Minkowski's second theorem - Alchetron, the free social encyclopedia

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

g ( x ) = inf { λ ∈ R : x ∈ λ K } .

Conversely, given a norm g on ℝⁿ we define K to be

K = { x ∈ R n : g ( x ) ≤ 1 } .

Let Γ be a lattice in ℝⁿ. The successive minima of K or g on Γ are defined by setting the kth 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 of the theorem

The successive minima satisfy

2 n n ! vol ⁡ ( R n / Γ ) ≤ λ 1 λ 2 ⋯ λ n vol ⁡ ( K ) ≤ 2 n vol ⁡ ( R n / Γ ) .

References

Minkowski's second theorem Wikipedia

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Contents

Setting

Statement of the theorem

References