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Two finiteness theorems in the Minkowski theory of reduction

Published online by Cambridge University Press:  09 April 2009

P. W. Aitchison
P. W. Aitchison
Affiliation:
Department of Mathematics, University of ManitobaWinnipeg, Canada
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Minkowski proved two important finiteness theorems concerning the reduction theory of positive definite quadratic forms (see [6], p. 285 or [7], §8 and §10). A positive definite quadratic form in n variables may be considered as an ellipsoid in n-dimensional Euclidean space, Rn, and then the two results can be investigated more generally by replacing the ellipsoid by any symmetric convex body in Rn. We show here that when n≧3 the two finiteness theorems hold only in the case of the ellipsoid. This is equivalent to showing that Minkowski's results do not hold in a general Minkowski space, namely in a euclidean space where the unit ball is a general symmetric convex body instead of the sphere or ellipsoid.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 3 , November 1972 , pp. 336 - 351
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Aitchison, P. W. ‘A Characterisation of the Ellipsoid.’ J. Australian Math. Soc. 11 (1970), 385–394.CrossRefGoogle Scholar
[2]Benson, R. V., Euclidean Geometry and Convexity (McGraw-Hill, U.S.A., 1966).Google Scholar
[3]Bonnesen, T., and Fenchel, W.Theorie der knovexen Körper (Reprint by Chelsea, New York, 1948).Google Scholar
[4]Busemann, H., Geometry of Geodesics (Academic Press, New York, 1955).Google Scholar
[5]Cassels, J. W. S., An Introduction to the Geometry of Numbers (Springer-Verlag, Berlin, 1959).CrossRefGoogle Scholar
[6]Weyl, H., ‘On Geometry of Numbers’. Proc. Lond. Math. Soc. 47 (1942), 268–289.CrossRefGoogle Scholar
[7]van der Waerden, B. L., ‘Die Redukionstheorie der positiven quadratischen Formen.’ Acta Math. 96 (1956), 265–309.CrossRefGoogle Scholar
[8]Minkowski, H., Geometrie der Zahlen (Reprint by Chelsea, New York, 1953).Google Scholar