Abstract
The talk deals with a type of random geometry which combines convex and integral geometry together with probability theory and specifically the notion of point processes. In general, it consists in starting with a discrete set of random points in the Euclidean space, then proceeding with a deterministic geometric construction based on this input and studying the produced random object. We concentrate in particular on the smallest convex polytope containing the random point cloud, namely its convex hull. Such a model naturally appears in various fields, including computational geometry, image analysis or statistics of multivariate data. After giving some historical background, we intend to discuss a few recent asymptotic results when doing a close-up on the boundary of that random polytope. This includes limit distributions, extreme values or properties in the high dimensional setting. Along the way, we hope to provide a meaningful insight into the mathematical tools required, in both probability and geometry, and to build a tentative bridge to other domains, including partial differential equations.
The presentation is based on several joint works with Joe Yukich, Gauthier Quilan and Benjamin Dadoun.
