Partial solution of a conjecture of Schmutz
It is proved that every non-arithmetic lattice in the Euclidean plane has every non-zero length shorter than the corresponding length of the \(A_2\)-lattice, when the lengths are numerated according to size (no multiplicities!) and when both lattices have the same covolume. This is done by elementary means. The conjecture of Schmutz is that the same fact holds for every lattice of the same covolume which is not isometric to the \(A_2\)-lattice. It does not seem to be treatable in the arithmetic case.
scientific article; zbMATH DE number 938589