Packings of unequal circles in a convex set
The authors prove the following result. Suppose that \(T\) is a convex set in the Euclidean plane, and let \(K_1,\dots, K_n\) be \(n\geq 2\) circles packed in \(T\). Suppose further that \(K_i\) has radius \(r_i\), and that \[ {\min r_i\over\max r_i}\geq a_0, \] with \(a_0\) defined by \[ a^2_0:= {a(6)- a(7)\over a(5)- a(6)}, \] where \(a(p):= p\tan (\pi/p)\) is the area of the regular \(p\)-gon circumscribed to the unit circle. Then the density of the packing is at most \(\pi/\sqrt{12}\), namely, that of the closest packing of circles in the plane. They also obtain better bounds for \(2\) and \(3\) circles, that for the former being exact.
scientific article; zbMATH DE number 1784799