On packing spheres into containers

Abstract

In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and $d\ge 2$ we show that solutions to the container problem can not have a “simple structure” for large $n$. By this we in particular find that there exist arbitrary small $r>0$, such that packings in a smooth, $3$-dimensional convex body, with a maximum number of spheres of radius $r$, are necessarily not hexagonal close packings. This contradicts Kepler's famous statement that the cubic or hexagonal close packing “will be the tightest possible, so that in no other arrangement more spheres could be packed into the same container”.