Abstract
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.
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