{
  "abstract": "Let U be uniformly distributed on the unit sphere. We develop a self-contained forward and inverse theory for the random width w_K(U) and brightness b_K(U) of three-dimensional convex bodies. For every full-dimensional polytope, a global spherical co-area formula expresses the width density as a finite sum of angular apertures determined by the normal fan of its difference body; in particular, the density is piecewise real analytic with a finite geometrically determined critical set. This theory yields exact densities for the width of the regular tetrahedron, resolving a question of Finch, and for the regular truncated octahedron, together with the tetrahedral brightness law and the equivalent rhombic-dodecahedral width law. On the inverse side, second- and third-order polarized cosine-transform moments reconstruct finite labelled direction systems whenever the observed triangles span the cycle space of the correlation graph; signed-graph switching describes the unavoidable ambiguity. In contrast, equal three-dimensional intrinsic volumes do not determine either the width law or the brightness law, even for centrally symmetric bodies. Removing the spatial rank constraint gives a dimension-free identifiability theorem for centered multivariate folded-normal vectors: pairwise absolute moments and an anchored family of triple absolute moments, comprising |m - 1|^2 labelled observations for a complete correlation graph, determine the correlation matrix up to diagonal sign conjugacy without fourth-order moments. A harmonic decomposition further identifies the degree-two variance contribution as a constant multiple of the squared Frobenius norm of the traceless part of the weighted frame operator and explains why this contribution vanishes under irreducible symmetry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26202",
  "author_total": 1,
  "authors": [
    {
      "name": "Omri Abas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/89bs1fsek2xqq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26202",
  "fulltext_url": null,
  "id": "89bs1fsek2xqq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26202",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214072057",
  "pdf_url": "https://arxiv.org/pdf/2609.26202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/89bs1fsek2xqq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Random Width and Brightness: Polyhedral Density Theory, Reconstruction, and Gaussian Identifiability",
  "title_text": "Random Width and Brightness: Polyhedral Density Theory, Reconstruction, and Gaussian Identifiability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:49Z",
  "url": "https://research.draconitic.com/paper/89bs1fsek2xqq",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.5281/zenodo.21870870",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}