On mixed Hodge–Riemann relations for translation-invariant valuations and Aleksandrov–Fenchel inequalities

Abstract

A version of the Hodge–Riemann relations for valuations was recently conjectured and proved in several special cases by [J. Kotrbatý, On Hodge–Riemann relations for translation-invariant valuations, preprint (2020), arXiv:2009.00310]. The Lefschetz operator considered there arises as either the product or the convolution with the mixed volume of several Euclidean balls. Here we prove that in (co-)degree one, the Hodge–Riemann relations persist if the balls are replaced by several different (centrally symmetric) convex bodies with smooth boundary with positive Gauss curvature. While these mixed Hodge–Riemann relations for the convolution directly imply the Aleksandrov–Fenchel inequality, they yield for the dual operation of the product a new inequality. This new inequality strengthens classical consequences of the Aleksandrov–Fenchel inequality for lower-dimensional convex bodies and generalizes some of the geometric inequalities recently discovered by [S. Alesker, Kotrbatý’s theorem on valuations and geometric inequalities for convex bodies, preprint (2020), arXiv:2010.01859].