Flight in a packing of disks (Q1196361)

The author considers the problem of packing of disks in the plane consisting of pairwise disjoint convex domains, none of which contains the origin 0. Let \(\lambda(r)\) be the length of the shortest path which evades the domains and connects 0 with a point at a distance \(r\) from 0. The question is the following: What can be said for the supremum of the function \(\lambda(r)\) taken over all packings whose members are domains of diameter at least 2? The author proves that there is a constant \(c\) such that, for \(r\geq 2\) \[ \lambda(r)\leq r^ 2+{1\over 2}\ln(r)+c \] and constructs a packing showing that, for \(r\geq\cot\bigl({\pi\over 9}\bigr)\) \[ \lambda(r)\geq r^ 2-\left({\pi\over 3}+{3\over\pi}\right)r+3.6. \] In the last section of the paper some interesting remarks connected with the examined problem can be found.