Paths on the Doubly Covered Region of a Covering of the Plane by Unit Discs

Abstract

Given a covering of the plane by closed unit discs $\mathcal{F}$ and two points $A$ and $B$ in the region doubly covered by $\mathcal{F}$, what is the length of the shortest path connecting them that stays within the doubly covered region? This is a problem of G. Fejes-Tóth and he conjectured that if the distance between $A$ and $B$ is $d$, then the length of this path is at most $\sqrt{2}d+O(1)$. In this paper we give a bound of $2.78d+O(1)$.