  On a compact Riemannian manifold $M$ of dimension $n$, we consider $n$
eigenfunctions of the Laplace operator $\Delta $ with eigenvalue $\lambda$. If
$M$ is homogeneous under a compact Lie group preserving the metric then we
prove that the average number of common zeros of $n$ eigenfunctions does not
exceed $c(n)\lambda^{n/2}{\rm vol}\,M$, the expression known from the
celebrated Weyl's law. Moreover, if the isotropy representation is irreducible,
then the estimate turns into equality. The constant $c(n)$ is explicitly given.
The method of proof is based on the application of Crofton's formula for the
sphere.
