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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?

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  • $\begingroup$ 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 $S^{n-1}$? $\endgroup$ – Anton Petrunin Nov 8 '11 at 17:54
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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).

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

http://people.cs.uchicago.edu/~klivans/reflection.pdf

http://people.cs.uchicago.edu/~klivans/volumes.pdf

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