Gilles Bonnet (Bochum)

Poisson hyperplane tessellation:
Asymptotic probabilities of the zero and typical cells.

vendredi 29 septembre 2017, 9h30 - 10h30

Salle du conseil, espace Turing

Let η be a Poisson hyperplane process in R^d, d ≥ 2. It induces a random tessellation, i.e. a collection of polytopes with pairwise disjoint interiors and whose union is the full euclidean space. The cell containing the origin is the so called zero cell Z_0. If η is stationary it induces an other random polytope, the so called typical cell Z_typ. The zero and typical cells form one of the most classical classes of random polytopes. In this talk we will describe properties of their distributions with a focus on asymptotic properties. We will address questions such as:
• How fast does the probability P ( Z_0 has n facets ) tends to 0 as n tend to infinity?
• How fast does the probability P ( Vol(Z_0) > a) tends to 0 as a tend to infinity?
• D.G. Kendall’s problem: What does big cells look like? If we condition on the event { Vol(Z_0) > a } does the normalized convex body Vol(Z_0)^(-1/d) Z_0 converges to a fixed convex body? Which one? In which sense? How fast?
• Small cells: Are small cells always simplices?
To answer these questions we will combines ideas from probability, integral geometry and convex geometry.