Minkowski's theorem

Minkowski's theorem is a result about points with integer coordinates (lattice points) inside certain shapes. It says that if you have a symmetric shape (like a circle or square centered at the origin) in the plane, and its area is large enough compared to the grid of integer points, then the shape must contain a lattice point other than the origin.

Important properties

• Applies to convex, centrally symmetric shapes (if $(x, y)$ is in the shape, so is $(-x, -y)$).

• If the area of the shape is greater than $4$, it must contain a nonzero lattice point.

• The theorem generalizes to higher dimensions, using volume and the concept of lattices.