Log-optimal (d+2)-configurations in d-dimensions

Abstract

We enumerate and classify all stationary logarithmic configurations of d+2 points on the unit (d-1)-sphere in d-dimensions. In particular, we show that the logarithmic energy attains its relative minima at configurations that consist of two orthogonal to each other regular simplexes of cardinality m and n. The global minimum occurs when m=n if d is even and m=n+1 otherwise. This characterizes a new class of configurations that minimize the logarithmic energy on the (d-1)-sphere for all d. The other two classes known in the literature, the regular simplex and the cross polytope, are both universally optimal configurations.