On the vacancy phenomenon in finite sphere packings
by
Wlodzimierz Kuperberg
Auburn University, Auburn, Alabama, U.S.A.

What is the minimum radius r_n(k) of a spherical container in Rⁿ that can hold k unit balls? The exact answer is known only in a few cases. For n > 2, all of the solved cases are essentially due to R.A. Rankin (1955), and they are limited to k £ 2n. However small this number of cases is, some of them display an interesting phenomenon: the value of r_n(k) can remain constant for several consecutive values of k (with n fixed). In other words, it can happen that if a spherical container is large enough to accommodate k unit balls, then there is still room in it for a few more. Specifically, Rankin's result is that r_n(n+2)=r_n(2n), which shows that, for sufficiently large n, the vacancy ratio [m/(k+m)], associated with the occurrence of r_n(k)=r_n(k+m), can be arbitrarily close to [1/2]. Here we prove that this ratio cannot reach [1/2], for any n. We then discuss the same phenomenon for packing balls, or translates of another body, in convex containers of various shapes.

Date received: March 11, 2005