Two consequences of Minkowski's 2n theorem

Abstract Consider the inequalities (a) ||⩽b,A∈ R r × n r , r positive vector (here |y| denotes the vector of absolute values of components of the vector y ) and x T Ax⩽λ,A positive semi-definite ∈ R n × n r , r 0 Both inequalities are guaranteed a nonzero integer solution x for every positive right-hand side ( b , α respectively). Such solutions will generally have a nonzero orthogonal projection X N ( A ) on the null space of A . We prove that a nonzero integer solution x exists with | x N ( A ) | bounded, for (a): ‖X N(A) ‖⩽ n−r volA b 1 …b r 1 (n−r) for (b): ‖X N(A) ‖⩽ 2 n volA λ r 2 K n 1 (n−r) where vol A= ϵ det 2 A IJ summing over all r × r submatrices A IJ , and K n is the volume of the Euclidean unit ball in R n .