Hodge-Riemann relations for valuations and geometric inequalities

Thomas Wannerer (Univ. Jena)

Abstract:
The Hodge-Riemann relations are a property of the cohomology of compact Kähler manifolds. Very recently, 
J. Kotrbatý has proved in several special cases an analogue of the Hodge-Riemann relations in a completely 
different setting, namely for smooth translation-invariant valuations (finitely additive functions on the 
family of convex compact subsets of Euclidean space satisfying a smoothness condition). As first observed 
by S. Alesker, these results directly imply both classical and new geometric inequalities between the mixed 
volumes of convex bodies. In this talk, we report on these developments and in particular on recent joint 
work with J. Kotrbatý.