Let $P$ be a convex 3d polyhedra / 2d polytope constrained by a set of linear inequalities $Ax<= b$.
1.How to define a uniform probability distribution over a polytope/polyhedra?
Let us say we have multiple polytopes/polyhedra with uniform probability distribution but non-identical due to fact the geometry is not constrained by the same set of inequalities. I
2. How to add two non-identical yet independent and uniform distributions defined over a polytope/polyhedra?
Relavance to the application: I would like to sample large set of points sampled from each of the convex polytopes and add points from each set among themselves to arrive at a sum distribution of higher probability. I need a theoretical tool to justify that my sampling is good enough to approximate the minkowski sum of polyhedra with an area/volume of higher probability.