  A version of the Hodge-Riemann relations for valuations was recently
conjectured and proved in several special cases by the first-named author. The
Lefschetz operator considered there arises as either the product or the
convolution with the mixed volume of several Euclidean balls. Here we prove
that in (co-)degree one the Hodge-Riemann relations persist if the balls are
replaced by several different (centrally symmetric) convex bodies with smooth
boundary with positive Gauss curvature. While these mixed Hodge-Riemann
relations for the convolution directly imply the Aleksandrov-Fenchel
inequality, they yield for the dual operation of the product a new inequality.
This new inequality strengthens classical consequences of the
Aleksandrov-Fenchel inequality for lower dimensional convex bodies and
generalizes some of the geometric inequalities recently discovered by S.
Alesker
