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.
Let K be a closed convex centrally symmetric body of positive finite volume in n-dimensional Euclidean space Rn.
The gauge[1] or distance[2][3] Minkowski functional g attached to K is defined by
λ ∈
: x ∈ λ
Conversely, given a norm g on Rn we define K to be
Let Γ be a lattice in Rn.
The successive minima of K or g on Γ are defined by setting the k-th successive minimum λk to be the infimum of the numbers λ such that λK contains k linearly-independent vectors of Γ.
The successive minima satisfy[4][5][6]
λ
A basis of linearly independent lattice vectors b1, b2, ..., bn can be defined by g(bj) = λj.
The lower bound is proved by considering the convex polytope 2n with vertices at ±bj/ λj, which has an interior enclosed by K and a volume which is 2n/n!λ1 λ2...λn times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by λj along each basis vector to obtain 2n n-simplices with lattice point vectors).
To prove the upper bound, consider functions fj(x) sending points x in
to the centroid of the subset of points in
for some real numbers
Then the coordinate transform
λ
λ
has a Jacobian determinant
λ
λ
λ
λ
(specifically the interior of
) is due to convexity and symmetry.
But lattice points in the interior of
, always expressible as a linear combination of
, so any two distinct points of
cannot be separated by a lattice vector.
must be enclosed in a primitive cell of the lattice (which has volume