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

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 $$\frac{2^n}{n!} \operatorname{vol}\left(\mathbb{R}^n/\Gamma\right) \le \lambda_1\lambda_2\cdots\lambda_n \operatorname{vol}(K)\le 2^n \operatorname{vol}\left(\mathbb{R}^n/\Gamma\right).$$

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 $ 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 $$x' = h(x) = \sum_{i=1}^{n} (\lambda_i -\lambda_{i-1}) f_i(x)/2 $$ has a Jacobian determinant $J = \lambda_1 \lambda_2 \ldots \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 \neq 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, \ldots b_{k-1} $ , so any two distinct points of $K' = h(K) = \{ x' \mid h(x) = x' \} $  cannot be separated by a lattice vector. Therefore, $K' $ must be enclosed in a primitive cell of the lattice (which has volume $\operatorname{vol}(\R^n/\Gamma) $ ), and consequently $\operatorname{vol} (K)/J = \operatorname{vol}(K') \le \operatorname{vol}(\R^n/\Gamma) $.