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

Theorem 1.1: for i.i.d. exponential weights of any rate, $\mu$ is Frechet differentiable on $\mathbb R^2\setminus\{0\}$ and $\mathcal B$ is strictly convex, so $\partial\mathcal B$ is a $C^1$ curve with a unique supporting line at each point. Theorem 1.2: differentiability (not strict convexity) holds for every Gamma law with positive shape and rate. Consequences for directional geodesics and Busemann functions follow from Damron-Hanson. It does not treat general continuous laws, higher dimensions, or curvature of the boundary.

Theorem 1.1: for i.i.d. exponential weights of any rate, $\mu$ is Frechet differentiable on $\mathbb R^2\setminus\{0\}$ and $\mathcal B$ is strictly convex, so $\partial\mathcal B$ is a $C^1$ curve with a unique supporting line at each point. Theorem 1.2: differentiability (not strict convexity) holds for every Gamma law with positive shape and rate. Consequences for directional geodesics and Busemann functions follow from Damron-Hanson. It does not treat general continuous laws, higher dimensions, or curvature of the boundary.