{
  "id": "1610.05642",
  "version": "v1",
  "published": "2016-10-18T14:35:27.000Z",
  "updated": "2016-10-18T14:35:27.000Z",
  "title": "Weak Compactness and Fixed Point Property for Affine Bi-Lipschitz Maps",
  "authors": [
    "C. S. Barroso",
    "V. Ferreira"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper we show that if $(y_n)$ is a seminormalized sequence in a Banach space which does not have any weakly convergent subsequence, then it contains a wide-$(s)$ subsequence $(x_n)$ which admits an equivalent convex basic sequence. This fact is used to characterize weak-compactness of bounded, closed convex sets in terms of the generic fixed point property ($\\mathcal{G}$-$FPP$) for the class of affine bi-Lipschitz maps. This result generalizes a theorem by Benavides, Jap\\'on Pineda and Prus previously proved for the class of continuous maps. We also introduce a relaxation of this notion ($\\mathcal{WG}$-$FPP$) and observe that a closed convex bounded subset of a Banach space is weakly compact iff it has the $\\mathcal{WG}$-$FPP$ for affine $1$-Lipschitz maps. Related results are also proved. For example, a complete convex bounded subset $C$ of a Hlcs $X$ is weakly compact iff it has the $\\mathcal{G}$-$FPP$ for the class of affine continuous maps $f\\colon C\\to X$ with weak-approximate fixed point nets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-18T14:35:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "affine bi-lipschitz maps",
      "fixed point property",
      "weak compactness",
      "equivalent convex basic sequence",
      "banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}