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

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

Conversely, given a norm g on ℝⁿ we define K to be

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

Let Γ be a lattice in ℝⁿ. The successive minima of K or g on Γ are defined by setting the kth 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

The successive minima satisfy

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

References

• Cassels, J. W. S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 45. Cambridge University Press. Zbl 0077.04801.
• Cassels, J. W. S. (1997). An Introduction to the Geometry of Numbers. Classics in Mathematics (Reprint of 1971 ed.). Springer-Verlag. ISBN 978-3-540-61788-4.
• 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.
• Siegel, Carl Ludwig (1989). Komaravolu S. Chandrasekharan (ed.). Lectures on the Geometry of Numbers. Springer-Verlag. ISBN 3-540-50629-2. Zbl 0691.10021.