On the centered co-circular central configurations for the n-body problem

For the power-law potential $n$-body problem, we study a special kind of central configurations where all the masses lie on a circle and the center of mass coincides with the center of the circle. It is also called the centered co-circular central configuration. We get some symmetry results for such central configurations. We show that for positive numbers $\alpha >0$ and integers $n\ge 3$ satisfying $\frac{1}{n}{\sum }_{j=1}^{n-1}{\csc }^{\alpha }\frac{j\pi }{n}\le 1+\frac{\alpha }{4}$, the regular $n$-gon with equal masses is the unique centered co-circular central configuration for the $n$-body problem with power-law potential $U_{\alpha }$. It quickly follows that for the Newtonian $n$-body problem (in the case $\alpha =1$) and $n\le 6$, the regular $n$-gon is the unique centered co-circular central configuration.