{
  "id": "1901.02956",
  "version": "v1",
  "published": "2019-01-09T22:31:59.000Z",
  "updated": "2019-01-09T22:31:59.000Z",
  "title": "The Bishop--Phelps--Bollobás property for Lipschitz maps",
  "authors": [
    "Rafael Chiclana",
    "Miguel Martin"
  ],
  "comment": "28 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollob\\'as property (Lip-BPB property). This property deals with the possibility to make a uniformly simultaneous approximation of a Lipschitz map $F$ and a pair of points at which $F$ almost attains its norm by a Lipschitz map $G$ and a pair of points such that $G$ strongly attains its norm at the new pair of points. We first show that if $M$ is a finite pointed metric space and $Y$ is a finite-dimensional Banach space, then the pair $(M,Y)$ has the Lip-BPB property, and that both finiteness are needed. Next, we show that if $M$ is a uniformly Gromov concave pointed metric space (i.e.\\ the molecules of $M$ form a set of uniformly strongly exposed points), then $(M,Y)$ has the Lip-BPB property for every Banach space $Y$. We further prove that this is the case of finite concave metric spaces, ultrametric spaces, and H\\\"older metric spaces. The extension of the Lip-BPB property from $(M,\\mathbb{R})$ to some Banach spaces $Y$, the relationship with absolute sums, and some results only valid for compact Lipschitz maps, are also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-09T22:31:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B04",
      "46B20",
      "26A16",
      "54E50"
    ],
    "keywords": [
      "lip-bpb property",
      "bishop-phelps-bollobás property",
      "banach space",
      "gromov concave pointed metric space",
      "finite concave metric spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}