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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

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