Non Hilbertian Lorentzian length spaces

In this note, the idea of finite-dimensional Lp spaces is extended to the setting of Lorentzian length spaces to provide an example of a constant Finsler manifold that is nowhere Minkowskian. Looking at the sectional curvature bounds of this example leads to the more general statement that non-Hilbertian, normed spaces have no sectional curvature bounds. This statement holds in both the Riemannian and Lorentzian cases. In addition, the Lorentzian Lp spaces serve as an example in the context of Lorentzian Gromov–Hausdorff convergence. They show that unbounded sectional curvature or geodesic regularity is generally not preserved in the Gromov-Hausdorff limit. This example also shows that Ricci bounds in the sense of the “timelike curvature dimension condition” as well as bounds on the volume, the timelike diameter, and the dimension are not enough to ensure Gromov pre-compactness in the category of Lorentzian length spaces.