{
  "abstract": "We prove a colorful quantitative Steinitz theorem in which the color classes may have different centers. If the convex hull of each of $2d$ sets in $\\R^d$ contains a translate of the Euclidean unit ball, then a rainbow convex hull contains a ball of radius $(3d)^{-2d^2}$ whose center belongs to the convex hull of the given centers. The main ingredient is an exact result for translates of a segment, proved by a lifting and a topological colorful Helly theorem. The same idea also yields a sharp colorful Helly theorem for translated cones: for $1 \\le k \\le d-1$, if every rainbow selection from $2d-k+1$ finite families of convex sets in $\\mathbb{R}^d$ has a $k$-dimensional cone in its intersection, then the intersection of one of the families contains such a cone.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21927",
  "author_total": 1,
  "authors": [
    {
      "name": "G. Ivanov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/37m6368ew547w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21927",
  "fulltext_url": null,
  "id": "37m6368ew547w",
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  "landing_url": "https://arxiv.org/abs/2609.21927",
  "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": "W7213886916",
  "pdf_url": "https://arxiv.org/pdf/2609.21927",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/37m6368ew547w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21927",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21927"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A colorful Steinitz theorem with different centers",
  "title_text": "A colorful Steinitz theorem with different centers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/37m6368ew547w",
  "venue": "arXiv",
  "versions": []
}