  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.
