I have had this question burning in my mind for awhile and was wondering if a more experienced geometer, statistician or probability theorist has any insight:
Given the facets of a polytope in $\mathbb{R}^2$, can we determine the probability that the given polytope is convex?
Can we extend this to $\mathbb{R}^3$ and eventually $\mathbb{R}^n$?
Starting with the $\mathbb{R}^2$ case, all facets will just be line segments which of course are convex. Then we could say any polytope would be some union of convex sets. But unions of convex sets need not be convex. This could possibly be solved using some advanced numerical methods but I want to look at it strictly from an analytical perspective and see if we can even begin to approach a solution.
Any input is appreciated! I hope we can figure something out!