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 be a closed convex centrally symmetric body of positive finite volume in -dimensional Euclidean space . The ''gauge'' or ''distance'' Minkowski functional attached to is defined by
$g(x) = \inf \left\ .$

Conversely, given a norm on we define to be
$K = \left\ .$

Let be a lattice in . The successive minima of or on are defined by setting the -th successive minimum to be the infimum of the numbers such that contains linearly-independent vectors of . We have .

Statement

The successive minima satisfySiegel (1989) p.57
$\frac \operatorname\left(\mathbb^n/\Gamma\right) \le \lambda_1\lambda_2\cdots\lambda_n \operatorname(K)\le 2^n \operatorname\left(\mathbb^n/\Gamma\right).$

Proof

A basis of linearly independent lattice vectors can be defined by .

The lower bound is proved by considering the convex polytope with vertices at , which has an interior enclosed by and a volume which is times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by along each basis vector to obtain -simplices with lattice point vectors).

To prove the upper bound, consider functions sending points in $K$ to the centroid of the subset of points in $K$ that can be written as $x + \sum_^ a_i b_i$ for some real numbers $a_i$. Then the coordinate transform $x' = h(x) = \sum_^ (\lambda_i -\lambda_) f_i(x)/2$ 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 $p-q = \sum_^k a_i b_i$(with $a_k \neq 0$) then $(h(p) - h(q)) = \sum_^k c_i b_i \in \lambda_k K$ 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 $b_1, b_2, \ldots b_$, so any two distinct points of $K' = h(K) = \$ cannot be separated by a lattice vector. Therefore, $K'$ must be enclosed in a primitive cell of the lattice (which has volume $\operatorname(\R^n/\Gamma)$), and consequently $\operatorname (K)/J = \operatorname(K') \le \operatorname(\R^n/\Gamma)$.

References

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* {{cite book , first=Carl Ludwig , last=Siegel , author-link=Carl Ludwig Siegel , title=Lectures on the Geometry of Numbers , publisher=Springer-Verlag , year=1989 , isbn=3-540-50629-2 , editor=Komaravolu S. Chandrasekharan , editor-link=Komaravolu S. Chandrasekharan , zbl=0691.10021

Hermann Minkowski