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Hyperbolicity Beyond Convexity in the Alexandrov-Fenchel Inequality

  • Leo Brauner
  • Oscar Ortega-Moreno
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Abstract

We establish an Alexandrov--Fenchel type inequality in which one of the reference bodies is replaced by a function whose Hessian satisfies a balancing condition on its eigenvalues, without being required to be positive semidefinite. The proof hinges on a new extension of Alexandrov's inequality for mixed discriminants. As applications, we derive a log-concavity principle for linear functionals of area measures, answering a question of Colesanti, Hug, and Saorín-Gómez, as well as Brunn--Minkowski type inequalities for intrinsic volumes of mean section bodies, answering a question of Schuster.

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Cite

BibTeXRIS
BibTeX
@misc{brauner2026hyperbolicity,
  title = {{Hyperbolicity Beyond Convexity in the Alexandrov-Fenchel Inequality}},
  author = {Leo Brauner and Oscar Ortega-Moreno},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08746},
  eprint = {2610.08746},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08746},
}
RIS
TY  - UNPB
TI  - Hyperbolicity Beyond Convexity in the Alexandrov-Fenchel Inequality
AU  - Leo Brauner
AU  - Oscar Ortega-Moreno
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.08746
UR  - https://arxiv.org/abs/2610.08746
AB  - We establish an Alexandrov--Fenchel type inequality in which one of the reference bodies is replaced by a function whose Hessian satisfies a balancing condition on its eigenvalues, without being required to be positive semidefinite. The proof hinges on a new extension of Alexandrov's inequality for mixed discriminants. As applications, we derive a log-concavity principle for linear functionals of area measures, answering a question of Colesanti, Hug, and Saorín-Gómez, as well as Brunn--Minkowski type inequalities for intrinsic volumes of mean section bodies, answering a question of Schuster.
ER  -