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Log-optimal spherical configuration

Banff International Research Station for Mathematical Innovation and Discovery

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 simplices of cardinality m and n, where m+n=d+2. 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. This is a joint work with Peter Dragnev.

Mathematics; Convex and discrete geometry; Statistical mechanics, structure of matter; Discrete mathematics

video/mp4

Author affiliation: University of Texas Rio Grande Valley

2020-04-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73877

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/