{
  "abstract": "We prove new bounds for Tverberg's theorem with tolerance. We show that $N = rt+Θ_{d,r}(t^{1/2-1/(2d)})$, where $N$ is the smallest number such that any set of $N$ points in $\\mathbb{R}^d$ has a partition into $r$ parts such that the convex hulls of the parts intersect even if we remove any $t$ of the points. We extend Tverberg's theorem with tolerance to families of hyperplanes in $\\mathbb{R}^d$, and show that for any set of $rt + O_{d,r}(t^{1/2-1/(2d)}\\sqrt{\\log (t+1)})$ hyperplanes in $\\mathbb{R}^d$ there exists a partition of them into $r$ parts such that the regression hulls of the parts intersect even if any $t$ hyperplanes are removed. Our bounds follow from establishing a connection between Tverberg-type results and discrepancy theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24775",
  "author_total": 2,
  "authors": [
    {
      "name": "Aleksey Lopez"
    },
    {
      "name": "Pablo Soberón"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6wv97hyz0pvz8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24775",
  "fulltext_url": null,
  "id": "6wv97hyz0pvz8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24775",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213996442",
  "pdf_url": "https://arxiv.org/pdf/2609.24775",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6wv97hyz0pvz8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24775",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24775"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Discrepancy theory, Tverberg's theorem, and regression depth",
  "title_text": "Discrepancy theory, Tverberg's theorem, and regression depth",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/6wv97hyz0pvz8",
  "venue": "arXiv",
  "versions": []
}