The main result of this paper is the description of translation invariant continuous valuations on convex sets. In particular, it provides an affirmative solution of P. McMullen's conjecture, and in a stronger form.

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We remind known and establish new properties of the Dieudonné and Moore determinants of quaternionic matrices. Using these linear algebraic results we develop a basic theory of plurisubharmonic functions of quaternionic variables.

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others. 0 Introduction.In this paper we obtain new results on the structure of the space of even translation invariant continuous valuations on convex sets. In particular we prove a version of hard Lefschetz theorem for them and introduce certain natural duality operator which establishes an isomorphism between the space of such valuations on a linear space V and on its dual V * (with an appropriate twisting). Then we obtain an explicit geometric classification of unitarily invariant translation invariant continuous valuations on a Hermitian space C n . This classification is used to deduce new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.