{
  "id": "2309.08364",
  "version": "v1",
  "published": "2023-09-15T12:44:48.000Z",
  "updated": "2023-09-15T12:44:48.000Z",
  "title": "On some isoperimetric inequalities for the Newtonian capacity",
  "authors": [
    "Michiel van den Berg"
  ],
  "comment": "13 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "It is shown that (i) for non-empty, compact, convex sets in $\\R^d,\\,d\\ge 3$ with a $C^2$ boundary the Newtonian capacity is bounded from above by $(d-2)M(K)$, where $M(K)>0$ is the integral of the mean curvature over the boundary of $K$ with equality if $K$ is a ball, (ii) for compact, convex sets in $\\R^d,\\,d\\ge 3$ with non-empty interior the Newtonian capacity is bounded from above by $\\frac{(d-2)P(K)^2}{d|K|}$ with equality if $K$ is a ball. Here $P(K)$ is the perimeter of $K$ and $|K|$ is its measure. A quantitative refinement of the latter inequality in terms of the Fraenkel asymmetry is also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-15T12:44:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "newtonian capacity",
      "isoperimetric inequalities",
      "inequality",
      "convex sets",
      "fraenkel asymmetry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}