{
  "abstract": "Let $Ω\\subset \\mathbb{R}^N$ be a bounded convex domain and let $H_Ω\\leq 0$ denote its variational mean curvature, in the sense of Barozzi, Gonzalez and Tamanini [PAMS, 1987]. We prove that $-H_Ω$ is harmonic convex, that is $1/H_Ω$ is a convex function in $Ω$, and that the exponent $-1$ is optimal in the class of convex domains. This strictly improves the concavity of $H_Ω$ obtained by Alter and Caselles [Nonlinear Anal., 2009], who also proved, by geometric methods, a closely related property for the classical mean curvature under Minkowski interpolation of convex bodies. Our approach is analytic: the result is deduced from a concavity property of the solution of the singular $1$-Laplace Dirichlet problem $$\\begin{cases} -Δ_1 u = {u^{-γ}} & \\quad \\hbox{in $Ω$}, \\\\ u = 0 &\\quad \\hbox{on $\\partial Ω$}, \\end{cases}$$ for $γ\\in (0,1]$, whose unique positive solution is $u=(-H_Ω)^{-1/γ}$. We show that $u^γ$ is concave, and we obtain this by proving that the solutions $u_p$ of the approximating problems $-Δ_p u_p = u_p^{-γ}$ are $\\frac{p-1+γ}{p}$-concave on every bounded convex domain, without any boundary regularity, and then letting $p\\to 1^+$. Explicit examples show that no Hölder continuity of $H_Ω$ up to $\\partialΩ$ holds uniformly in the class of convex domains with $C^{1,1}$ boundary.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28321",
  "author_total": 3,
  "authors": [
    {
      "name": "Marco Gallo"
    },
    {
      "name": "Francesco Petitta"
    },
    {
      "name": "Marco Squassina"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6krx7sppst5sv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28321",
  "fulltext_url": null,
  "id": "6krx7sppst5sv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28321",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223094746",
  "pdf_url": "https://arxiv.org/pdf/2610.28321",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6krx7sppst5sv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28321",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28321"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Optimal power concavity of the variational mean curvature via singular $1$-Laplace problems",
  "title_text": "Optimal power concavity of the variational mean curvature via singular $1$-Laplace problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/6krx7sppst5sv",
  "venue": "arXiv",
  "versions": []
}