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

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. T...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society. Series B 2023-02, Vol.10 (5), p.155
Main Authors: Peter D. Dragnev, Oleg R. Musin
Format: Article
Language:English
Online Access:Get full text
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Summary: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 \mathbb {S}^{d-1} for all d. The other two classes known in the literature, the regular simplex (d+1 points on \mathbb {S}^{d-1}) and the cross-polytope (2d points on \mathbb {S}^{d-1}), are both universally optimal configurations.
ISSN:2330-0000
DOI:10.1090/btran/118