Here is the proof of Minkowski's theorem:
1) We can translate any part of the region into
the central 2x2 square translating by an even integer distance horizontally
and shifting by an even number vertically.
2) Because of the area condition, there will be overlap. This means
that two points A - (2a,2b) = B - (2c,2d) where a,b,c,d are integers.
3) Combining 1) and 2) means that A-B = (2n,2m), where n,m are integers.
3) Symmetry and the fact that B is in the region implies that also (-B) is in the region.
4) Convexity implies that (A + (-B))/2 is in the region. But this is (n,m), a lattice point.
The following pictures illustrate the part of the proof which translates everything into the middle 2x2 square.
We rotate the region and at each time have shifted all the parts outside the middle 2x2 square into the 2x2 middle
square: