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.