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

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

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

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

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

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

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

Proof

A basis of linearly independent lattice vectors b₁ , b₂ , ... b_n can be defined by g(b_j) = λ_j .

The lower bound is proved by considering the convex polytope 2n with vertices at ±b_j/ λ_j, which has an interior enclosed by K and a volume which is 2ⁿ/n!λ₁ λ₂...λ_n times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by λ_j along each basis vector to obtain 2ⁿ n-simplices with lattice point vectors).

To prove the upper bound, consider functions f_j(x) sending points x in ${\textstyle K}$ to the centroid of the subset of points in ${\textstyle K}$ that can be written as ${\textstyle x+\sum _{i=1}^{j-1}a_{i}b_{i}}$ for some real numbers ${\textstyle a_{i}}$. Then the coordinate transform ${\textstyle x'=h(x)=\sum _{i=1}^{n}(\lambda _{i}-\lambda _{i-1})f_{i}(x)/2}$ has a Jacobian determinant ${\textstyle J=\lambda _{1}\lambda _{2}\ldots \lambda _{n}/2^{n}}$. If ${\textstyle p}$ and ${\textstyle q}$ are in the interior of ${\textstyle K}$ and ${\textstyle p-q=\sum _{i=1}^{k}a_{i}b_{i}}$(with ${\textstyle a_{k}\neq 0}$) then ${\textstyle (h(p)-h(q))=\sum _{i=0}^{k}c_{i}b_{i}\in \lambda _{k}K}$ with ${\textstyle c_{k}=\lambda _{k}a_{k}/2}$ , where the inclusion in ${\textstyle \lambda _{k}K}$ (specifically the interior of ${\textstyle \lambda _{k}K}$) is due to convexity and symmetry. But lattice points in the interior of ${\textstyle \lambda _{k}K}$ are, by definition of ${\textstyle \lambda _{k}}$, always expressible as a linear combination of ${\textstyle b_{1},b_{2},\ldots b_{k-1}}$, so any two distinct points of ${\textstyle K'=h(K)=\lbrace x'|h(x)=x'\rbrace }$ cannot be separated by a lattice vector. Therefore, ${\textstyle K'}$ must be enclosed in a primitive cell of the lattice (which has volume ${\textstyle {\text{vol}}(\mathbb {R} ^{n}/\Gamma )}$) , and consequently ${\textstyle {\text{vol}}(K)/J={\text{vol}}(K')\leq {\text{vol}}(\mathbb {R} ^{n}/\Gamma )}$.

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.