Skip to content
DraconiticResearch
Search Sources API
201,818 studies31 sourcesUpdated 49 min ago
PreprintCC BY 4.0arXiv

Affine variational functionals, isoperimetric problems and stability

  • Francisco Marín Sola
Open at sourcePDFDOICite

Abstract

We study three affine variational quantities built on the affine Lp energy of Lutwak, Yang and Zhang: the affine Poincaré--Sobolev constants Λ𝒜p,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 Λ𝒜p,q. Following the approach of Haddad, Jiménez and Montenegro to affine Sobolev inequalities, our methods rest on the Lp Busemann--Petty centroid inequality. Proving a stability version of it, we obtain quantitative forms of the isoperimetric inequalities for Λ𝒜p,q, the affine torsional rigidity and the affine p-capacity, together with a stability result for the affine Kohler-Jobin inequality on convex sets.

Related

  1. Isoperimetric Problems on cyclic polygons
    Abd Raouf Chouikha
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. Efficiency Optimality without Pathwise Differentiability: A Variational Theory for Marginal-Integral Functionals
    Shuoxun Xu, Xinzhou Guo
    arXivPreprintCC BY-NC-ND 4.0Full textStatisticsarXiv
  3. Quantitative Stability for Anisotropic Gaussian Isoperimetric Inequality
    Sean McCurdy, Kuan-Ting Yeh
    arXivPreprintCC BY-NC-SA 4.0Full textAnalysis & PDEsarXiv
  4. Lp-Integral Affine Surface Area-Variational Capacity Inequalities
    Yibin Feng, Ding Zhang
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  5. PERSISTENCE OF THE WIENER SAUSAGE: SAMPLING STABILITY AND LIMIT THEOREMS FOR BETTI-CURVE FUNCTIONALS OF DRIFTED PLANAR BROWNIAN MOTION
    Tristan Guillaume
    HALPreprintPublisher termsFull textProbabilityHAL

Cite

BibTeXRIS
BibTeX
@misc{sola2026affine,
  title = {{Affine variational functionals, isoperimetric problems and stability}},
  author = {Francisco Marín Sola},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05403},
  eprint = {2610.05403},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05403},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Affine variational functionals, isoperimetric problems and stability
AU  - Francisco Marín Sola
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05403
UR  - https://arxiv.org/abs/2610.05403
AB  - We study three affine variational quantities built on the affine L^p energy of Lutwak, Yang and Zhang: the affine Poincaré--Sobolev constants Λ^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^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^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.
ER  -