A reverse Thomson problem on the unit circle

Abstract:

Let $x_1$, $x_2$, …, $x_n$ be $n$ points on the sphere $S^2$. Determining the value $\inf \sum _{1\leq k<j\leq n}|x_k-x_j|^{-1}$, is a long-standing open problem in discrete geometry, which is known as Thomson’s problem. In this paper, we propose a reverse problem on the sphere $S^{d-1}$ in $d$-dimensional Euclidean space, which is equivalent to establish the reverse Thomson inequality. In the planar case, we establish two variants of the reverse Thomson inequality.

In addition, we give a proof to the minimal logarithmic energy of $x_1$, $x_2$, …, $x_n$ and two dimensional Thomson’s problem on the unit circle for all integer $n\geq 2$.