{
  "abstract": "We study three affine variational quantities built on the affine $L^p$ energy of Lutwak, Yang and Zhang: the affine Poincaré--Sobolev constants $Λ^{\\mathcal{A}}_{p,q}(Ω)$ of a bounded open set, the associated affine torsional rigidity, and the affine $p$-capacity of a compact set. We show that ellipsoids are the only extremal sets in their corresponding isoperimetric inequalities, and that the affine capacity of a compact set of positive volume is attained by a potential solving the affine $p$-Laplace equation. We also prove an affine Kohler-Jobin inequality: among sets with prescribed affine torsional rigidity, only ellipsoids minimize $Λ^{\\mathcal{A}}_{p,q}$. Following the approach of Haddad, Jiménez and Montenegro to affine Sobolev inequalities, our methods rest on the $L_p$ Busemann--Petty centroid inequality. Proving a stability version of it, we obtain quantitative forms of the isoperimetric inequalities for $Λ^{\\mathcal{A}}_{p,q}$, the affine torsional rigidity and the affine $p$-capacity, together with a stability result for the affine Kohler-Jobin inequality on convex sets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05403",
  "author_total": 1,
  "authors": [
    {
      "name": "Francisco Marín Sola"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5nsy180e5zn3s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05403",
  "fulltext_url": null,
  "id": "5nsy180e5zn3s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05403",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7220713900",
  "pdf_url": "https://arxiv.org/pdf/2610.05403",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5nsy180e5zn3s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05403",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05403"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Affine variational functionals, isoperimetric problems and stability",
  "title_text": "Affine variational functionals, isoperimetric problems and stability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/5nsy180e5zn3s",
  "venue": "arXiv",
  "versions": []
}