{
  "abstract": "We prove a colorful quantitative Helly theorem for volume with the optimal number $2d$ of colors. If every rainbow intersection from $2d$ finite families of convex sets in $\\R^d$ has volume at least one, then the intersection of one family has volume at least $d^{-O(d^2)}$. We also prove a colorful quantitative Steinitz theorem for origin-centered ellipsoids of different shapes. The proof uses a common normalization of positive operators and a lift that produces two rainbow bases with large determinants.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25671",
  "author_total": 1,
  "authors": [
    {
      "name": "Grigory Ivanov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fgjd6f06thjy2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25671",
  "fulltext_url": null,
  "id": "fgjd6f06thjy2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25671",
  "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",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214171071",
  "pdf_url": "https://arxiv.org/pdf/2609.25671",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fgjd6f06thjy2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.25671",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25671"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A colorful quantitative Helly theorem for volume",
  "title_text": "A colorful quantitative Helly theorem for volume",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/fgjd6f06thjy2",
  "venue": "arXiv",
  "versions": []
}