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

Let K be a bounded, open convex set in euclidean n-space R<SUB>n</SUB>, symmetric in the origin 0. Further let L be a lattice in R<SUB>n</SUB> containing 0 and put m<SUB>4</SUB>=infimum u<SUB>i</SUB> i=1,2,.....,n; extended over all positive real numbers u<SUB>i</SUB> for which u<SUB>i</SUB>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 m<SUB>1</SUB>m<SUB>2</SUB>...m<SUB>n</SUB>V(K)≦ 2<SUP>n</SUP>d(L) Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1]