Strict convexity and differentiability of the planar exponential first-passage limit shape

We prove that the limit shape of undirected nearest-neighbor first-passage percolation on $\Z^2$ with independent exponential edge weights is strictly convex and has a $C^1$ boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.