Consider a d-dimensional simplex whose vertices are random points chosen independently according to the standard Gaussian distribution on ℝd. We prove that the expected angle sum of this random simplex equals the angle sum of the regular simplex of the same dimension d.
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References
E. Bosetto, “Systems of stochastically independent and normally distributed random points in the Euclidean space E3,” Beiträge Algebra Geom., 40, No. 2, 291–301 (1999).
D. V. Feldman and D. A. Klain, “Angles as probabilities,” Amer. Math. Monthly, 116, No. 8, 732–735 (2009).
S. Finch, “Random Gaussian tetrahedra,” ar**v:1005.1033 (2010).
S. Ivanov, Private communication, 2018.
Z. Kabluchko, C. Thäle, and D. Zaporozhets, “Beta polytopes and Poisson polyhedra: f-vectors and angles,” ar**v:1805.01338 (2018).
Z. Kabluchko and D. Zaporozhets, “Absorption probabilities for Gaussian polytopes, and regular spherical simplices,” ar**v:1704.04968 (2017).
C. A. Rogers, “The packing of equal spheres,” Proc. London Math. Soc. (3), 8, 609–620 (1958).
L. Schläfli, “Theorie der vielfachen Kontinuität,” in: Gesammelte mathematische Abhandlungen (1950), pp. 167–387.
R. Schneider and W. Weil, Stochastic and Integral Geometry, Springer-Verlag, Berlin (2008).
A. M. Vershik and P. V. Sporyshev, “Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem,” Selecta Math. Soviet., 11, No. 2, 181–201 (1992).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 476, 2018, pp. 79–91.
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Kabluchko, Z., Zaporozhets, D. Angles of the Gaussian Simplex. J Math Sci 251, 480–488 (2020). https://doi.org/10.1007/s10958-020-05107-2
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DOI: https://doi.org/10.1007/s10958-020-05107-2