Average number of zeros and mixed symplectic volume of Finsler sets

Let $X$ be an $n$-dimensional manifold and $V_1, \ldots, V_n \subset C^\infty(X, \mathbb R)$ finite-dimensional vector spaces with Euclidean metric. We assign to each $V_i$ a Finsler ellipsoid, i.e., a family of ellipsoids in the fibers of the cotangent bundle of $X$. We prove that the average number of isolated common zeros of $f_1 \in V_1, \ldots, f_n \in V_n$ 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 $V_i$ and 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 for a generalization of Crofton formula to the product of spheres.