{
  "id": "2409.05744",
  "version": "v1",
  "published": "2024-09-09T15:59:41.000Z",
  "updated": "2024-09-09T15:59:41.000Z",
  "title": "No-dimensional Helly's theorem in uniformly convex Banach spaces",
  "authors": [
    "G. Ivanov"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We study the ``no-dimensional'' analogue of Helly's theorem in Banach spaces. Specifically, we obtain the following no-dimensional Helly-type results for uniformly convex Banach spaces: Helly's theorem, fractional Helly's theorem, colorful Helly's theorem, and colorful fractional Helly's theorem. The combinatorial part of the proofs for these Helly-type results is identical to the Euclidean case as presented in \\cite{adiprasito2020theorems}. The primary difference lies in the use of a certain geometric inequality in place of the Pythagorean theorem. This inequality can be explicitly expressed in terms of the modulus of convexity of a Banach space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-09T15:59:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A05",
      "52A35"
    ],
    "keywords": [
      "uniformly convex banach spaces",
      "no-dimensional hellys theorem",
      "no-dimensional helly-type results",
      "colorful fractional hellys theorem",
      "primary difference lies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}