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Minkowski's theorem
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(Theorem)
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Let $\mathcal{L} \in \mathbb{R}^2$ be a lattice in the sense of number theory, i.e. a 2-dimensional free group over ${\mathbb{Z}}$ which generates $\mathbb{R}^2$ over $\mathbb{R}$ . Let $w_1,w_2$ be generators of the lattice $\mathcal{L}$ . A set $\mathcal{F}$ of the form $$\mathcal{F}=\{(x,y)\in\mathbb{R}^2: (x,y)=\alpha w_1+\beta w_2,\quad 0\leq \alpha < 1,\quad 0\leq \beta <1 \}$$ is usually called a fundamental domain or fundamental parallelogram for the lattice $\mathcal{L}$ .
Theorem 1 (Minkowski's Theorem) Let $\mathcal{L}$ be an arbitrary lattice in $\mathbb{R}^2$ and let $\Delta$ be the area of a fundamental parallelogram. Any convex region $\mathfrak{K}$ symmetrical about the origin and of area greater than $4\Delta$ contains points of the lattice $\mathcal{L}$ other than the origin.
More generally, there is the following $n$ -dimensional analogue.
Theorem 2 Let $\mathcal{L}$ be an arbitrary lattice in $\mathbb{R}^n$ and let $\Delta$ be the area of a fundamental parallelopiped. Any convex region $\mathfrak{K}$ symmetrical about the origin and of volume greater than $2^n\Delta$ contains points of the lattice $\mathcal{L}$ other than the origin.
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"Minkowski's theorem" is owned by alozano.
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See Also: lattice in  , proof of Minkowski's bound
| Other names: |
Minkowski's theorem on convex bodies |
| Also defines: |
Minkowski's theorem, fundamental parallelogram |
| Keywords: |
Minkowski, convex |
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Cross-references: volume, points, contains, origin, convex region, area, theorem, domain, generators, generates, free group, number theory, lattice
There are 8 references to this entry.
This is version 5 of Minkowski's theorem, born on 2003-08-15, modified 2005-11-24.
Object id is 4601, canonical name is MinkowskisTheorem.
Accessed 9719 times total.
Classification:
| AMS MSC: | 11H06 (Number theory :: Geometry of numbers :: Lattices and convex bodies) |
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Pending Errata and Addenda
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