  For a connected network on Poisson points in the plane, consider the
route-length $D(r,\theta) $ between a point near the origin and a point near
polar coordinates $(r,\theta)$, and suppose $E D(r,\theta) = O(r)$ as $r \to
\infty$. By analogy with the shape theorem for first-passage percolation, for a
translation-invariant and ergodic network one expects $r^{-1} D(r, \theta)$ to
converge as $r \to \infty$ to a constant $\rho(\theta)$. It turns out there are
some subtleties in making a precise formulation and a proof. We give one
formulation and proof via a variant of the subadditive ergodic theorem wherein
random variables are sometimes infinite.
