Placing and moving spheres in the gaps of a cylinder packing (Q1179482)

The closeness of a packing is defined as the reciprocal of the supremum of the radius of a ball contained in the complement of the packing of \(\mathbb{R}^ 3\) with infinite circular cylinders of radius 1 equals \(\rho^{-1}=3+2\sqrt 3\), and, up to an isometry, the packing of maximum closeness is unique. The authors prove: If in a packing of \(\mathbb{R}^ 3\) no two cylinders are parallel, then, for any two balls of radius \(\rho\) non-overlapping with any of the cylinders, each of the balls can move between the cylinders to assume the other ball's place, without overlapping with any of the cylinders during the motion. This theorem can be strengthened concerning the position of the cylinders (Remark 1); it can also be generalized to \(n\) dimensions (Remark 2).