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

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