Minkowski's second theorem

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

Setting

Let K be a closed convex centrally symmetric body of positive finite volume in n-dimensional Euclidean space Rⁿ. The gauge or distance function g attached to K is defined by

$g(x)=\inf\{\lambda \in \mathbb {R} :x\in \lambda K\}.$

Conversely, given a quadratic form q on Rⁿ we define K to be

$K=\{x\ in\mathbb {R} ^{n}:q(x)\leq 1\}.$

Let Γ be a lattice in Rⁿ. The successive minima of K, g or q on Γ are defined by setting the k-th successive minimum λ_k to be the infimum of the numbers λ such that λK contains k linearly independent vectors of Γ. We have 0 < λ₁ ≤ λ₂ ≤ ... ≤ λ_n < ∞.

Statement of the theorem

Then

${\frac {2^{n}}{n!}}\mathrm {vol} (R^{n}/\Gamma )\leq \lambda _{1}\lambda _{2}\cdots \lambda _{n}\mathrm {vol} (K)\leq 2^{n}\mathrm {vol} (R^{n}/\Gamma ).$

References

• Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer-Verlag. pp. 180–185. ISBN 0-387-94655-1. Zbl 0859.11003.
• Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. p. 6. ISBN 3-540-54058-X. Zbl 0754.11020.