I would like to compute volumes of irregular d-dimensional, convex shapes. In principle, I know the equations of hyperplanes that bound the polygon.
In d=2, I could proceed with an algorithm as follows: I can easily compute the N vertices of my polygon by intersections of bounding lines, then I can compute the areas of the N triangles defined by one edge and a fixed random internal point (https://en.wikipedia.org/wiki/Heron%27s_formula or equivalent) and sum the total area.
Does exist something similar in d>2 (or even alternative), that is computationally fast?
It's ok also a good approximations, I don't really need the exact volume.
I thought about something similar to https://en.wikipedia.org/wiki/Pick's_theorem, using a multidimensional lattices, but I'm not sure it is the best idea by a computational point of view.