "Martin Fraas",
    "Evans M. Harrell II"
  ],
  "comment": "LaTeX, 12 pages, with 1 eps figure",
  "doi": "10.1016/j.physleta.2007.03.067",
  "categories": [
    "math-ph",
    "math.AC",
    "math.MP"
  ],
  "abstract": "The problem of maximizing the $L^p$ norms of chords connecting points on a closed curve separated by arclength $u$ arises in electrostatic and quantum--mechanical problems. It is known that among all closed curves of fixed length, the unique maximizing shape is the circle for $1 \\le p \\le 2$, but this is not the case for sufficiently large values of $p$. Here we determine the critical value $p_c(u)$ of $p$ above which the circle is not a local maximizer finding, in particular, that $p_c(\\frac12 L)=\\frac52$. This corrects a claim made in \\cite{EHL}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-05T20:11:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "isoperimetric inequality",
      "critical exponent",
      "closed curve",
      "chords connecting points",
      "unique maximizing shape"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}