Minkowski's Theorem with Curvature Limitations (I)

Extract

The well-known theorem of Minkowski, [1], [2], states that;

(M) a plane convex region, symmetric about the origin O, includes a lattice point other than O if its area is greater than 4.

By a lattice point we shall understand a point in the plane, both of whose coordinates are rational integers. In connection with (M) a critical region is defined to be a convex symmetric region of area 4, which includes no lattice point other than O. One such region is the open square S={(x, y)| |x| < 1, |y|<1} an infinite set of critical regions is formed by the parallelograms bounded by the lines y = x+1, y = x-1, y-1 = k (x-l), y+1 = k(x+l), 0≤-k<∞. Finally, there is a critical hexagon H, bounded by the following six lines: y-1 = -(x-l), y+1 = -(x+1), y-1 = mx, y+1 = mx, y = (1/m)(x-1), y = (l/m)(x+l), where m = tan π/12. All the vertex angles of H are equal to 2π/3.