Minimal requirements for Minkowski's theorem in the plane I

Abstract

Let K be a closed convex set in the Euclidean plane, with area A(K), which contains in its interior only one point 0 of the integer lattice. If K has other than one or three chords bisected by 0, it is shown that A(K) ≤ 4. Also, if K has three such chords, A(K) ≤ 4.5. The results are generalised to any lattice in the plane.