  In this note, the idea of finite dimensional $L^p$ spaces is transferred to
Lorentzian length spaces to provide an example that is locally nowhere
Minkowskian. Looking at the sectional curvature bounds of this example leads to
the more general statement that normed spaces in which the norm does not come
from an inner product, have no sectional curvature bounds. This statement holds
in the Riemannian and Lorentzian cases. In addition, the Lorentzian $L^p$ space
can be used as an example in the context of Lorentzian Gromov-Hausdorff
convergence, to show that unbounded sectional curvature or geodesic regularity
is in general not preserved in the GH limit, and as an example of a sequence of
uniform bounded length spaces which are not GH pre-compact.
