{
  "abstract": "We study the simultaneous Busemann-Petty and Shephard volume comparison problem: whether comparison of the volumes of all central hyperplane sections and all orthogonal hyperplane projections determines the ordering of the volumes of two convex bodies. For every $n\\geq5$, we construct origin-symmetric convex bodies of revolution $K,L\\subset {\\mathbb R}^n$ such that every central hyperplane section and every orthogonal hyperplane projection of $K$ has strictly smaller volume than the corresponding section or projection of $L$, while $|K|>|L|$. For $n\\leq4$, the affirmative solution of the Busemann--Petty problem shows that the section inequalities alone imply $|K|\\leq|L|$. Without origin symmetry, we construct such counterexamples in every dimension $n\\geq2$, with one body a nontrivial translate of a Euclidean ball and the other a noncentrally symmetric body of constant brightness.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29505",
  "author_total": 1,
  "authors": [
    {
      "name": "Artem Zvavitch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f6ngxq44nhkpr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29505",
  "fulltext_url": null,
  "id": "f6ngxq44nhkpr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29505",
  "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.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7204931170",
  "pdf_url": "https://arxiv.org/pdf/2608.29505",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f6ngxq44nhkpr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29505",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29505"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Simultaneous Busemann-Petty and Shephard Volume Comparisons",
  "title_text": "Simultaneous Busemann-Petty and Shephard Volume Comparisons",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/f6ngxq44nhkpr",
  "venue": "arXiv",
  "versions": []
}