In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond modulo isometries to vectors in the Euclidean Weyl chamber. We can hence assign vector valued lengths to segments. Our main result is a system of homoge- neous linear inequalities, which we call the generalized triangle inequalities or stability inequalities, describing the restrictions on the vector valued side lengths of oriented polygons. It is based on the mod 2 Schubert calculus in the real Grassmannians G=P for maximal parabolic subgroups P.

We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, as- pherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity com- ponent, then the local isometry orbits in M are roughly fibers of a fiber bundle. A corollary is that if M has an open, dense, locally homogeneous subset, then M is locally homogeneous.