KotrbatATIN SMALL LETTER Y WITH ACUTE's theorem on valuations and geometric inequalities for convex bodies

Very recently J. KotrbatATIN SMALL LETTER Y WITH ACUTE[14] has proven general inequalities for translation invariant smooth valuations formally analogous to the Hodge-Riemann bilinear relations in Kahler geometry. The goal of this note is to prove several inequalities for mixed volumes of convex bodies using KotrbatATIN SMALL LETTER Y WITH ACUTE's theorem as the main tool.