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A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains

  • Jiahuan Li
  • Xi-Nan Ma
  • Guohuan Qiu
  • Paolo Salani
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Abstract

We prove the strict log-concavity of the positive first eigenfunction −u of the 2-Hessian equation and the strict 1/2-convexity of the solution for the corresponding torsion problem in smooth bounded uniformly convex domains in ℝn. As applications, we establish the associated Brunn--Minkowski inequalities. We also show that this transformed-convexity phenomenon fails for 3-Hessian equations by constructing, in dimension four, a smooth uniformly convex domain whose admissible zero-boundary solution has a nonconvex sublevel set.

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Cite

BibTeXRIS
BibTeX
@misc{li2026brunnminkowski,
  title = {{A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains}},
  author = {Jiahuan Li and Xi-Nan Ma and Guohuan Qiu and Paolo Salani},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09175},
  eprint = {2609.09175},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09175},
}
RIS
TY  - UNPB
TI  - A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains
AU  - Jiahuan Li
AU  - Xi-Nan Ma
AU  - Guohuan Qiu
AU  - Paolo Salani
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.09175
UR  - https://arxiv.org/abs/2609.09175
AB  - We prove the strict log-concavity of the positive first eigenfunction −u of the 2-Hessian equation and the strict 1/2-convexity of the solution for the corresponding torsion problem in smooth bounded uniformly convex domains in ℝ^n. As applications, we establish the associated Brunn--Minkowski inequalities. We also show that this transformed-convexity phenomenon fails for 3-Hessian equations by constructing, in dimension four, a smooth uniformly convex domain whose admissible zero-boundary solution has a nonconvex sublevel set.
ER  -