Minkowski's second theorem

Template:Short description 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 Template:Mvar be a closed convex centrally symmetric body of positive finite volume in Template:Mvar-dimensional Euclidean space Template:Math. The gauge or distance Minkowski functional Template:Math attached to Template:Math is defined by $$g(x)=inf\{\lambda \in ℝ:x\in \lambda K\}.$$

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

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

Statement

The successive minima satisfy $$\frac{2^n}{n!}\mathrm{vol}({ℝ}^n/\mathrm{\Gamma })\le {\lambda }_1{\lambda }_2\cdots {\lambda }_n\mathrm{vol}(K)\le 2^n\mathrm{vol}({ℝ}^n/\mathrm{\Gamma }).$$

Proof

A basis of linearly independent lattice vectors Template:Math can be defined by Template:Math (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 Template:Math with vertices at Template:Math, which has an interior enclosed by Template:Mvar and a volume which is Template:Math times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by Template:Math along each basis vector to obtain Template:Math [[Simplex|Template:Mvar-simplices]] with lattice point vectors), the integer being the index of the $\mathbb{Z}$-span of the family $(b_i{)}_i$ as a subgroup of the lattice.

To prove the upper bound, consider functions Template:Math sending points Template:Mvar 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 $$x^{\prime }=h(x)=\sum _{i=1}^n({\lambda }_i-{\lambda }_{i-1})f_i(x)/2$$ has a Jacobian determinant $J={\lambda }_1{\lambda }_2\dots {\lambda }_n/2^n$. If $p$ and $q$ are in the interior of $K$ and $p-q={\sum }_{i=1}^k{a}_i{b}_i$(with $a_k\ne 0$) 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 $b_1,b_2,\dots b_{k-1}$, so any two distinct points of $K^{\prime }=h(K)=\{x^{\prime }\mid h(x)=x^{\prime }\}$ cannot be separated by a lattice vector. Therefore, $K^{\prime }$ must be enclosed in a primitive cell of the lattice (which has volume $\mathrm{vol}({ℝ}^n/\mathrm{\Gamma })$), and consequently $\mathrm{vol}(K)/J=\mathrm{vol}(K^{\prime })\le \mathrm{vol}({ℝ}^n/\mathrm{\Gamma })$.

References

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