Is there a decomposition of $S^2$ into $k$ (geodesically) convex polyhedra that are congruent to each other? What about $S^n$ for $n>1$?
- A polyhedron is defined as an area enclosed by a piecewise-geodesic simple closed curve.
- Decomposition is meant in the usual sense of polyhedral decomposition.
- (Reworded) We impose a strict form of convexity which, in this case, means that no individual polyhedron contains a pair of antipodal points. (So that, in particular, $k>2$.) The idea is that for any two points in a polyhedron, the distance minimizing geodesic is unique and lies completely within the polyhedron.
- Congruence just means the existence of an isometry.
- Can this be done with $k=4$ in the case of $S^2$? (Or $k=n+2$ for $S^n$?)
I strongly suspect that the answer is no. This prompts the question what's the best one can do (e.g., are there any interesting polyhedral decompositions of $S^2$ that satisfy all but one of these requirements?)