On the logarithmic energy of points on \(\mathbb{S}^2\) (Q2680334)

This article is concerned with the study of the smallest value of the energy \[ \mathcal{E}_{\log}(n)=\sum_{i, j=1, i\neq y}^n \log\frac{1}{\|x_i-x_j\|}, \] where \(x_1, x_2, \dots, x_n\) lie on the unit sphere \(\mathbb{S}^2\subset \mathbb{R}^3\). The author provides a simple renormalization approach that results in a purely local problem involving superpositions of Gaussians. Connections and improvements on recent results in the study of \(\mathcal{E}_{\log}\) are presented.