{
  "abstract": "We study the computation of mixed volumes of convex sets given as rational subspace sections of bodies described by membership oracles. The sets need not have nonempty interior. For positive multiplicities $α_1+\\cdots+α_k=n$, we give a randomized algorithm that computes a relative $\\varepsilon$-approximation with probability at least $1-δ$. Its running time is polynomial in Gurvits' factor $Γ_α=(n^n/n!)\\prod_iα_i!/α_i^{α_i}$, the input presentation length, and $\\varepsilon^{-1}$, and linear in $1+\\logδ^{-1}$, with degree independent of $k$. The running-time bound holds for every choice of the random bits, and zero mixed volumes are detected exactly. We extend the capacity-scaling and sampling method of Narayanan--Roy by using rational projections and a smooth mixed Monge--Ampère representation. In particular, the algorithm runs in randomized polynomial time when the number of distinct convex sets is bounded above by a constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08248",
  "author_total": 1,
  "authors": [
    {
      "name": "Hariharan Narayanan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/42hzxt2z8ez7g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08248",
  "fulltext_url": null,
  "id": "42hzxt2z8ez7g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08248",
  "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",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7221002721",
  "pdf_url": "https://arxiv.org/pdf/2610.08248",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/42hzxt2z8ez7g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.08248",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08248"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Approximation algorithm for mixed volumes of a constant number of convex bodies",
  "title_text": "Approximation algorithm for mixed volumes of a constant number of convex bodies",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/42hzxt2z8ez7g",
  "venue": "arXiv",
  "versions": []
}