{
  "id": "1307.3806",
  "version": "v4",
  "published": "2013-07-15T02:06:13.000Z",
  "updated": "2013-08-01T20:38:32.000Z",
  "title": "A necessary and sufficient condition on the stability of the infimum of convex functions",
  "authors": [
    "Iosif Pinelis"
  ],
  "comment": "8 pages. Version 2: two references added; discussion expanded, including now Example 6. Version 3: discussion further expanded, including now a 5-line proof in the case when f and f_n are real-valued. Version 4: Theorem 9 is added about the equivalence of the sequential and topological variants of the infimum-stability",
  "categories": [
    "math.OC",
    "math.CA"
  ],
  "abstract": "Let us say that a convex function f\\colon C\\to[-\\infty,\\infty] on a convex set C\\subseteq\\R is infimum-stable if, for any sequence (f_n) of convex functions f_n\\colon C\\to[-\\infty,\\infty] converging to f pointwise, one has \\inf_C f_n\\to\\inf_C f. A simple necessary and sufficient condition for a convex function to be infimum-stable is given. The same condition remains necessary and sufficient if one uses Moore--Smith nets (f_\\nu) in place of sequences (f_n). This note is motivated by certain applications to stability of measures of risk/inequality in finance/economics.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-08-01T20:38:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A51",
      "90C25",
      "49J45",
      "49K05",
      "49K30"
    ],
    "keywords": [
      "convex function",
      "sufficient condition",
      "condition remains necessary",
      "moore-smith nets",
      "simple necessary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.3806P"
    }
  }
}