{
  "id": "2002.11878",
  "version": "v1",
  "published": "2020-02-27T02:25:38.000Z",
  "updated": "2020-02-27T02:25:38.000Z",
  "title": "Subsets of rectifiable curves in Banach spaces: sharp exponents in Schul-type theorems",
  "authors": [
    "Matthew Badger",
    "Sean McCurdy"
  ],
  "comment": "49 pages, 4 figures",
  "categories": [
    "math.CA",
    "math.FA",
    "math.MG"
  ],
  "abstract": "The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space $\\ell_2$ by Schul in 2007. In this paper, we establish sharp extensions of Schul's necessary and sufficient conditions for a bounded set $E\\subset\\ell_p$ to be contained in a rectifiable curve from $p=2$ to $1<p<\\infty$. While the necessary and sufficient conditions coincide when $p=2$, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when $p\\neq 2$. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the Analyst's TSP in general metric spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-27T02:25:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "26A16",
      "28A80",
      "46B20",
      "65D10"
    ],
    "keywords": [
      "rectifiable curve",
      "banach spaces",
      "sharp exponents",
      "schul-type theorems",
      "analysts traveling salesman problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}