A reverse Thomson problem on the unit circle.

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. [ABSTRACT FROM AUTHOR]