Minkowski's second theorem

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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

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Let K be a closed convex centrally symmetric body of positive finite volume in n-dimensional Euclidean space Rn. The gauge[1] or distance[2][3] 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 k-th successive minimum λk to be the infimum of the numbers λ such that λK contains k linearly-independent vectors of Γ. We have 0 < λ1λ2 ≤ ... ≤ λn < ∞.

Statement

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The successive minima satisfy[4][5][6]  

Proof

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A basis of linearly independent lattice vectors b1, b2, ..., bn can be defined by g(bj) = λj.

The lower bound is proved by considering the convex polytope 2n with vertices at ±bj/ λj, which has an interior enclosed by K and a volume which is 2n/n!λ1 λ2...λn times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by λj along each basis vector to obtain 2n n-simplices with lattice point vectors).

To prove the upper bound, consider functions fj(x) sending points x in   to the centroid of the subset of points in   that can be written as   for some real numbers  . Then the coordinate transform   has a Jacobian determinant  . If   and   are in the interior of   and  (with  ) then   with  , where the inclusion in   (specifically the interior of  ) is due to convexity and symmetry. But lattice points in the interior of   are, by definition of  , always expressible as a linear combination of  , so any two distinct points of   cannot be separated by a lattice vector. Therefore,   must be enclosed in a primitive cell of the lattice (which has volume  ), and consequently  .

References

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  1. ^ Siegel (1989) p.6
  2. ^ Cassels (1957) p.154
  3. ^ Cassels (1971) p.103
  4. ^ Cassels (1957) p.156
  5. ^ Cassels (1971) p.203
  6. ^ Siegel (1989) p.57
  • Cassels, J. W. S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 45. Cambridge University Press. Zbl 0077.04801.
  • 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. Vol. 165. Springer-Verlag. pp. 180–185. ISBN 0-387-94655-1. Zbl 0859.11003.
  • Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. p. 6. ISBN 3-540-54058-X. Zbl 0754.11020.
  • Siegel, Carl Ludwig (1989). Komaravolu S. Chandrasekharan (ed.). Lectures on the Geometry of Numbers. Springer-Verlag. ISBN 3-540-50629-2. Zbl 0691.10021.