Macroemulsion

In mathematics, Minkowski's second theorem is a result in the Geometry of numbers about the values taken by a quadratic form 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

$g(x)=\inf \{\lambda \in \mathbb{ℝ}:x\in \lambda K\}.$

Conversely, given a quadratic form q on Rⁿ we define K to be

$K=\{x\in {\mathbb{ℝ}}^n:q(x)\le 1\}.$

Let Γ be a lattice in Rⁿ. The successive minima of K, g or q 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 of the theorem

The successive minima satisfy

$\frac{2^n}{n!}\mathrm{v}\mathrm{o}\mathrm{l}({\mathbb{ℝ}}^n/\mathrm{\Gamma })\le {\lambda }_1{\lambda }_2\cdots {\lambda }_n\mathrm{v}\mathrm{o}\mathrm{l}(K)\le 2^n\mathrm{v}\mathrm{o}\mathrm{l}({\mathbb{ℝ}}^n/\mathrm{\Gamma }).$

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534