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.

Statement

Suppose that Γ is a lattice in n-dimensional Euclidean space Rⁿ and K is a convex centrally symmetric body. The k-th successive minimum λ_k is defined to be the inf of the numbers λ such that λK contains k linearly independent vectors of Γ. We have 0 < λ₁ ≤ λ₂ ≤ ... ≤ λ_n < ∞. 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.