{
  "abstract": "For positive matrices $A,B$ and $0<t<1$, Bourin asked whether the crossed-Heinz expression \\[ C_t(A,B)=A^tB^{1-t}+B^tA^{1-t} \\] satisfies \\[ \\UIN{C_t(A,B)}\\le \\UIN{A+B} \\] for every unitarily invariant norm. The most direct route to this inequality is to compare $C_t(A,B)$ with the ordinary Heinz expression \\[ H_t(A,B)=A^tB^{1-t}+A^{1-t}B^t, \\] because the classical Heinz inequality gives $\\UIN{H_t(A,B)}\\le\\UIN{A+B}$. This route, however, encounters a genuine dimensional obstruction: the stronger comparison $\\UIN{C_t(A,B)}\\le\\UIN{H_t(A,B)}$ is known to fail for positive definite $3\\times3$ matrices, and hence, by block-diagonal embedding, in every dimension $n\\ge3$. We first determine the exact threshold for this obstruction. For every pair of positive $2\\times2$ matrices and every $0<t<1$, we prove the weak-majorization relation \\[ s(C_t(A,B))\\prec_w s(H_t(A,B)). \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04567",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohammad Sababheh"
    },
    {
      "name": "Jean-Christophe Bourin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ezs3mmawd4gjv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04567",
  "fulltext_url": null,
  "id": "ezs3mmawd4gjv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04567",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7220745098",
  "pdf_url": "https://arxiv.org/pdf/2610.04567",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ezs3mmawd4gjv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04567",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04567"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Crossed-Heinz Inequalities: The Two-Dimensional Threshold and Higher-Dimensional Stability",
  "title_text": "Crossed-Heinz Inequalities: The Two-Dimensional Threshold and Higher-Dimensional Stability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/ezs3mmawd4gjv",
  "venue": "arXiv",
  "versions": []
}