by Achill Schürmann.

**Series:** 2005-18, Preprints

- MSC:
- 52C17 Packing and covering in $n$ dimensions
- 05B40 Packing and covering
- 01A45 17th century

**Abstract:**

In Euclidean $d$-spaces, 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 $d\geq 2$ we show that sequences of solutions to the container

problem can not have a ``simple structure''.

By this we in particular find that there exist arbitrary small $r>0$

such that packings with spheres of radius $r$

into a smooth $3$-dimensional convex body

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''.

**Keywords:**

sphere packings, Kepler, container problem