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Explore the fascinating world of high-dimensional random polytopes in this 30-minute lecture by Eliza O'Reilly from the Hausdorff Center for Mathematics. Delve into the study of n i.i.d. points chosen uniformly from the unit sphere in R^d and examine the asymptotic behavior of the (d−1)-dimensional faces, or facets, of their convex hull. Discover how known asymptotic formulas in fixed dimension d provide insights into sphere approximation and random spherical Delaunay tessellations as the number of points n grows. Learn about the groundbreaking work by O'Reilly and Gilles Bonnet, which generalizes these results to cases where both n and d tend to infinity. Uncover the different regimes imposed by high-dimensional geometry and their impact on the asymptotic behavior of facets. Gain a deeper understanding of the limiting shapes of these polytopes in high dimensions through the presented asymptotic formulas.
