The author establishes that over a \(C^{2,1}\) manifold the exponentional map of any Lipschitz connection or spray determines a local Lipeomorphism and that, furthermore, reversible convex neighborhoods do exist. To that end he uses the method of Picard-Lindelöf 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 as 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, it is proved that the local Lorentzian(-Finsler) length maximization property of casual geodesics in the class of absolutely continuous casual curves holds already for \(C^{1,1}\) spacetime metrics. Finally, the author studies the local existence of convex functions and shows that arbitrarily small globally hyperbolic convex normal neighborghoods do exist.