My question concerns the existence of a nice (deterministic?) method/algorithm for calculating the distribution of surface areas for two-dimensional projections of an arbitrary polytope (or convex approximation of a polytope). Less optimistically, a method of finding the minimum, maximum, and perhaps, mean surface area of the polytope's projections.

It is a relatively straightforward procedure to calculate a given two-dimensional surface projection along some orientational vector, and then calculate the approximate surface area of the projection (or its convex hull). But, beyond statistical sampling or methods related to simulated annealing, I'm having trouble imagining how to go about characterizing the full set of projections along all arbitrary vectors... and I haven't had any luck with a literature search (so far).

Note - This question is directly related to computations one might like to perform for - Characterizing a tumbling convex polytope from the surface areas of its two-dimensional projections. I hope this follow-up post is appropriate...