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.