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An elementary proof of Minkowski's second inequality

Published online by Cambridge University Press:  09 April 2009

I. Danicic
I. Danicic
Affiliation:
University College of WalesAberystwyth Gt. Britain
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Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 10 , Issue 1-2 , August 1969 , pp. 177 - 181
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Mordell, L. J., ‘On some arithmetical results in the Geometry of Numbers’, Compositio Math. 1 (1934).Google Scholar
[2]Davenport, H., ‘Minkowski's inequality for the minima associated with a convex body’, Quarterly J. of Math. (10) 38, (1939), 119–121.CrossRefGoogle Scholar
[3]Davenport, H., ‘Indefinite quadratic forms in many variables(II)’, Proc. Lond. Math. Soc. (3) 8 (1958), 109–126.CrossRefGoogle Scholar
[4]Weyl, H., ‘On Geometry of Numbers’, Proc. Lond. Math. Soc. (2) 47 (1942), 268–289.CrossRefGoogle Scholar
[5]Minkowski, H., Geometrie der Zahlen (Teubner 1896) chapter 5.Google Scholar
[6]Bambah, R. P., Woods, A. C. and Zassenhaus, H., ‘Three proofs of Minkowski's second inequality in the geometry of numbers’, J. Aust. Math. Soc. 5 (1965), 453–462.CrossRefGoogle Scholar