Problems of optimal configurations of points on the sphere

In this paper, problems of evenly spaced configuration of points on the sphere are discussed. After introducing the minimax optimal problem and the minimum energy problem on the sphere, a new method of obtaining an optimal configuration is proposed under an utterly different principle. The new method is called a spherical adjustment method. It is first shown that the spherical adjustment method is useful for generating a global optimal configuration of a small number of points. Then, it is shown that the new method is also useful for generating local optimal configurations of points for a wide range of the number of points. It is also shown that, by carefully selecting a good configuration as an initial condition, our new method is able to improve the equilibrium configurations of points, for instance, which are generated by Markov Chain Monte Carlo method under a repulsive interaction potential.