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 $ℝ n$ . The gauge^[1] or distance^[2]^[3] 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 $ℝ n$ we define $K$ to be

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

Let $Γ$ be a lattice in $ℝ n$ . 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 $0 < λ 1 \leq λ 2 \leq ... \leq λ n < \infty$ .

Statement of the theorem

The successive minima satisfy^[4]^[5]^[6]

{\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

↑ Siegel (1989) p.6
↑ Cassels (1957) p.154
↑ Cassels (1971) p.103
↑ Cassels (1957) p.156
↑ Cassels (1971) p.203
↑ 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.

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