Here is the simplest and most precise description of the central problem in this collection of pages. We have an n dimensional vector space with metric (positive definite quadratic form Q). We have k (k≤n) independent vectors xi each defining a half space {z | z・xi ≤ 0}. Sn−1 in this space is {x | x・x = 1}. Its n−1 measure (content) is πn/2n/(n/2)!. What is the content of the intersection of Sn−1 and the k half spaces?

The following digression follows a paper claiming to compute the content of a ‘spherical tetrahedron’ given either edge lengths, or dihedral angles. That is less general than I want and also supposes different ‘givens’. It is nonetheless quite interesting. It is arcane and requires the dilogarithm function to be calculated on its radius of convergence. A digression into dilogarithms.

This program is a slavish copy of the formulae given here in terms of the dihedral angles. It compiled without error the first time and got the right answer for the tetrahedron with 6 right dihedral angles. I have learned to distrust such programs. Indeed when just one of those angles is modified then the volume is proportional to that angle and the program gives the wrong volume.