{
  "id": "2305.14155",
  "version": "v1",
  "published": "2023-05-23T15:21:23.000Z",
  "updated": "2023-05-23T15:21:23.000Z",
  "title": "On a Blaschke-Santaló-type inequality for $r$-ball bodies",
  "authors": [
    "Károly Bezdek"
  ],
  "comment": "5 pages. arXiv admin note: text overlap with arXiv:1810.11886",
  "categories": [
    "math.MG"
  ],
  "abstract": "Let ${\\mathbb E}^d$ denote the $d$-dimensional Euclidean space. The $r$-ball body generated by a given set in ${\\mathbb E}^d$ is the intersection of balls of radius $r$ centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke-Santal\\'o-type inequality for $r$-ball bodies: for all $0<k< d$ and for any set of given $d$-dimensional volume in ${\\mathbb E}^d$ the $k$-th intrinsic volume of the $r$-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-23T15:21:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ball body",
      "blaschke-santaló-type inequality",
      "dimensional euclidean space",
      "th intrinsic volume",
      "discrete optimization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}