A stability property of the densest circle packing (Q1107830)

A packing \({\mathcal P}\) of congruent circles in the plane is said to be uniformly stable if there is \(\epsilon >0\) such that no finite subset of the circles can be rearranged such that each circle is moved within a distance less than \(\epsilon\) and the rearranged circles together with the rest form a packing not congruent to the original one. The main result of this paper is that the packing \({\mathcal C}\) obtained from the densest circle packing of congruent circles by removing one circle is uniformly stable (with \(\epsilon =1/40\) if the circles are of unit diameter).