The relationships between regular polygon central configurations and masses for Newtonian N-body problems

Perko and Walter (1985) [9] proved that N (N⩾4) point masses located at the vertices of a regular N-polygon form a central configuration if and only if all the masses are equal. In this Letter, we give a simpler proof of their theorem, and in addition, we show that if N point masses are located at the vertices of a regular (N+1)-polygon, they do not form a central configuration for any value of masses.