Two consequences of Minkowski's \(2^ n\) theorem (Q1357754)
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scientific article; zbMATH DE number 1021698
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Two consequences of Minkowski's \(2^ n\) theorem |
scientific article; zbMATH DE number 1021698 |
Statements
Two consequences of Minkowski's \(2^ n\) theorem (English)
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13 January 1998
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Let \(A\) be an \(r \times n\) matrix with real entries of rank \(r < n\) and let \(b\) be a positive vector. Consider \(|Ax|\leq b\). Let \(A\) be a positive semi-definite \(n \times n\) matrix of rank \(r < n\), and let \(\alpha > 0\). Consider \(x^TAx = \alpha\). In each case the authors show that there is a nonzero integer vector \(x\) satisfying the inequality and such that the length of the orthogonal projection of \(x\) on the null space of \(A\) is within certain bounds.
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lattice points
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convex sets
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linear inequality
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quadratic form
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Minkowski's \(2^ n\) theorem
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0.747955858707428
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0.7328826189041138
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0.7266165614128113
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