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

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- by Zakhar Kabluchko
- Proc. Amer. Math. Soc.
**148**(2020), 3079-3086 - 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 $4$.## References

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## Bibliographic Information

**Zakhar Kabluchko**- Affiliation: Institut für Mathematische Stochastik, Westfälische Wilhelms-Universität Münster, Orléans-Ring 10, 48149 Münster, Germany
- MR Author ID: 696619
- ORCID: 0000-0001-8483-3373
- Email: zakhar.kabluchko@uni-muenster.de
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 3079-3086 - MSC (2010): Primary 52A22, 60D05; Secondary 52B11
- DOI: https://doi.org/10.1090/proc/14934
- MathSciNet review: 4099794