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 Poincar\'e duality and the hard Lefschetz theorem. In this article, a version of the Hodge-Riemann relations for the Alesker algebra is conjectured, and the conjecture is proved in two particular situations: for even valuations, and for 1-homogeneous valuations. The latter result is then used to deduce a special case of the Aleksandrov-Fenchel inequality. Finally, mixed versions of the hard Lefschetz theorem and of the Hodge-Riemann relations are conjectured, and it is shown that the Aleksandrov-Fenchel inequality follows from the latter in its full generality.