{
  "id": "1910.06146",
  "version": "v1",
  "published": "2019-10-14T13:59:29.000Z",
  "updated": "2019-10-14T13:59:29.000Z",
  "title": "Volume of the Minkowski sums of star-shaped sets",
  "authors": [
    "Matthieu Fradelizi",
    "Zsolt Lángi",
    "Artem Zvavitch"
  ],
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "abstract": "For a compact set $A \\subset {\\mathbb R}^d$ and an integer $k\\ge1$, let us denote by $$ A[k] = \\left\\{a_1+\\cdots +a_k: a_1, \\ldots, a_k\\in A\\right\\}=\\sum_{i=1}^k A$$ the Minkowski sum of $k$ copies of $A$. A theorem of Shapley, Folkmann and Starr (1969) states that $\\frac{1}{k}A[k]$ converges to the convex hull of $A$ in Hausdorff distance as $k$ tends to infinity. Bobkov, Madiman and Wang (2011) conjectured that the volume of $\\frac{1}{k}A[k]$ is non-decreasing in $k$ , or in other words, in terms of the volume deficit between the convex hull of $A$ and $\\frac{1}{k}A[k]$, this convergence is monotone. It was proved by Fradelizi, Madiman, Marsiglietti and Zvavitch (2016) that this conjecture holds true if $d=1$ but fails for any $d \\geq 12$. In this paper we show that the conjecture is true for any star-shaped set $A \\subset {\\mathbb R}^d$ for arbitrary dimensions $d \\ge 1$ under the condition $k \\ge d-1$. In addition, we investigate the conjecture for connected sets and present a counterexample to a generalization of the conjecture to the Minkowski sum of possibly distinct sets in ${\\mathbb R}^d$, for any $d \\geq 7$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-14T13:59:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A40",
      "52A38",
      "60E15"
    ],
    "keywords": [
      "minkowski sum",
      "star-shaped set",
      "convex hull",
      "conjecture holds true",
      "arbitrary dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}