Average number of zeros and mixed symplectic volume of Finsler sets

Let X be an n-dimensional manifold and is equal to the mixed symplectic volume of these Finsler ellipsoids. If X is a homogeneous space of a compact Lie group and all vector spaces V-i together with their Euclidean metrics are invariant, then the average numbers of zeros satisfy the inequalities, similar to Hodge inequalities for intersection numbers of divisors on a projective variety. This is applied to the eigenspaces of Laplace operator of an invariant Riemannian metric. The proofs are based on a construction of the ring of normal densities on X, an analogue of the ring of differential forms. In particular, this construction is used to carry over the Crofton formula to the product of spheres.