Pierre Calka, Université de Rouen - 26 Feb 2021
Title. Typical and extremal results for random convex hull interfaces
Abstract. In this talk, we consider the random polytope defined as the convex hull of a point set constituted with independent and uniformly distributed points in a smooth convex body K of R^d. When the size of the input goes to infinity, the random polytope approximates K and we are interested in estimating the fluctuations around this limit shape.
The question leads us to studying the distributions of several functionals, including the distance to the boundary of K and the area, of the so-called typical facet of the random polytope. Our main results are extreme value convergences for the maximal area of a facet and for the Hausdorff distance from the random polytope to K when K is a ball. This, in turn, provides estimates for the radial and longitudinal fluctuations of the boundary of the random polytope. Surprisingly, the rates are similar to those observed for a large variety of random interfaces in probability theory (random cluster models, polynuclear growth model...).
This is joint work with J. E. Yukich (Lehigh University, USA).