{
  "abstract": "In convex geometry, a number of inequalities for the volume of coordinate projections of convex bodies share a common structural property. We formalize them as inequalities of log-submodularity type, providing a definition that stems from the notion of a submodular function. The Loomis-Whitney inequality and some of its generalizations are the most prominent examples. Some inequalities of log-submodularity type are valid for zonoids but not for general convex bodies. They have a natural meaning in statistics and information theory. We propose a new inequality of log-submodularity type for zonoids. In dimension 3, our inequality is equivalent to the log-submodularity conjecture for zonoids formulated by Fradelizi, Madiman, Meyer, and Zvavitch, which is true in dimension $3$ and false in higher dimensions. In contrast, our inequality is true in every dimension. We reduce our inequality to copositivity of a special polynomial, which we call the hypercube polynomial. For 4-dimensional zonoids, we verify the copositivity of the respective hypercube polynomial algebraically by providing a sum-of-squares certificate. In every dimension, a proof of copositivity of the hypercube polynomial that uses probability theory and well-known matrix inequalities has been produced by LLMs and brought to our attention. We present a detailed and easy-to-read version of this proof.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14909",
  "author_total": 3,
  "authors": [
    {
      "name": "Gennadiy Averkov"
    },
    {
      "name": "Katherina von Dichter"
    },
    {
      "name": "Ivan Soprunov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/08t70xy2j0nxq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14909",
  "fulltext_url": null,
  "id": "08t70xy2j0nxq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.14909",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7203687994",
  "pdf_url": "https://arxiv.org/pdf/2608.14909",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/08t70xy2j0nxq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2608.14909",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14909"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube",
  "title_text": "On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/08t70xy2j0nxq",
  "venue": "arXiv",
  "versions": []
}