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

TL;DR

This paper extends the Hodge-Riemann relations for valuations to a broader class of convex bodies, leading to new inequalities that strengthen classical convex geometry results and generalize recent discoveries.

Contribution

It proves that mixed Hodge-Riemann relations hold for convolution with diverse convex bodies, resulting in new inequalities that extend and strengthen existing geometric inequalities.

Findings

Hodge-Riemann relations hold for convex bodies with smooth boundary and positive curvature

New inequalities for the product operation strengthen classical convex geometry results

Convolution-based relations imply the Aleksandrov-Fenchel inequality

Abstract

A version of the Hodge-Riemann relations for valuations was recently conjectured and proved in several special cases by the first-named author. 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