Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere

We consider the problem of finding an N -point configuration on the sphere S d ⊂ R d + 1 with the smallest absolute maximum value over S d of its total potential. The potential induced by each point y in a given configuration at a point x ∈ S d is f x - y 2 , where f is continuous on [0, 4] and completely monotone on (0, 4], and x - y is the Euclidean distance between points x and y . We show that any sharp point configuration ω ¯ N on S d , which is antipodal or is a spherical design of an even strength is a solution to this problem. We also prove that the absolute maximum over S d of the potential of any such configuration ω ¯ N is attained at points of ω ¯ N .