Two finiteness theorems in the Minkowski theory of reduction
1972 ◽
Vol 14
(3)
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pp. 336-351
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Keyword(s):
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.
1970 ◽
Vol 11
(4)
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pp. 385-394
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Keyword(s):
1968 ◽
Vol 8
(1)
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pp. 56-62
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Keyword(s):
2009 ◽
Vol 52
(3)
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pp. 361-365
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Keyword(s):
1998 ◽
Vol 20
(2)
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pp. 143-153
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2007 ◽
Vol 38
(2)
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pp. 159-165
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