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

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 m₄=infimum u_i i=1,2,.....,n; extended over all positive real numbers u_i for which u_iK 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 m₁m₂...m_nV(K)≦ 2ⁿd(L) Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].