{
  "id": "1609.06033",
  "version": "v1",
  "published": "2016-09-20T06:40:18.000Z",
  "updated": "2016-09-20T06:40:18.000Z",
  "title": "Mappings of preserving $n$-distance one in $n$-normed spaces",
  "authors": [
    "Xujian Huang",
    "Dongni Tan"
  ],
  "comment": "11 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We give a positive answer to the Aleksandrov problem in $n$-normed spaces under the surjectivity assumption. Namely, we show that every surjective mapping preserving $n$-distance one is affine, and thus is an $n$-isometry. This is the first time to solve the Aleksandrov problem in $n$-normed spaces with only surjective assumption even in the usual case $n=2$. Finally, when the target space is $n$-strictly convex, we prove that every mapping preserving two $n$-distances with an integer ratio is an affine $n$-isometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-20T06:40:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A03",
      "51K05"
    ],
    "keywords": [
      "normed spaces",
      "aleksandrov problem",
      "integer ratio",
      "first time",
      "usual case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}