{
  "abstract": "Let $0 < α\\leq 1/2$. We say that a finite point set $P$ in $\\mathbb{R}^d$ is $α$-split by a hyperplane $h$ if each of the closed half-spaces determined by $h$, contains at least $α|P|$ of the points of $P$. We further say $P$ is $α$-split by a $k$-dimensional flat $τ$ if $P$ is $α$-split by any hyperplane through $τ$. In the standard notation (which coincides with Tukey depth for $k= 0$), the $k$-flat $τ$ has depth $α$ with respect to $P$. We establish interesting Helly-type theorems for splitting families of finite point sets in $\\mathbb{R}^d$. Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for point and hyperplanes, our results extend to splitting families of point sets by collections of $k$-flats of arbitrary dimensionality $ 0 \\leq k \\leq d-1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02180",
  "author_total": 2,
  "authors": [
    {
      "name": "Lidor Portal"
    },
    {
      "name": "Natan Rubin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8kmjmxmyny970.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02180",
  "fulltext_url": null,
  "id": "8kmjmxmyny970",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02180",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.02180",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8kmjmxmyny970.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.02180",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02180"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Helly-Type Theorems for Splitting Point Sets",
  "title_text": "Helly-Type Theorems for Splitting Point Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/8kmjmxmyny970",
  "venue": "arXiv",
  "versions": []
}