Editing Geometry of numbers (section)

==Minkowski's results==
{{Main article|Minkowski's theorem}}
Suppose that <math>\Gamma</math> is a [[Lattice (group)|lattice]] in <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math> and <math>K</math> is a convex centrally symmetric body.
[[Minkowski's theorem]], sometimes called Minkowski's first theorem, states that if <math>\operatorname{vol} (K)>2^n \operatorname{vol}(\mathbb{R}^n/\Gamma)</math>, then <math>K</math> contains a nonzero vector in <math>\Gamma</math>.

{{Main article|Minkowski's second theorem}}

The successive minimum <math>\lambda_k</math> is defined to be the [[Infimum|inf]] of the numbers <math>\lambda</math> such that <math>\lambda K</math> contains <math>k</math> linearly independent vectors of <math>\Gamma</math>.
Minkowski's theorem on [[successive minima]], sometimes called [[Minkowski's second theorem]], is a strengthening of his first theorem and states that<ref>Cassels (1971) p. 203</ref>
:<math>\lambda_1\lambda_2\cdots\lambda_n \operatorname{vol} (K)\le 2^n \operatorname{vol} (\mathbb{R}^n/\Gamma).</math>