We enumerate and classify all stationary logarithmic configurations of d + 2 points on the unit sphere in 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 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 Sd−1 for all d. The other two classes known in the literature, the regular simplex (d + 1 points on Sd−1) and the cross-polytope (2d points on Sd−1), are both universally optimal configurations.
Dragnev, Peter, and Oleg Musin. "Log-optimal (𝑑+ 2)-configurations in 𝑑–dimensions." Transactions of the American Mathematical Society, Series B 10.05 (2023): 155-170. https://doi.org/10.1090/btran/118
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Transactions of the American Mathematical Society, Series B