{
  "id": "2411.14195",
  "version": "v1",
  "published": "2024-11-21T15:04:58.000Z",
  "updated": "2024-11-21T15:04:58.000Z",
  "title": "On $k$-convex hulls",
  "authors": [
    "Davide Ravasini"
  ],
  "comment": "7 pages",
  "categories": [
    "math.MG"
  ],
  "abstract": "For every integer $k\\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\\subset\\mathbb{R}^n$ with $\\text{diam} Q_{k-1}(K)\\geq R\\cdot\\text{diam} Q_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull of $K$. The purpose of this short note is to show that this result due to E. Kopeck\\'{a} is impossible to obtain if one additionally requires that all isometric images of $K$ satisfy the same inequality. To this end, we introduce the dual construction to the $k$-convex hull of $K$, which we call $k$-cross approximation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-21T15:04:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A20",
      "52A23"
    ],
    "keywords": [
      "convex hull",
      "symmetric convex body",
      "isometric images",
      "short note",
      "cross approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}