Equipartitions of surfaces of convex spatial regions

Let S be the surface of a 3D convex region. Let S' be a subset of S. We shall refer to S' as geodesically convex wrto S if the following condition holds: If A and B are two points in S', the shortest path in S between A and B lies entirely in S'.

Question: It is easy to see that if S is the surface of a sphere and n is any integer, S cen be partitioned into n equal area pieces that are also geodesically convex in S - these pieces are separated by great circle arcs between 2 antipodal points on S. Is the sphere the only convex surface (surface of a convex 3D region) for which such an equipartition exists for any n? If not, how does one characterize such surfaces? Prolate and oblate spheroids too seem to allow such equipartitions. Beyond that, not sure.

If S is the surface of a cube, I don't see how it can be cut into 3 geodesically convex pieces all of same area.