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Minkowski's fundamental inequality for reduced positive quadratic forms(II)

Published online by Cambridge University Press:  09 April 2009

E. S. Barnes
E. S. Barnes
Affiliation:
Vice-Chancellor's OfficeUniversity of Adelaide, South Australia
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Abstract

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A convex polytope D (α) was defined in Barnes (1978) as the set of Minkowski-reduced forms with prescribed diagonal coefficients α1, α2,…αn. A local minimum of the determinant D(f) over D(α) must occur at a vertex of D(α). Here a criterion is obtained for a given vertex to provide a local minimum, completely analogous to Voronoï's criterion for a perfect form to be extreme.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 1 , February 1979 , pp. 1 - 6
Copyright
Copyright © Australian Mathematical Society 1979

References

Barnes, E. S. (1978), ‘On Minkowski's fundamental inequality for reduced positive quadratic forms (I)’, J. Austral. Math. Soc. 26, 46–52.CrossRefGoogle Scholar
Barnes, E. S. (1957), ‘On a theorem of Voronoï’, Proc. Cambridge Phil. Soc. 53, 537–539.CrossRefGoogle Scholar
Stiemke, E. (1915), ‘Über positive Lösungen homogener linearer Gleichungen’, Math. Ann. 76, 340–342.CrossRefGoogle Scholar
Voronoï, G. (1907), ‘Sur quelques propriétés des formes quadratiques positives parfaites’, J. reine angew. Math. 133, 97–178.Google Scholar