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

## TL;DR

This paper derives exact formulas for the expected sum of solid angles at vertices of random simplices in dimensions 3 and 4, with vertices sampled uniformly from the sphere or ball, extending to beta simplices.

## Contribution

It provides explicit expected angle-sum formulas for random simplices in dimensions 3 and 4, including cases with vertices in the sphere or ball, generalizing to beta simplices.

## Key findings

- Expected angle sum in 3D is 1/8.
- Expected angle sum in 4D is 539/(288π^2) - 1/6.
- Results extend to beta simplices in these dimensions.

## 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 $\frac 18$ if $d=3$ and $\frac{539}{288\pi^2}-\frac 16$ 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 $4$.