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 Rⁿ. The gauge or distance Minkowski functional g attached to K is defined by [math]\displaystyle{ g(x) = \inf \left\{\lambda \in \mathbb{R} : x \in \lambda K \right\} . }[/math]

Conversely, given a norm g on Rⁿ we define K to be [math]\displaystyle{ K = \left\{ x \in \R^n : g(x) \le 1 \right\} . }[/math]

Let Γ be a lattice in Rⁿ. 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 < λ₁ ≤ λ₂ ≤ ... ≤ λ_n < ∞.

Statement

The successive minima satisfy [math]\displaystyle{ \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). }[/math]

Proof

A basis of linearly independent lattice vectors b₁, b₂, ..., 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 2ⁿ/n!λ₁ λ₂...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 2ⁿ n-simplices with lattice point vectors).

To prove the upper bound, consider functions f_j(x) sending points x in [math]\displaystyle{ K }[/math] to the centroid of the subset of points in [math]\displaystyle{ K }[/math] that can be written as [math]\displaystyle{ x + \sum_{i=1}^{j-1} a_i b_i }[/math] for some real numbers [math]\displaystyle{ a_i }[/math]. Then the coordinate transform [math]\displaystyle{ x' = h(x) = \sum_{i=1}^{n} (\lambda_i -\lambda_{i-1}) f_i(x)/2 }[/math] has a Jacobian determinant [math]\displaystyle{ J = \lambda_1 \lambda_2 \ldots \lambda_n/2^n }[/math]. If [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] are in the interior of [math]\displaystyle{ K }[/math] and [math]\displaystyle{ p-q = \sum_{i=1}^k a_i b_i }[/math](with [math]\displaystyle{ a_k \neq 0 }[/math]) then [math]\displaystyle{ (h(p) - h(q)) = \sum_{i=0}^k c_i b_i \in \lambda_k K }[/math] with [math]\displaystyle{ c_k = \lambda_k a_k /2 }[/math], where the inclusion in [math]\displaystyle{ \lambda_k K }[/math] (specifically the interior of [math]\displaystyle{ \lambda_k K }[/math]) is due to convexity and symmetry. But lattice points in the interior of [math]\displaystyle{ \lambda_k K }[/math] are, by definition of [math]\displaystyle{ \lambda_k }[/math], always expressible as a linear combination of [math]\displaystyle{ b_1, b_2, \ldots b_{k-1} }[/math], so any two distinct points of [math]\displaystyle{ K' = h(K) = \{ x' \mid h(x) = x' \} }[/math] cannot be separated by a lattice vector. Therefore, [math]\displaystyle{ K' }[/math] must be enclosed in a primitive cell of the lattice (which has volume [math]\displaystyle{ \operatorname{vol}(\R^n/\Gamma) }[/math]), and consequently [math]\displaystyle{ \operatorname{vol} (K)/J = \operatorname{vol}(K') \le \operatorname{vol}(\R^n/\Gamma) }[/math].

References

• Cassels, J. W. S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. 45. Cambridge University Press. 
• Cassels, J. W. S. (1997). An Introduction to the Geometry of Numbers. Classics in Mathematics (Reprint of 1971 ed.). Springer-Verlag. ISBN 978-3-540-61788-4. 
• Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. 165. Springer-Verlag. pp. 180–185. ISBN 0-387-94655-1. 
• Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. 1467 (2nd ed.). Springer-Verlag. p. 6. ISBN 3-540-54058-X. 
• Siegel, Carl Ludwig (1989). Komaravolu S. Chandrasekharan. ed. Lectures on the Geometry of Numbers. Springer-Verlag. ISBN 3-540-50629-2.