{
  "id": "1804.08212",
  "version": "v1",
  "published": "2018-04-23T01:18:39.000Z",
  "updated": "2018-04-23T01:18:39.000Z",
  "title": "On the Banach-Mazur distance to cross-polytope",
  "authors": [
    "Konstantin Tikhomirov"
  ],
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "abstract": "Let $n\\geq 3$, and let $B_1^n$ be the standard $n$-dimensional cross-polytope (i.e. the convex hull of standard coordinate vectors and their negatives). We show that there exists a symmetric convex body $\\mathcal G_m$ in ${\\mathbb R}^n$ such that the Banach--Mazur distance $d(B_1^n,\\mathcal G_m)$ satisfies $d(B_1^n,\\mathcal G_m)\\geq n^{5/9}\\log^{-C}n$, where $C>0$ is a universal constant. The body $\\mathcal G_m$ is obtained as a typical realization of a random polytope in ${\\mathbb R}^n$ with $2m:=2n^C$ vertices (for a large constant $C$). The result improves upon an earlier estimate of S.Szarek which gives $d(B_1^n,\\mathcal G_m)\\geq c n^{1/2}\\log n$ (with a different choice of $m$). This shows in a strong sense that the cross-polytope (or the cube $[-1,1]^n$) cannot be an \"approximate\" center of the Minkowski compactum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-23T01:18:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "banach-mazur distance",
      "standard coordinate vectors",
      "symmetric convex body",
      "dimensional cross-polytope",
      "convex hull"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}