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

Conversely, given a norm g on Rⁿ we define K to be $$K=\{x\in {ℝ}^n:g(x)\le 1\}.$$

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

The successive minima satisfy $$\frac{2^n}{n!}\mathrm{vol}({ℝ}^n/\Gamma )\le {\lambda }_1{\lambda }_2\cdots {\lambda }_n\mathrm{vol}(K)\le 2^n\mathrm{vol}({ℝ}^n/\Gamma ).$$

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 $K$ to the centroid of the subset of points in $K$ that can be written as $x+{\sum }_{i=1}^{j-1}{a}_i{b}_i$ for some real numbers $a_i$. Then the coordinate transform $$x^{\prime }=h(x)=\sum _{i=1}^n({\lambda }_i-{\lambda }_{i-1})f_i(x)/2$$ has a Jacobian determinant $J={\lambda }_1{\lambda }_2\dots {\lambda }_n/2^n$. If $p$ and $q$ are in the interior of $K$ and $p-q={\sum }_{i=1}^k{a}_i{b}_i$(with $a_k\ne 0$) then $$(h(p)-h(q))=\sum _{i=0}^k{c}_i{b}_i\in {\lambda }_{k}K$$ with $c_k={\lambda }_k{a}_k/2$, where the inclusion in ${\lambda }_{k}K$ (specifically the interior of ${\lambda }_{k}K$) is due to convexity and symmetry. But lattice points in the interior of ${\lambda }_{k}K$ are, by definition of ${\lambda }_k$, always expressible as a linear combination of $b_1,b_2,\dots b_{k-1}$, so any two distinct points of $K^{\prime }=h(K)=\{x^{\prime }\mid h(x)=x^{\prime }\}$ cannot be separated by a lattice vector. Therefore, $K^{\prime }$ must be enclosed in a primitive cell of the lattice (which has volume $\mathrm{vol}({ℝ}^n/\Gamma )$), and consequently $\mathrm{vol}(K)/J=\mathrm{vol}(K^{\prime })\le \mathrm{vol}({ℝ}^n/\Gamma )$.