Fejes Toth Lecture

A stationary (but not necessarily isotropic) Poisson line field in $\mathbb{R}^2$ decomposes the plane into convex polygonal cells. The zero cell $Z_0$ is the cell containing the origin, the typical cell $Z$ is a random polygon which is obtained by picking one of the infinitely many cells at random. It is known that the expected number of vertices, $\mathbb{E} f_0(Z)$, of $Z$ is always $4$, whereas for the zero cell we have $\mathbb{E} f_0(Z_0)\in [4,\pi^2/2]$. The bounds are attained for Poisson line fields having special direction distributions, and they are related to sharp affine invariant geometric inequalities. More generally, in the talk we explain connections between properties of random tessellations and stability results for geometric inequalities in $\mathbb{R}^d$. Other examples arise, for instance, in the study of asymptotic shapes of large cells or large $k$-faces in random tessellations.