{
  "id": "1506.05237",
  "version": "v1",
  "published": "2015-06-17T08:25:52.000Z",
  "updated": "2015-06-17T08:25:52.000Z",
  "title": "Diameter 2 properties and convexity",
  "authors": [
    "Trond A. Abrahamsen",
    "Peter Hájek",
    "Olav Nygaard",
    "Jarno Talponen",
    "Stanimir Troyanski"
  ],
  "comment": "15 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We present an equivalent midpoint locally uniformly rotund (MLUR) renorming $X$ of $C[0,1]$ on which every weakly compact projection $P$ satisfies the equation $\\|I-P\\| = 1+\\|P\\|$ ($I$ is the identity operator on $X$). As a consequence we obtain an MLUR space $X$ with the properties D2P, that every non-empty relatively weakly open subset of its unit ball $B_X$ has diameter 2, and the LD2P+, that for every slice of $B_X$ and every norm 1 element $x$ inside the slice there is another element $y$ inside the slice of distance as close to 2 from $x$ as desired. An example of an MLUR space with the D2P, the LD2P+, and with convex combinations of slices of arbitrary small diameter is also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-17T08:25:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B04",
      "46B20"
    ],
    "keywords": [
      "properties",
      "mlur space",
      "non-empty relatively weakly open subset",
      "arbitrary small diameter",
      "equivalent midpoint locally uniformly rotund"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150605237A"
    }
  }
}