N=3 Particles

3 particles which lie on the sphere will lie on a common plane as well and so they lie on a circle. The problem of minimizing the potential of points on a circle is solved by choosing equally spaced points. (This can be proved by showing that the potential function is convex and comparing any minimal configuration with an appropriately chosen reflection or rotation). The resulting potential is minimized for the largest possible circle. Thus, the minimal configuration is given by choosing equally spaced points on any great circle. In the list below we take the points on the equator.

Potential = Sqrt[3]

The convex hull of 3 points is not a three dimensional figure, so we give no picture, nor projections.

The points:

{{1,0,0}, {Cos[2 Pi/3] , Sin[2 Pi/3] ,0}, {Cos[4 Pi/3] , Sin[4 Pi/3] ,0}}