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

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.