On the Separation of Logarithmic Points on the Sphere

On the Separation of Logarithmic Points on the Sphere P. D. Dragnev

In this article we consider the distribution of N points on the unit sphere, whose mutual distances have maximal geometric mean. The problem is reduced to a weighted energy problem for circular symmetric weights. As a result a new improved separation conditions on the optimal con guration is derived.

Abstract.

x1. Introduction The distribution of points on the sphere, which satisfy some optimal conditions, is a fascinating subject that has interested researchers for a long time. Ever since the ancient Greek and the Platonic solids, passing through Isaak Newton and the Thirteen Sphere Problem, and in recent times the discovery of fullerenes, patterns on the sphere have been a source of many comprehensive investigations throughout the centuries. In this article we shall focus our attention to a particular problem, concerning the so called logarithmic points on the sphere. Sometimes referred to as elliptic Fekete points [20], these points maximize the geometric mean of their mutual distances. Indeed, if we de ne the logarithmic energy of a collection of N points !N := fx1 ; x2 ; : : : ; xN g on the unit sphere S 2 := fx 2 IR3 : jxj = 1g to be X 1 E0 (!N ) := log (1) i 0 ( < 0) the -extremal points minimize (maximize) the -energy

E (!N ) :=

X

1

: j x x j i j i