On a conjecture of Schmutz

In this work we discuss Schmutz’s conjecture that in dimension 2 to 8 the distinct norms that occur in the lattices with the best known sphere packings are strictly greater than those in any other lattice of the same covolume. We see that the ternary conjecture is not true. However, it seems that there is but one exception: one lattice, where for one length the conjecture fails. Received: 11 February 2008, Revised: 20 May 2008