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^[1] or distance^[2][3] Minkowski functional g attached to K is defined by

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

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

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

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 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^[4][5][6]

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

References

1. Siegel (1989) p.6
2. Cassels (1957) p.154
3. Cassels (1971) p.103
4. Cassels (1957) p.156
5. Cassels (1971) p.203
6. Siegel (1989) p.57

• Cassels, J.W.S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. 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. 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. 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.