{
  "id": "2005.13839",
  "version": "v1",
  "published": "2020-05-28T08:29:48.000Z",
  "updated": "2020-05-28T08:29:48.000Z",
  "title": "Estimating the average of functions with convexity properties by means of a new center",
  "authors": [
    "Bernardo González Merino"
  ],
  "comment": "20 pages",
  "categories": [
    "math.FA",
    "math.MG"
  ],
  "abstract": "In this article we show the following result: if $C$ is an n-dimensional convex and compact set, $f:C\\rightarrow[0,\\infty)$ is concave, and $\\phi:[0,\\infty)\\rightarrow[0,\\infty)$ is a convex function with $\\phi(0)=0$, then \\[ \\int_C\\phi(f(x))dx \\leq \\max_{R}\\int_R\\phi(g(x))dx. \\] The maximum above ranges over the set of truncated cones $R$ with the same volume than $C$ and $g$ is an affine function which becomes zero at one of the bases of $R$ with $g(x_{R,g})=f(x_{C,f})$. Moreover, given such $C$ and $f$, we introduce a new point $x_{C,f}\\in C$, which divides evenly the mass distribution of $C$. As a consequence, we obtain a generalization of the type of results [Thm. 1.2, GM] and [Lem. 1.1, MP]. Besides, we also obtain some new estimates on the volume of particular sections of a convex set passing through $x_{C,f}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-28T08:29:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convexity properties",
      "compact set",
      "n-dimensional convex",
      "convex function",
      "estimating"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}