Three proofs of Minkowski's second inequality in the geometry of numbers

Source abstract

Let K be a bounded, open convex set in euclidean n -space R n , symmetric in the origin 0. Further let L be a lattice in R n containing 0 and put extended over all positive real numbers u i for which u i K contains i linearly independent points of L . Denote the Jordan content of K by V ( K ) and the determinant of L by d ( L ). Minkowski's second inequality in the geometry of numbers states that Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].