On the Fejes Tóth problem about the sum of angles between lines

Bilyk, Dmitriy; Matzke, Ryan

doi:10.1090/proc/14263

On the Fejes Tóth problem about the sum of angles between lines
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by Dmitriy Bilyk and Ryan W. Matzke

Proc. Amer. Math. Soc. 147 (2019), 51-59

DOI: https://doi.org/10.1090/proc/14263

Published electronically: October 18, 2018

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Abstract:

In 1959 Fejes Tóth posed a conjecture that the sum of pairwise nonobtuse angles between $N$ unit vectors in $\mathbb {S}^d$ is maximized by periodically repeated elements of the standard orthonormal basis. We obtain new improved upper bounds for this sum, as well as for the corresponding energy integral. We also provide several new approaches to the only settled case of the conjecture: $d=1$.

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