  We establish that over a C^{2,1} manifold the exponential map of any
Lipschitz connection or spray determines a local Lipeomophism and that,
furthermore, reversible convex normal neighborhoods do exist. To that end we
use the method of Picard-Lindelof approximation to prove the strong
differentiability of the exponential map at the origin and hence a version of
Gauss' Lemma which does not require the differentiability of the exponential
map. Contrary to naive differential degree counting, the distance functions are
shown to gain one degree and hence to be C^{1,1}. As an application to
mathematical relativity, it is argued that the mentioned differentiability
conditions can be considered the optimal ones to preserve most results of
causality theory. This theory is also shown to be generalizable to the Finsler
spacetime case. In particular, we prove that the local Lorentzian(-Finsler)
length maximization property of causal geodesics in the class of absolutely
continuous causal curves holds already for C^{1,1} spacetime metrics. Finally,
we study the local existence of convex functions and show that arbitrarily
small globally hyperbolic convex normal neighborhoods do exist.
