# 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}$$^{j−1}$a$_i$b$_i${\textstyle x+\sum _{i=1}^{j-1}a_{i}b_{i}} for some real numbers a$_i${\textstyle a_{i}}. Then the coordinate transform $x'=h(x)=\sum _{i=1}^{n}(\lambda _{i}-\lambda _{i-1})f_{i}(x)/2$ has a Jacobian determinant J=λ$_1$λ$_2$…λ$_n$/2$^n${\textstyle J=\lambda _{1}\lambda _{2}\ldots \lambda _{n}/2^{n}}. If p{\textstyle p} and q{\textstyle q} are in the interior of K{\textstyle K} and p−q=∑$_{i=1}$$^k$a$_i$b$_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$_k$a$_k$/2{\textstyle c_{k}=\lambda _{k}a_{k}/2}, where the inclusion in λ$_k$K{\textstyle \lambda _{k}K} (specifically the interior of λ$_k$K{\textstyle \lambda _{k}K}) is due to convexity and symmetry. But lattice points in the interior of λ$_k$K{\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 )}.