On Fejes T\'oth's conjectured maximizer for the sum of angles between lines

TL;DR

This paper investigates Fejes Tóth's conjecture on the optimal arrangement of lines in space to maximize the sum of angles, proving the conjecture's validity and uniqueness in certain limiting cases.

Contribution

It links Fejes Tóth's conjecture to a family of optimization problems involving angle powers, establishing optimality and uniqueness in the limit cases.

Findings

Conjecture is equivalent to a unique optimizer for all powers greater than 1.

Optimality and uniqueness are proven for the limiting case of infinite angles.

Results extend to infinitely many lines with finitely many distinct directions.

Abstract

Choose $N$ unoriented lines through the origin of ${\bf R}^{d+1}$. The sum of the angles between these lines is conjectured to be maximized if the lines are distributed as evenly as possible amongst the coordinate axes of some orthonormal basis for ${\bf R}^{d+1}$. For $d \ge 2$ we embed the conjecture into a one-parameter family of problems, in which we seek to maximize the sum of the $\alpha$-th power of the renormalized angles between the lines. We show the conjecture is equivalent to this same configuration becoming the {\em unique} optimizer (up to rotations) for all $\alpha>1$. We establish both the asserted optimality and uniqueness in the limiting case $\alpha =\infty$ of mildest repulsion. The same conclusions extend to $N=\infty$, provided we assume only finitely many of the lines are distinct.