{
  "id": "2010.11814",
  "version": "v1",
  "published": "2020-10-22T15:49:20.000Z",
  "updated": "2020-10-22T15:49:20.000Z",
  "title": "Bounds for the $p$-angular distance and characterizations of inner product spaces",
  "authors": [
    "Mario Krnic",
    "Nicusor Minculete"
  ],
  "comment": "accepted for publication in Mediterranean Journal of Mathematics",
  "categories": [
    "math.FA"
  ],
  "abstract": "Based on a suitable improvement of a triangle inequality, we derive new mutual bounds for $p$-angular distance $\\alpha_p[x,y]=\\big\\Vert \\Vert x\\Vert^{p-1}x- \\Vert y\\Vert^{p-1}y\\big\\Vert$, in a normed linear space $X$. We show that our estimates are more accurate than the previously known upper bounds established by Dragomir, Hile and Maligranda. Next, we give several characterizations of inner product spaces with regard to the $p$-angular distance. In particular, we prove that if $|p|\\geq |q|$, $p\\neq q$, then $X$ is an inner product space if and only if for every $x,y\\in X\\setminus \\{0\\}$, $${\\alpha_p[x,y]}\\geq \\frac{{\\|x\\|^{p}+\\|y\\|^{p} }}{\\|x\\|^{q}+\\|y\\|^{q} }\\alpha_q[x,y].$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-22T15:49:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A30",
      "46C15",
      "26D15"
    ],
    "keywords": [
      "inner product space",
      "angular distance",
      "characterizations",
      "mutual bounds",
      "normed linear space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}