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

On mixed Hodge-Riemann relations for translation-invariant valuations and Aleksandrov-Fenchel inequalities