Before revision:  These positive results hold for any L_p metric when P_i's are point sets of the same size, and also hold for the case when P_i's have different sizes in the L_\infty metric (i.e., for the center persistence diagram problem).
After revision:  These positive results hold for any L_p metric when P_i's are point sets of the same size, and also hold for the case when P_i's have different sizes in the L_\infty metric (i.e., for the Center Persistence Diagram problem).