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

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

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

${\frac {2^{n}}{n!}}\mathrm {vol} (\mathbb {R} ^{n}/\Gamma )\leq \lambda _{1}\lambda _{2}\cdots \lambda _{n}\mathrm {vol} (K)\leq 2^{n}\mathrm {vol} (\mathbb {R} ^{n}/\Gamma ).$

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