A Class of Geometric Lattices

and that the same holds if any n— 1 of the signs are replaced by strict inequality. Those unimodular lattices, such as Z , which have only the origin in common with the open cube shall be called critical, as shall the corresponding matrices. Minkowski conjectured, and Hajos [ l ] proved in 1938, that a critical lattice must contain one of the points (5»i, • • • , 5,-»), i = 1, • • • , n. If A is critical then so is any matrix obtained from it by permuting rows and post-multiplication by integral unimodular matrices: such matrices will be called equivalent to A. An induction argument shows that Hajos' theorem is the same as the assertion: A is critical if and only if it is equivalent to a matrix with ones on the diagonal and all zeros above, Siegel [3] tried to prove Minkowski's conjecture by showing that, if A is critical, then each point other than 0 of the lattice corresponding to A has at least one coordinate in Z*, the set of nonzero integers. If we consider the set of matrices A denned by the property