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Estranged facets and k-facets of Gaussian random point sets
- Part of
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- Journal:
- Journal of Applied Probability / Volume 62 / Issue 3 / September 2025
- Published online by Cambridge University Press:
- 02 April 2025, pp. 859-875
- Print publication:
- September 2025
-
- Article
- Export citation
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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 with Gaussian random polytopes in this setting. First, we show that the expected number of facets is equal to
$C(\alpha)^{d+o(d)}$, where 
$C(\alpha)$ is some constant which depends on the constant of proportionality 
$\alpha$. 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 the number of points is 2d, we determine the constant C such that the expected number of pairs of estranged facets is equal to 
$C^{d+o(d)}$.
