{
  "abstract": "We revisit the classic 1959 paper of L. D. Landau on possible singularities of the integral associated to a Feynman graph. Focusing on the real setup, we relate it to convex geometry by representing a collection $p$ of incoming momenta as the weighted normals of a convex polytope $Q$ via the Minkowski problem. Then, a regular polyhedral subdivision $\\mathcal{P}$ of $Q$ exhibits $p$ as a Landau singularity labelled by the dual graph of $\\mathcal{P}$ with masses being the areas of the faces. Positivity of the Landau/Feynman/Schwinger multipliers is interpreted as strict convexity of a PL-function. This gives an interesting class of ``polyhedral'' Landau singularities. In the planar case going back to the original 1959 paper, Landau graphs can also be identified with plane webs of Gaiotto-Moore-Witten that provide a language dual to that of regular polygonal subdivisions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27508",
  "author_total": 1,
  "authors": [
    {
      "name": "Mikhail Kapranov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0w2erz29316m6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27508",
  "fulltext_url": null,
  "id": "0w2erz29316m6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27508",
  "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",
      "hep-th"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214221924",
  "pdf_url": "https://arxiv.org/pdf/2609.27508",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0w2erz29316m6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2609.27508",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27508"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Landau singularities and convex geometry",
  "title_text": "Landau singularities and convex geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/0w2erz29316m6",
  "venue": "arXiv",
  "versions": []
}