Log-optimal configurations on the sphere

In this article we consider the distribution of $N$ points on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$ interacting via logarithmic potential. A characterization theorem of the stationary configurations is derived when $N=d+2$ and two new log-optimal configurations minimizing the logarithmic energy are obtained for six points on $\mathbb{S}^3$ and seven points on $\mathbb{S}^4$. A conjecture on the log-optimal configurations of $d+2$ points on $\mathbb{S}^{d-1}$ is stated and three auxiliary results supporting the conjecture are presented.