Given an irregular convex die, I would like to calculate the probability of it landing in every single face. I've worked on the 2-dimensional case and I have made some progress by considering distances from the boundary of the die to its center of mass/gravity. However, my method cannot be easily generalized to higher dimensions.
I've considered the possibility of projecting the die on a sphere centred on its center of mass/gravity and comparing area ratios but this method seems to fail, as some dice have faces with clearly 0 probability and yet that's not the outcome of the ratios (for instance, in 2 dimensions, consider a triangle with a very obtuse angle and with very different side lengths and you will easily find an example). Any ideas? Thank you in advance.
Of course, I'm under the assumption that there is gravity. Also, I would neglect the friction of the air, but NOT the bounces with the ground.