Minkowski's second theorem


In mathematics, Minkowski's second theorem is a result in the geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental cell.

Setting

Let be a closed convex centrally symmetric body of positive finite volume in -dimensional Euclidean space. The gauge or distance Minkowski functional attached to is defined by
Conversely, given a norm on we define to be
Let be a lattice in. The successive minima of or on are defined by setting the th successive minimum to be the infimum of the numbers such that contains linearly-independent vectors of. We have.

Statement

The successive minima satisfy

Proof

A basis of linearly independent lattice vectors can be defined by .
The lower bound is proved by considering the convex polytope with vertices at, which has an interior enclosed by and a volume which is times an integer multiple of a primitive cell of the lattice.
To prove the upper bound, consider functions sending points in to the centroid of the subset of points in that can be written as for some real numbers. Then the coordinate transform has a Jacobian determinant. If and are in the interior of and then with , where the inclusion in is due to convexity and symmetry. But lattice points in the interior of are, by definition of, always expressible as a linear combination of, so any two distinct points of cannot be separated by a lattice vector. Therefore, must be enclosed in a primitive cell of the lattice , and consequently.