Work

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

Year: 2026

Type: preprint

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 $\mathbb{R}^{n}$. 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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.09175

Authors: Li, Jiahuan, Ma, Xi-Nan, Qiu, Guohuan, Salani, Paolo

Institutions: University of Science and Technology of China, University of Florence

Language: English

Topic: Mathematical Inequalities and Applications

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", European Commission, National Natural Science Foundation of China, Ministero dell'Università e della Ricerca

Awards: G2701445157, G4488344675, G5245466801, G598502428, G972786147

OpenAlex ID: W7212182882

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Cited by: 3

Related to: 10