  Some foundational results on the geometry of Lorentz-Minkowski spaces and
Finsler spacetimes are obtained. We prove that the local light cone structure
of a reversible Finsler spacetime with more than two dimensions is
topologically the same as that of Lorentzian spacetimes: at each point we have
just two strictly convex causal cones which intersect only at the origin.
Moreover, we prove a reverse Cauchy-Schwarz inequality for these spaces and a
corresponding reverse triangle inequality. The Legendre map is proved to be a
diffeomorphism in the general pseudo-Finsler case provided the dimension is
larger than two.
