{
  "id": "1907.06122",
  "version": "v1",
  "published": "2019-07-13T20:16:39.000Z",
  "updated": "2019-07-13T20:16:39.000Z",
  "title": "Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions",
  "authors": [
    "Thomas Beck",
    "Barbara Brandolini",
    "Krzysztof Burdzy",
    "Antoine Henrot",
    "Jeffrey J. Langford",
    "Simon Larson",
    "Robert G. Smits",
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.CA",
    "math.FA",
    "math.MG"
  ],
  "abstract": "Let $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and let $f:\\Omega \\rightarrow \\mathbb{R}$ be a positive, subharmonic function (i.e. $\\Delta f \\geq 0$). Then $$ \\frac{1}{|\\Omega|} \\int_{\\Omega}{f dx} \\leq \\frac{c_n}{ |\\partial \\Omega| } \\int_{\\partial \\Omega}{ f d\\sigma},$$ where $c_n \\leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \\geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ \\Omega_2 \\subset \\Omega_1 \\subset \\mathbb{R}^n$: $$ \\frac{|\\partial \\Omega_1|}{|\\Omega_1|} \\frac{| \\Omega_2|}{|\\partial \\Omega_2|} \\leq n.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-13T20:16:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher dimensions",
      "hermite-hadamard inequalities",
      "convex domain",
      "optimal constant satisfies",
      "sharp geometric inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}