{
  "id": "2106.15763",
  "version": "v1",
  "published": "2021-06-30T00:57:54.000Z",
  "updated": "2021-06-30T00:57:54.000Z",
  "title": "Lipschitz mappings, metric differentiability, and factorization through metric trees",
  "authors": [
    "Behnam Esmayli",
    "Piotr Hajłasz"
  ],
  "categories": [
    "math.MG",
    "math.CA"
  ],
  "abstract": "Given a Lipschitz map $f$ from a cube into a metric space, we find several equivalent conditions for $f$ to have a Lipschitz factorization through a metric tree. As an application we prove a recent conjecture of David and Schul. The techniques developed for the proof of the factorization result yield several other new and seemingly unrelated results. We prove that if $f$ is a Lipschitz mapping from an open set in $\\mathbb{R}^n$ onto a metric space $X$, then the topological dimension of $X$ equals $n$ if and only if $X$ has positive $n$-dimensional Hausdorff measure. We also prove an area formula for length-preserving maps between metric spaces, which gives, in particular, a new formula for integration on countably rectifiable sets in the Heisenberg group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-30T00:57:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "30L99",
      "51F30",
      "28A78",
      "53C17",
      "53C23",
      "54F45",
      "54F50"
    ],
    "keywords": [
      "lipschitz map",
      "metric tree",
      "metric differentiability",
      "metric space",
      "dimensional hausdorff measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}