2023/07/02 by Brett Leroux, Leroux, Brett, Luis Rademacher +1
Mathematics · #52A22 #52B05 #60C05 #60D05 #62H10 #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2307.00687
openalex publication_date 2023/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Gaussian random polytopes have received a lot of attention especially in the case where the dimension is fixed and the number of points goes to infinity. Our focus is on the less studied case where the dimension goes to infinity and the number of points is proportional to the dimension d. We study several natural quantities associated to Gaussian random polytopes in this setting. First, we show that the expected number of facets is equal to C(α)d+o(d) where C(α) is some constant which depends on the constant of proportionality α. We also extend this result to the expected number of k-facets. We then consider the more difficult problem of the asymptotics of the expected number of pairs of estranged facets of a Gaussian random polytope. When n=2d, we determined the constant C so that the expected number of pairs of estranged facets is equal to Cd+o(d).