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 $ℝ^{n}$ . The gauge or distance Minkowski functional $g$ attached to $K$ is defined by $g (x) = inf {λ \in ℝ : x \in λ K} .$

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

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 $0 < λ 1 \leq λ 2 \leq ... \leq λ n < \infty$ .

Statement

The successive minima satisfy $\frac{2^{n}}{n!} vol (ℝ^{n} / Γ) \leq λ_{1} λ_{2} \dots λ_{n} vol (K) \leq 2^{n} vol (ℝ^{n} / Γ) .$

Proof

A basis of linearly independent lattice vectors $b 1, b 2, ..., b n$ 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 $2 n$ with vertices at $\pm b j / λ j$ , 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), the integer being the index of the $ℤ$ -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 $K$ 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 $x^{'} = h (x) = \sum_{i = 1}^{n} (λ_{i} - λ_{i - 1}) f_{i} (x) / 2$ has a Jacobian determinant $J = λ_{1} λ_{2} \dots λ_{n} / 2^{n}$ . 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 $(h (p) - h (q)) = \sum_{i = 0}^{k} c_{i} b_{i} \in λ_{k} K$ with $c_{k} = λ_{k} a_{k} / 2$ , where the inclusion in $λ_{k} K$ (specifically the interior of $λ_{k} K$ ) is due to convexity and symmetry. But lattice points in the interior of $λ_{k} K$ are, by definition of $λ_{k}$ , always expressible as a linear combination of $b_{1}, b_{2}, \dots b_{k - 1}$ , so any two distinct points of $K^{'} = h (K) = {x^{'} ∣ h (x) = x^{'}}$ cannot be separated by a lattice vector.

The entire space $ℝ^{n}$ is tiled by translated copies of a primitive cell $ℝ^{n} = ⋃_{v \in Γ} ((ℝ^{n} / Γ) + v)$ . The intersections $K'_{v} := K^{'} \cap ((ℝ^{n} / Γ) + v)$ with $K^{'}$ have a total volume in sum that is less than the volume of a single primite cell $ℝ^{n} / Γ$ and therefore $vol (K) \cdot J = vol (K^{'}) \leq vol (ℝ^{n} / Γ) .$ To see this, consider two intersections $K'_{u}$ and $K'_{v}$ with $u, v \in Γ$ and their translated versions $K'_{u} - u$ and $K'_{v} - v$ that are both contained in the primitive cell located at the origin. If they would intersect at some point $y \in (ℝ^{n} / Γ)$ , then the two original points $y + u \in K'_{u} \subseteq K^{'}$ and $y + v \in K'_{v} \subseteq K^{'}$ would be separated by a lattice vector, which can be seen by considering the difference $(y + u) - (y + v) = u - v \in Γ$ . This cannot be the case and therefore, translating every of the intersections into the primitive cell located at the origin does not lead to any intersections, which shows the claim.