On some functionals for which the ball is a saddle shape

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.

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BibTeX RIS

BibTeX

@misc{masiello2026some,
  title = {{On some functionals for which the ball is a saddle shape}},
  author = {Alba Lia Masiello and Gloria Paoli and Francesco Salerno},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17083},
  eprint = {2609.17083},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17083},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On some functionals for which the ball is a saddle shape
AU  - Alba Lia Masiello
AU  - Gloria Paoli
AU  - Francesco Salerno
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17083
UR  - https://arxiv.org/abs/2609.17083
AB  - 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.
ER  -

Abstract

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