On Hodge-Riemann relations for translation-invariant valuations

The Alesker product turns the space of smooth translation-invariant valuations on convex bodies into a commutative associative unital algebra, satisfying Poincaré duality and hard Lefschetz theorem. In this article, version Hodge-Riemann relations for algebra is conjectured, conjecture proved in two particular situations: even valuations, 1-homogeneous valuations. latter result then used to deduce special case Aleksandrov-Fenchel inequality. Finally, mixed versions theorem are it shown that inequality follows from its full generality.