Joining Two Pairs of Planar Points by Disjoint Arcs: The Sharp $\sqrt{2}$ Length Bound

Abstract

Problem F16 of Croft, Falconer, and Guy asks for the least worst-case length needed to join prescribed pairs of points by pairwise disjoint planar arcs, when the distance within each pair is at most one. The book suggests that the answer for two pairs is $\sqrt2$. We prove this exactly. The lower bound is a crossing argument in a square. The upper bound follows from a sharp ellipse lemma. As a consequence, the value proposed in the book for three pairs, $(\sqrt3+1)/2$, cannot be correct under the literal formulation, because the constants are nondecreasing in the number of pairs.