Let $C$ be a pointed polyhedral cone in $\mathbb{R}^n$ and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. Given a description of the supporting hyperplanes of $C$ is there an algorithm for computing the spherical measure of $C \cap S^{n-1}$? I suppose you could randomly generate points uniformly distributed on the unit sphere, and test each point to see if it is in $C$. Is there a better way?

## 2 Answers

This is in general as hard as computing the volume of an Euclidean polytope, but there are reductions for even dimensional polytopes to volumes of lower-dimensional things (of which there may, of course, be an exponential number). See http://www.math.ru.nl/~heckman/Heck_7.pdf (he mostly talks about the hyperbolic case, but the spherical case is identical).

For very special classes of cones, there are combinatorial formulas related to these questions, e.g.

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