A remark on translated sets

Extract

Let Λ denote the integral lattice in n-dimensional Euclidean space. A classical theorem of Minkowski's states that any bounded closed convex region K symmetrical in the origin 0 and with volume 2ⁿ contains a point of Λ other than 0. There will be a lattice point other than 0 in the interior of K except when K has certain forms, of which we will denote an arbitrary one by K*. An example of a K* is the cube |x_i| ≤ 1 (i = 1, 2,..., n), and more generally a famous theorem of Hajós (3) states that if K* is a parallelepiped it is defined (except for integral unimodular transformations of the x's) by inequalities of the form|x₁| ≤ 1, |a₂₁x₁ + x₂| ≤ 1, …, |a_n1x₁ + … + a_{n, n-1}x_n-1+x_n| ≤ 1.