{
  "id": "1808.07794",
  "version": "v1",
  "published": "2018-08-23T15:10:52.000Z",
  "updated": "2018-08-23T15:10:52.000Z",
  "title": "The Hermite-Hadamard inequality in higher dimensions",
  "authors": [
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "The Hermite-Hadamard inequality states that the average value of a convex function on an interval is bounded from above by the average value of the function at the endpoints of the interval. We provide a generalization to higher dimensions: let $\\Omega \\subset \\mathbb{R}^n$ be a convex domain and let $f:\\Omega \\rightarrow \\mathbb{R}$ be a convex function, then $$ \\frac{1}{|\\Omega|} \\int_{\\Omega}{f ~d \\mathcal{H}^n} \\leq \\frac{n^{n+1}}{|\\partial \\Omega|} \\int_{\\partial \\Omega}{f~d \\mathcal{H}^{n-1}}.$$ The constant $n^{n+1}$ is presumably far from optimal, however, it cannot be replaced by 1 in general. We also show, for some universal constant $c>0$, if $\\Omega \\subset \\mathbb{R}^2$ is simply connected with smooth boundary, $f:\\Omega \\rightarrow \\mathbb{R}_{}$ is subharmonic, i.e. $\\Delta f \\geq 0$, and $f \\big|_{\\partial \\Omega} \\geq 0$, then $$ \\int_{\\Omega}{f~ d \\mathcal{H}^2} \\leq c \\cdot \\mbox{inradius}(\\Omega) \\int_{\\partial \\Omega}{ f ~d\\mathcal{H}^{1}}.$$ We also prove that every domain $\\Omega \\subset \\mathbb{R}^n$ whose boundary is 'flat' at a certain scale $\\delta$ admits a Hermite-Hadamard inequality for all subharmonic functions with a constant depending only on the dimension, the measure $|\\Omega|$ and the scale $\\delta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-23T15:10:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher dimensions",
      "average value",
      "convex function",
      "hermite-hadamard inequality states",
      "subharmonic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}