  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.
