Kahler Packages on Valuations on Convex Sets and Their Applications

A valuation is a finitely additive measure on the class of all convex compact subsets of Rn. Over the past two decades, a number of structures has been discovered on the space of translation invariant smooth valuations. Recently, these findings culminated in a construction of two Kahler packages, each corresponding to a distinct multiplicative structure on valuations, connected by a Fourier-type transform. The Hodge-Riemann relations for one of these structures imply the classical Alexandrov-Fenchel inequality for convex bodies, while for the second leads to new geometric inequalities.