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

Bambah, R. P. ; Woods, Alan ; Zassenhaus, Hans (1965) Three proofs of Minkowski's second inequality in the geometry of numbers Journal of the Australian Mathematical Society, 5 (4). pp. 453-462. ISSN 1446-7887

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Related URL: http://dx.doi.org/10.1017/S1446788700028482

Abstract

Let K be a bounded, open convex set in euclidean n-space Rn, symmetric in the origin 0. Further let L be a lattice in Rn containing 0 and put m4=infimum ui i=1,2,.....,n; extended over all positive real numbers ui for which uiK 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...mnV(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].

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