{
  "id": "math/0508097",
  "version": "v5",
  "published": "2005-08-04T18:39:23.000Z",
  "updated": "2006-03-20T19:41:29.000Z",
  "title": "Lipschitz extension constants equal projection constants",
  "authors": [
    "Marc A. Rieffel"
  ],
  "comment": "16 pages. Three very minor mathematical typos corrected. Intended for the proceedings of GPOTS05",
  "journal": "Contemporary Math. 414 (2006) 147-162",
  "categories": [
    "math.FA",
    "math.MG"
  ],
  "abstract": "For a Banach space $V$ we define its Lipschitz extension constant, $\\cL\\cE(V)$, to be the infimum of the constants $c$ such that for every metric space $(Z,\\rho)$, every $X \\subset Z$, and every $f: X \\to V$, there is an extension, $g$, of $f$ to $Z$ such that $L(g) \\le cL(f)$, where $L$ denotes the Lipschitz constant. The basic theorem is that when $V$ is finite-dimensional we have $\\cL\\cE(V) = \\cP\\cC(V)$ where $\\cP\\cC(V)$ is the well-known projection constant of $V$. We obtain some direct consequences of this theorem, especially when $V = M_n(\\bC)$. We then apply techniques for calculating projection constants, involving averaging projections, to calculate $\\cL\\cE((M_n(\\bC))^{sa})$. We also discuss what happens if we also require that $\\|g\\|_{\\infty} = \\|f\\|_{\\infty}$.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2006-03-20T19:41:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "26A16"
    ],
    "keywords": [
      "lipschitz extension constants equal projection",
      "extension constants equal projection constants",
      "well-known projection constant",
      "metric space",
      "lipschitz constant"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8097R"
    }
  }
}