Stacked central configurations with a homogeneous potential in $\mathbb{R}^3$

In this paper we generalize some results in \cite{Yu2021} concerning stacked central configurations. We can deal with the general homogeneous potential $U_{\alpha}$ (containing the vortex case) in $\mathbb{R}^3$. We give the admissible set of $\alpha$ for a convex central configuration (with respect to the Newtonian potential i.e. $\alpha=3$). We discuss some properties of the regular $n$-gon co-circular central configurations. We also find that the stacked property is particular for central configurations by studying the S-balanced configuration case.