On the Fejes T\'oth Problem about the Sum of Angles Between Lines

TL;DR

This paper advances the understanding of Fejes Toth's conjecture by providing improved bounds on the sum of angles between vectors and exploring new methods, especially for the one-dimensional case.

Contribution

The paper introduces new upper bounds for the sum of angles and energy integrals, and offers novel approaches to the conjecture in the one-dimensional case.

Findings

Improved upper bounds for angle sums and energy integrals.

New methods for analyzing the one-dimensional case.

Enhanced understanding of the conjecture's structure.

Abstract

In 1959 Fejes T\'oth posed a conjecture that the sum of pairwise non-obtuse 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: $d=1$.