On Jordan angles and triangle inequality in Grassmannian

We obtain the following version of Lidskii theorem. Let L, M, N be p-dimensional subspaces in R^n. Let \psi_j be the angles between L and M, let \phi_j be the angles between M and N, and let \theta_j be the angles between L and N. Consider the orbit of the vector \psi with respect to permutations of coordinates and inversions of axises. Let Z be the convex hull of this orbit. Then \theta is an element of the polyhedron \phi + Z. We discuss similar theorems for other symmetric spaces. We obtain formula for geodesic distance for any invariant Finsler metrics on a classical Riemannian symmetric space.