A Random Thought on Convex Partitions...

Just recording a thought: If we partition a convex planar region into n pieces requiring ONLY that the SUM of moment of inertias of the n pieces is minimized (we can alternatively consider ' higher moments' - ie integrals of m*higher powers of distance or even m*exp(r) with distance measured from center of mass), we could get nice rounded pieces - indeed, even requiring "area AND perimeter" equal might result in all pieces being long and thin. Perhaps this moment based approach might help in facility location.