ABSTRACT
A major result in the geometry of numbers is Minkowski’s second convex bodies theorem. This result gives bounds for the product of the successive minima of a convex body with respect to a lattice in terms of the volume of the body and the determinant (covolume) of the lattice. Here by convex body we mean a non-empty compact and convex subset of ℝ n such that r x is an interior point whenever x is in the subset and |r| < 1, and by lattice we mean a discrete https://www.w3.org/1998/Math/MathML"> Z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138747022/abb99e87-ffb7-4196-a4c4-acb0033e3d6a/content/eq2931.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -module spanning ℝ n . If C is a convex body and Λ is a lattice, the successive minima λi (C, Λ) for https://www.w3.org/1998/Math/MathML"> i = 1 , … , n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138747022/abb99e87-ffb7-4196-a4c4-acb0033e3d6a/content/eq2932.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are defined by https://www.w3.org/1998/Math/MathML"> λ i ( C , Λ ) = inf r > 0 { r : r C ∩ Λ contains i linearly independent vectors} . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138747022/abb99e87-ffb7-4196-a4c4-acb0033e3d6a/content/eq2933.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
