Minkowski's theorem

The Minkowski's lattice point theorem (after Hermann Minkowski ) hits a geometric statement about the location of grid points in certain amounts. If one around the zero point of the lattice symmetric, convex and bounded set exceeds a certain size, it must contain, in addition to the zero point further points of the grid.

Statement of the theorem

Be a lattice in, limited, convex and symmetrical to the zero point. Then applies, so C contains in addition to the zero point another grid point ( and because of the symmetry even two). The volume of the lattice is defined as the volume of a " ground loop ".

Example

An example of a regular grid in the x. Since a grid mesh is here formed by two unit vectors, the volume of this lattice is 1 According to the record, there is not a subset of, the limited, convex and symmetric to the zero point, a surface area > of 4 and next to zero no other lattice point contains.

For squares around the zero point, this can be easy to see, for such a square with area greater than 4 has an edge length greater than 2 and have thus contains the eight grid points. However, this is true for every bounded, centrally symmetric, convex set, so irregular they may be.

Applications

It results in a variety of applications of Minkowski's lattice point theorem, from the approximation of real numbers by fractions ( Dirichlet's unit theorem ) to "practical" issues such as the question of how far a bullet will fly into a ( regularly planted ) forest.