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. It is named so because it is a strengthening of Minkowski's theorem.

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 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

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

Proof

A basis of linearly independent lattice vectors b₁, b₂, ..., b_n can be defined by g(b_j) = λ_j (warning : it may be not a basis of the lattice, but only a basis of the ambient space).

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), the integer being the index of the $\mathbb {Z}$-span of the family $(b_{i})_{i}$ as a subgroup of the lattice.

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 $$x'=h(x)=\sum _{i=1}^{n}(\lambda _{i}-\lambda _{i-1})f_{i}(x)/2$$ 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 $$(h(p)-h(q))=\sum _{i=0}^{k}c_{i}b_{i}\in \lambda _{k}K$$ 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 ${\textstyle K'=h(K)=\{x'\mid h(x)=x'\}}$ cannot be separated by a lattice vector. Therefore, ${\textstyle K'}$ must be enclosed in a primitive cell of the lattice (which has volume ${\textstyle \operatorname {vol} (\mathbb {R} ^{n}/\Gamma )}$), and consequently ${\textstyle \operatorname {vol} (K)/J=\operatorname {vol} (K')\leq \operatorname {vol} (\mathbb {R} ^{n}/\Gamma )}$.

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.