TITLE: Algorithm for finding the volume of a convex polytope
QUESTION [41 upvotes]: It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing them into simplices, but gave it up and went with a Monte Carlo estimation scheme instead. Bonus question: How to find the surface area of those same convex bodies?
EDIT: To answer David's question: the data set is a Voronoi tessellation of an n-dimensional volume (n usually 4) with a periodic boundary (like a torus). So I have the coordinates of the vertices of the convex bodies as well as the connectivity of all the facets, faces, etc. For the Monte Carlo I mentioned, I did convert everything to half-spaces, so I think that was not very difficult.

REPLY [6 votes]: An implementation of randomized algorithms using hit-and-run and comparison with exact software presented in Bueler, Enge and Fukuda is presented here.
It works efficiently in high dimensions (say few hudrend) but it only computes an estimation, not the exact volume.
The software is available on github.