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.
Contents
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
Conversely, given a norm g on ℝn we define K to be
Let Γ be a lattice in ℝn. The successive minima of K or g on Γ are defined by setting the kth successive minimum λk to be the infimum of the numbers λ such that λK contains k linearly-independent vectors of Γ. We have 0 < λ1 ≤ λ2 ≤ ... ≤ λn < ∞.
Statement of the theorem
The successive minima satisfy
References
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