{
  "abstract": "In the present paper, we study the maximization problem of the $k$-Torsional rigidity under quermassintegral constraint, with particular emphasis on the role of the ball. Our main result shows a phenomenon which, as far as we know, has not previously been observed: the ball, rather than being an extremal shape, is a saddle point. More precisely, for suitable quermassintegral constraints, we show that the $k$-torsional rigidity admits both increasing and decreasing perturbations around the ball. The proof is based on a second-order analysis of the corresponding variational problem. To the best of our knowledge, this is the first use of a second-order shape derivative approach in the study of variational problems for $k$-Hessian operators. We also establish an analogous saddle-point behavior for a related curvature functional under a volume constraint, highlighting a structural distinction between the intermediate Hessian regime and the classical Laplace and Monge-Ampère cases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17083",
  "author_total": 3,
  "authors": [
    {
      "name": "Alba Lia Masiello"
    },
    {
      "name": "Gloria Paoli"
    },
    {
      "name": "Francesco Salerno"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/664w2jg4s0yfa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17083",
  "fulltext_url": null,
  "id": "664w2jg4s0yfa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17083",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.17083",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/664w2jg4s0yfa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17083",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17083"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On some functionals for which the ball is a saddle shape",
  "title_text": "On some functionals for which the ball is a saddle shape",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/664w2jg4s0yfa",
  "venue": "arXiv",
  "versions": []
}