Geodesics in Planar Poisson Road Random Metric

We study the structure of geodesics in the fractal random metric constructed by Kendall from a self-similar Poisson process of roads (that is, lines with speed limits) in R2$\mathbb {R}2$. In particular, we prove a conjecture of Kendall stating that geodesics do not pause en route, that is, use roads of arbitrary small speed except at their endpoints. It follows that the geodesic frame of R2,T$\left(\mathbb {R}2,T\right)$ is the set of points on roads. We also consider geodesic stars and hubs, and give a complete description of the local structure of geodesics around points on roads. Notably, we prove that leaving a road by driving off-road is never geodesic.