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Explore the fascinating world of beta polytopes in this advanced mathematics lecture. Delve into the definition of beta polytopes as convex hulls of independent and identically distributed samples from the beta density on the d-dimensional unit ball. Examine beta' polytopes, defined similarly but using the beta' density on d-dimensional Euclidean space. Discover various stochastic geometry models 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 expected internal and external angles of beta and beta' simplices. Gain insights into the explicit computation of these angles, based on several research papers in the field of random polytopes and stochastic geometry.
