Hyperbolicity Beyond Convexity in the Alexandrov-Fenchel Inequality

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.