On the angle sum of lines

Abstract

What is the maximum of the sum of the pairwise (non-obtuse) angles formed by n lines in the Euclidean 3-space? This question was posed by Fejes Toth in (Acta Math Acad Sci Hung 10:13–19, 1959). Fejes Toth solved the problem for $${n \leq 6}$$ , and proved the asymptotic upper bound $${n^{2} \pi /5}$$ as $${n \to \infty}$$ . He conjectured that the maximum is asymptotically equal to $${n^{2} \pi /6}$$ as $${n \to \infty}$$ . The main result of this paper is an upper bound on the sum of the angles of n lines in the Euclidean 3-space that is asymptotically equal to $${3n^{2} \pi /16}$$ as $${n \to \infty}$$ .