{
  "id": "2401.12112",
  "version": "v1",
  "published": "2024-01-22T16:50:23.000Z",
  "updated": "2024-01-22T16:50:23.000Z",
  "title": "A quantitative version of the Steinhaus theorem",
  "authors": [
    "Alex Iosevich",
    "Jonathan Pakianathan"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "The classical Steinhaus theorem (\\cite{Steinhaus1920}) says that if $A \\subset {\\Bbb R}^d$ has positive Lebesgue measure than $A-A=\\{x-y: x,y \\in A\\}$ contains an open ball. We obtain some quantitative lower bounds on the size of this ball and in some cases, relate it to natural geometric properties of $\\partial A$. We also study the process $K_n =\\frac{1}{2}(K_{n-1} - K_{n-1})$ when $K_0$ is a compact subset of $\\mathbb{R}^d$ and determine various aspects of its convergence to $Conv(K_1)$, the convex hull of $K_1$. We discuss some connections with convex geometry, Weyl tube formula and the Kakeya needle problem. \\noindent {\\it Keywords: Measure theory, Steinhaus theorem, Convex geometry, Weyl tube formula.} \\noindent 2020 {\\it Mathematics Subject Classification:} Primary: 28A75, 52A27. Secondary: 52A30, 53A07.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-22T16:50:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "52A27",
      "52A30",
      "53A07"
    ],
    "keywords": [
      "steinhaus theorem",
      "quantitative version",
      "weyl tube formula",
      "convex geometry",
      "kakeya needle problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}