A form of Alexandrov-Fenchel inequality (Q541360)

Based on polarity for convex bodies, the authors introduce a class of domains called \(k^*\)-convex, as natural generalization of the usual notion of convex body and related to the Hessian equation on the unit sphere by the distance function. The authors derive a form of the Aleksandrov-Fenchel inequality for such domains, and they even prove a uniqueness theorem for the Hessian equation (yielding also a generalization of the classical Aleksandrov-Fenchel-Jessen theorem). They use an interesting mixture of methods (using, e.g., hyperbolic polynomial theory).