Marking out Geodesically convex regions on surfaces of polyhedrons

We refer to any subset S of the surface of any convex solid C as being geodesically convex if for any two points on S, the shortest line on the surface of C that joins them is contained entirely in S - and if there are more than one such shortest path, at least one is entirely contained in S.

Question: Consider any convex 3d solid. We need to mark out its surface into n regions all of equal area such that each region is geodesically convex. First of all one needs to decide if a given convex polyhedron's surface can be so marked out for a specified n - an algorithmic decision question.

It appears that the surfaces of all convex solids of revolution about an axis can be marked out into n such regions for any n.

It seems unlikely that the surface of a regular tetrahedron can be marked out into 5 geodesically convex equal area regions. For even a cube, it is not clear if such a marking out of the surface can be done for any n.

Note: This question was posted at mathoverflow a few days back and simply disappeared!