{
  "abstract": "Grünbaum asked whether every planar convex body admits, for every $0\\leq t\\leq 1/4$, two orthogonal lines cutting it into pieces with cyclically ordered areas $t,t,1/2-t,1/2-t$. Bárány posed the analogous question for well-behaved planar measures and conjectured that the answer there is negative. We confirm Bárány's conjecture in a particularly robust form: for every fixed $0<t<1/4$ we construct smooth, strictly positive, centrally symmetric, strongly log-concave measures arbitrarily close to the standard Gaussian for which the prescribed partition does not exist. In contrast, we prove that the partition exists for every $t$ whenever the measure is invariant under an orientation-reversing affine involution. We also exhibit a $96$-point counterexample for which no pair of perpendicular lines produces cyclic counts $8,8,40,40$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16757",
  "author_total": 1,
  "authors": [
    {
      "name": "Leonardo Martínez-Sandoval"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8a6cy6hw7m6v0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16757",
  "fulltext_url": null,
  "id": "8a6cy6hw7m6v0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16757",
  "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",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7213393397",
  "pdf_url": "https://arxiv.org/pdf/2609.16757",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8a6cy6hw7m6v0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2609.16757",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16757"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Counterexamples and symmetry for uneven orthogonal mass partitions in the plane",
  "title_text": "Counterexamples and symmetry for uneven orthogonal mass partitions in the plane",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/8a6cy6hw7m6v0",
  "venue": "arXiv",
  "versions": []
}