{
  "abstract": "We prove the strict log-concavity of the positive first eigenfunction \\(-u\\) of the \\(2\\)-Hessian equation and the strict $1/2$-convexity of the solution for the corresponding torsion problem in smooth bounded uniformly convex domains in $\\mathbb{R}^{n}$. As applications, we establish the associated Brunn--Minkowski inequalities. We also show that this transformed-convexity phenomenon fails for \\(3\\)-Hessian equations by constructing, in dimension four, a smooth uniformly convex domain whose admissible zero-boundary solution has a nonconvex sublevel set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09175",
  "author_total": 4,
  "authors": [
    {
      "name": "Jiahuan Li"
    },
    {
      "name": "Xi-Nan Ma"
    },
    {
      "name": "Guohuan Qiu"
    },
    {
      "name": "Paolo Salani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4qw1jqjtydkz7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09175",
  "fulltext_url": null,
  "id": "4qw1jqjtydkz7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09175",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7212182882",
  "pdf_url": "https://arxiv.org/pdf/2609.09175",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4qw1jqjtydkz7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09175",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09175"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains",
  "title_text": "A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/4qw1jqjtydkz7",
  "venue": "arXiv",
  "versions": []
}