On a conjecture of Schmutz (Q998953)

Motivated by related observations from the hyperbolic world, Paul Schmutz conjectured in the mid nineties of last century that among all Euclidean lattices in dimension three and with covolume 1 there is one lattice which is extremal in the sense that for every natural number \(k\) its \(k\)-th length (counted without multiplicities) is strictly larger than the \(k\)-th length of any other lattice. By asymptotic considerations, this would have to be the normalized root lattice \(A_2.\) The author gives an example of a lattice violating this conjecture and suggests that this phenomenon is the only counterexample to be found. This suggestion in particular holds for the lattice giving the counterexample to Schmutz' original conjecture. It is furthered by yet unpublished results of Willging.