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Angle sums of random simplices in dimensions $ 3$ and $ 4$


Author: Zakhar Kabluchko
Journal: Proc. Amer. Math. Soc. 148 (2020), 3079-3086
MSC (2010): Primary 52A22, 60D05; Secondary 52B11
DOI: https://doi.org/10.1090/proc/14934
Published electronically: February 26, 2020
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Abstract: Consider a random $ d$-dimensional simplex whose vertices are $ d+1$ random points sampled independently and uniformly from the unit sphere in $ \mathbb{R}^d$. We show that the expected sum of solid angles at the vertices of this random simplex equals $ \frac {1}{8}$ if $ d=3$ and $ \frac {539}{288\pi ^2}-\frac {1}{6}$ if $ d=4$. The angles are measured as proportions of the full solid angle which is normalized to be $ 1$. Similar formulae are obtained if the vertices of the simplex are uniformly distributed in the unit ball. These results are special cases of general formulae for the expected angle sums of random beta simplices in dimensions $ 3$ and $ 4$.


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Additional Information

Zakhar Kabluchko
Affiliation: Institut für Mathematische Stochastik, Westfälische Wilhelms-Universität Münster, Orléans-Ring 10, 48149 Münster, Germany
Email: zakhar.kabluchko@uni-muenster.de

DOI: https://doi.org/10.1090/proc/14934
Keywords: Random polytopes, random simplices, solid angles, sum of angles, beta distributions
Received by editor(s): May 31, 2019
Received by editor(s) in revised form: November 12, 2019
Published electronically: February 26, 2020
Additional Notes: The author was supported by the German Research Foundation under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics - Geometry - Structure.
Communicated by: David Levin
Article copyright: © Copyright 2020 American Mathematical Society