{
  "id": "1806.01197",
  "version": "v1",
  "published": "2018-06-04T16:39:33.000Z",
  "updated": "2018-06-04T16:39:33.000Z",
  "title": "Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem",
  "authors": [
    "Matthew Badger",
    "Lisa Naples",
    "Vyron Vellis"
  ],
  "comment": "70 pages, 4 figures",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a $(1/s)$-H\\\"older continuous map $f:[0,1]\\rightarrow l^2$, with $s>1$. Our results are motivated by and generalize the \"sufficient half\" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in $\\mathbb{R}^N$ or $l^2$ in terms of a quadratic sum of linear approximation numbers called Jones' beta numbers. The original proof of the Analyst's Traveling Salesman Theorem depends on a well-known metric characterization of rectifiable curves from the 1920s, which is not available for higher-dimensional curves such as H\\\"older curves. To overcome this obstacle, we reimagine Jones' non-parametric proof and show how to construct parameterizations of the intermediate approximating curves $f_k([0,1])$. We then find conditions in terms of tube approximations that ensure the approximating curves converge to a H\\\"older curve. As an application, we provide sufficient conditions to guarantee fractional rectifiability of pointwise doubling measures in $\\mathbb{R}^N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-04T16:39:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "26A16",
      "28A80",
      "30L05",
      "65D10"
    ],
    "keywords": [
      "analysts traveling salesman theorem",
      "hölder curves",
      "parameterizations",
      "approximating curves",
      "well-known metric characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 70,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}