{
  "id": "math/0101262",
  "version": "v2",
  "published": "2001-01-31T20:43:55.000Z",
  "updated": "2004-05-10T20:24:56.000Z",
  "title": "A general theory of almost convex functions",
  "authors": [
    "S. J. Dilworth",
    "Ralph Howard",
    "James W. Roberts"
  ],
  "comment": "40 pages with 5 postscript figures. Minor errors and typographical errors corrected",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\Delta_m$ be the standard $m$-dimensional simplex of non-negative $m+1$ tuples that sum to unity and let $S$ be a nonempty subset of $\\Delta_m$. A real valued function $h$ defined on a convex subset of a real vector space is $S$-almost convex iff for all $(t_0,...,t_m)\\in S$ and $x_0,...,x_m\\in C$ the inequality h(t_0 x_0+ ... +t_m x_m)\\leq 1+ t_0 h(x_0)+ ... +t_m h(x_m) holds. A detailed study of the properties of $S$-almost convex functions is made, including the constriction of the extremal (i.e. pointwise largest bounded) $S$-almost convex function on simplices that vanishes on the vertices. In the special case that $S$ is the barycenter of $\\Delta_m$ very explicit formulas are given for the extremal function and its maximum. This is of interest as the extremal function and its maximum give the best constants in various geometric and analytic inequalities and theorems.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-05-10T20:24:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26B25",
      "52A27",
      "39B72",
      "41A44",
      "51M16",
      "52A21"
    ],
    "keywords": [
      "convex function",
      "general theory",
      "extremal function",
      "real vector space",
      "nonempty subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......1262D"
    }
  }
}