Minkowski's second theorem
# 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 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 $R^n$ 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 < $λ_1$ ≤ $λ_2$ ≤ ... ≤ $λ_n$ < ∞.

## 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 $b_1,$ $b_2,$ ..., $b_n$ 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 $2^n/n!λ_1$ $λ_2...λ_n$ 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 $2^n$ n-simplices with lattice point vectors).

To prove the upper bound, consider functions $f_j(x)$ sending points x in K{\textstyle K} to the centroid of the subset of points in K{\textstyle K} that can be written as $x+∑_{i=1}^{j−1}a_ib_i{\textstyle$ x+\sum _{i=1}^{j-1}a_{i}b_{i}} for some real numbers $a_i{\textstyle$ 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 $J=λ_1λ_2…λ_n/2^n{\textstyle$ J=\lambda _{1}\lambda _{2}\ldots \lambda _{n}/2^{n}}. If p{\textstyle p} and q{\textstyle q} are in the interior of K{\textstyle K} and $p−q=∑_{i=1}^ka_ib_i{\textstyle$ p-q=\sum _{i=1}^{k}a_{i}b_{i}}(with $a_k≠0{\textstyle$ a_{k}\neq 0}) then

$(h(p)-h(q))=\sum _{i=0}^{k}c_{i}b_{i}\in \lambda _{k}K$

with $c_k=λ_ka_k/2{\textstyle$ c_{k}=\lambda _{k}a_{k}/2}, where the inclusion in $λ_kK{\textstyle$ \lambda _{k}K} (specifically the interior of $λ_kK{\textstyle$ \lambda _{k}K}) is due to convexity and symmetry. But lattice points in the interior of $λ_kK{\textstyle$ \lambda _{k}K} are, by definition of $λ_k{\textstyle$ \lambda _{k}}, always expressible as a linear combination of $b_1,$ $b_2,$ $…b_{k−1}{\textstyle$ b_{1}, b_{2}, \ldots b_{k-1}}, so any two distinct points of $K^′=h(K)={x^′∣h(x)=x^′}{\textstyle$ K'=h(K)=\{x'\mid h(x)=x'\}} cannot be separated by a lattice vector. Therefore, $K^′{\textstyle$ K'} must be enclosed in a primitive cell of the lattice (which has volume $vol⁡(R^n/Γ){\textstyle$ \operatorname {vol} (\mathbb {R} ^{n}/\Gamma )}), and consequently $vol⁡(K)/J=vol⁡(K^′)≤vol⁡(R^n/Γ){\textstyle$ \operatorname {vol} (K)/J=\operatorname {vol} (K')\leq \operatorname {vol} (\mathbb {R} ^{n}/\Gamma )}.

