A reverse Thomson problem on the unit circle

Abstract

Let x 1 x_1 , x 2 x_2 , …, x n x_n be n n points on the sphere S 2 S^2 . Determining the value inf ∑ 1 ≤ k > j ≤ n | x k − x j | − 1 \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 S^{d-1} in d 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_1 , x 2 x_2 , …, x n x_n and two dimensional Thomson’s problem on the unit circle for all integer n ≥ 2 n\geq 2 .