Let $C$ be a pointed polyhedral cone in $\mathbb{R}^n$ and let $S^{n1}$ 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 $C \cap S^{n1}$? 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?

$\begingroup$ Is the base of the cone at the origin? $\endgroup$ – Igor Rivin Nov 8 '11 at 14:51

$\begingroup$ @Igor: I call that bit of a cone the vertex, and I call the other end (when there is one) the base. $\endgroup$ – Andreas Blass Nov 8 '11 at 14:59

$\begingroup$ @Andreas: yes, of course, base was a poorly chosen term... $\endgroup$ – Igor Rivin Nov 8 '11 at 15:06

$\begingroup$ Yes, the vertex is the origin $\endgroup$ – Brian Lins Nov 8 '11 at 15:20

$\begingroup$ Things may depend on how the set of supporting planes is described. Is it given as normal plane to a convex hull of some finite set in $S^{n1}$? $\endgroup$ – Anton Petrunin Nov 8 '11 at 17:54
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 lowerdimensional 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.