Let
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
be a convex polygon in the plane and suppose that
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
is positioned in the coordinate system in such a way that
![\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),](/media/m/2/2/4/224ac8fad397280c4996b7e67104b2ae.png)
where the
![Q_i](/media/m/0/6/c/06c3347a281d624861ac22da912fe6e2.png)
denote the quadrants of the plane. Prove that if
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
contains no nonzero lattice point, then the area of
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
is less than
%V0
Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that
$$\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),$$
where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$