{
  "id": "2406.13774",
  "version": "v1",
  "published": "2024-06-19T18:59:31.000Z",
  "updated": "2024-06-19T18:59:31.000Z",
  "title": "Chessboard and level sets of continuous functions",
  "authors": [
    "Michał Dybowski",
    "Przemysław Górka"
  ],
  "comment": "19 pages",
  "categories": [
    "math.GN",
    "math.CO"
  ],
  "abstract": "We show the following result: Let $f \\colon I^n \\to \\mathbb{R}^{n-1}$ be a continuous function. Then, there exist $p \\in \\mathbb{R}^{n-1}$ and compact subset $S \\subset f^{-1}\\left[\\left\\{p\\right\\}\\right]$ which connects some opposite faces of the $n$-dimensional unit cube $I^n$. We give an example that shows it cannot be generalized to path-connected sets. We also provide a discrete version of this result which is inspired by the $n$-dimensional Steinhaus Chessboard Theorem. Additionally, we show that the latter one and the Brouwer Fixed Point Theorem are simple consequences of the main result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-19T18:59:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D05",
      "54C05",
      "05C15",
      "51M99"
    ],
    "keywords": [
      "continuous function",
      "level sets",
      "dimensional steinhaus chessboard theorem",
      "brouwer fixed point theorem",
      "dimensional unit cube"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}