I'm not sure of the correct name of this old problem, but it's been around since at least the mid 1800's. (There's a legend about George Boole's discussing this with a teacher.): Does there exist a convex polyhedron P in 3-space such that, if it has constant positive 3D density, then when its center of gravity is projected orthogonally onto the planes of its faces, at least one such projection lies inside the face. (I.e., there is a face F such that if P is placed on F on a horizontal plane under gravity, it won't fall over.) An old heuristic nonexistence proof is that if P existed, then placing P on an infinite plane under gravity would lead to a perpetual motion machine. (I don't find this argument persuasive even in the realm of physics.) But in any case, does anyone know whether this problem has been settled? Reference(s) ? (And what is it called?) Thanks, Dan