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

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 $4$.