{
  "id": "math/0204171",
  "version": "v3",
  "published": "2002-04-13T05:03:39.000Z",
  "updated": "2002-04-24T20:01:25.000Z",
  "title": "The Beckman-Quarles theorem for continuous mappings from R^n to C^n",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "9 pages, added proofs of technical lemmas",
  "categories": [
    "math.MG"
  ],
  "abstract": "Let \\phi((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say that f:R^n -> C^n preserves distance d>=0 if for each x,y \\in R^n \\phi(x,y)=d^2 implies \\phi(f(x),f(y))=d^2. We prove that if x,y \\in R^n (n>=3) and |x-y|=(\\sqrt{2+2/n})^k \\cdot (2/n)^l (k,l are non-negative integers) then there exists a finite set {x,y} \\subseteq S(x,y) \\subseteq R^n such that each unit-distance preserving mapping from S(x,y) to C^n preserves the distance between x and y. It implies that each continuous map from R^n to C^n (n>=3) preserving unit distance preserves all distances.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2002-04-24T20:01:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51M05"
    ],
    "keywords": [
      "beckman-quarles theorem",
      "continuous mappings",
      "preserving unit distance preserves",
      "preserves distance",
      "finite set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......4171T"
    }
  }
}