We consider the random metric constructed by Kendall on $\mathbb{R}^d$ from a self-similar Poisson process of roads, i.e. lines with a speed limit. Intuitively, the process generates a random road network in $\mathbb{R}^d$ that one can travel on, respecting the speed limits; and this induces a random metric $T$ on $\mathbb{R}^d$, for which the distance between points is given by the optimal connection time.
In this talk, I will present some fractal properties of the random metric space $\left(\mathbb{R}^d,T\right)$. In particular, although it is almost surely homeomorphic to the usual Euclidean $\mathbb{R}^d$, its Hausdorff dimension is given by $(\gamma-1)d/(\gamma-d)$, where $\gamma>d$ is a parameter of the model. This fractal property, which is reminiscent of the Brownian sphere, confirms a conjecture of Kahn.
If time allows, I will also mention some multifractal properties of the metric space $\left(\mathbb{R}^d,T\right)$ equipped with the Lebesgue measure, which in particular distinguish it from the Brownian sphere equipped with its volume measure.
