Random Polytopes

Explore the fascinating world of beta polytopes in this third lecture of a series on random polytopes. Delve into the mathematical intricacies of these structures, defined as convex hulls of independent and identically distributed samples from the beta density on the d-dimensional unit ball. Examine the related beta' polytopes, constructed from samples of the beta' density in d-dimensional Euclidean space. Discover various models in stochastic geometry that can be reduced to beta and beta' polytopes, including random cones in a half-space, the Poisson zero cell, and the typical Poisson-Voronoi cell. Learn how to express functionals of these models through the expected internal and external angles of beta and beta' simplices. If time permits, gain insights into the explicit computation of these angles. This 44-minute lecture, presented by Zakhar Kabluchko at the Hausdorff Center for Mathematics, is based on several research papers and provides an in-depth exploration of this advanced mathematical topic.