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Zakhar Kabluchko, University of Münster, Germany, Expected f-vector of the Poisson Zero Cell
September 21, 2020 | 3:00 pm - 4:00 pm EDT
The Poisson hyperplane process describes, roughly speaking, infinitely many hyperplanes thrown uniformly at random into the d-dimensional Euclidean space. The hyperplanes dissect the space into countably many cells. The a.s. unique cell containing the origin is called the Poisson zero polytope. We prove an explicit combinatorial formula for the expected number of k-dimensional faces of the Poisson zero polytope. This number is expressed through the coefficients of the polynomial
$$
(1+ (d-1)^2 x^2) (1+(d-3)^2 x^2) (1+(d-5)^2 x^2) \ldots.
$$
We shall also discuss the analogue of the Sylvester four-point problem on the half-sphere as well as the following closely related problems. Sample n points $U_1,\ldots,U_n$ uniformly at random on the $d$-dimensional upper half-sphere. Let $C_n$ be the convex cone spanned by the vectors $U_1,\ldots,U_n$. What is the expected number of $k$-dimensonal faces of $C$? What is the expected solid angle of $C_n$?
Website: https://sites.google.com/view/paw-seminar
Host: Paata Ivanisvili pivanis@ncsu.edu