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Dive into the fascinating world of beta polytopes in this first lecture of a three-part series on random polytopes. Explore the definition of beta polytopes as convex hulls of independent and identically distributed samples from the beta density on the d-dimensional unit ball. Learn about beta' polytopes and their relation to the beta' density on d-dimensional Euclidean space. Discover various models of 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. Understand how functionals of these models can be expressed through expected internal and external angles of beta and beta' simplices. Gain insights into the computation of these angles and their significance in random polytope theory. Based on multiple research papers, this lecture provides a comprehensive introduction to the subject, setting the foundation for deeper exploration in subsequent talks.
