# Limit theorems for random polytopes

We construct a random polytope as the convex hull of a random set of points in Rd which is either a binomial or a Poisson point process. We are interested in the asymptotic behaviour of various geometric characteristics, including the number of k-dimensional faces and intrinsic volumes, when the size of the input goes to infinity. More precisely, we aim at obtaining second-order results with explicit limiting variance.

In the particular case of the convex hull of a binomial or Poisson point process in the unit-ball, the key idea is to apply a scaling transform to the whole picture and see the boundary of the random polytope (resp. its Voronoi flower) when the intensity of the point process goes to infinity as a so-called *it paraboloid hull process* (resp. *it paraboloid growth process*) in a product-space. Consequences of such approach include the calculation of explicit limit variances for the characteristics mentioned above as well as certain functional central limit theorems via the technique of *it stabilization*. Thanks to an inversion transformation, these results have also counterparts for the typical cell of a Poisson-Voronoi tessellation with large inradius. Finally, we will discuss extensions of the methods to more general settings, in particular when the unit-ball is replaced with a smooth convex body.

This talk will be a review of works due to Tomasz Schreiber & J. E. Yukich and joint works with them. It aims at emphasizing some of Tomasz Schreiber's most beautiful contributions to the topic.