Eliza O’Reilly: Facets of high dimensional random polytopes скачать в хорошем качестве

Eliza O’Reilly: Facets of high dimensional random polytopes

We consider the model of n i.i.d. points chosen uniformly from the unit sphere in R^d and study the asymptotic behavior of the (d−1)-dimensional faces, or facets, of the convex hull of these points. In fixed dimension d, known asymptotic formulas as the number of points n grows provide results on approximation of the sphere and random spherical Delaunay tessellations. In joint work with Gilles Bonnet, we generalize these results to the case where both the number of points n and the space dimension d are allowed to tend to infinity. The geometry of high dimensional space imposes different regimes for n and d with different asymptotic behavior of the facets. We obtain asymptotic formulas in each case, illuminating the limiting shapes of these polytopes in high dimensions.