Minkowski's Theorem
Theorem 1
Let ![$ \zeta$](img1.png) be a convex set in
![$ {\mathbb{R}}^2$](img2.png) , with ![$ \zeta$](img1.png) centrally
symmetric and
![$ area(\zeta)>4$](img3.png) . Then ![$ \zeta$](img1.png) must contain a lattice point
besides ![$ (0,0)$](img4.png) .
Definition 2
A set is convex whenever
![$ x,y\in\zeta$](img5.png) implies
![$ \overline{xy}\in\zeta$](img6.png) .
Definition 3
An object ![$ \zeta$](img1.png) is centrally symmetric if whenever
![$ (0,0)\in\zeta$](img7.png) and
![$ P\in\zeta$](img8.png) implies
![$ \overline{0P}\in\zeta$](img9.png) and
![$ -\overline{0P}\in\zeta$](img10.png) .
1. Take
2. Assign a copy of to each lattice point in an grid.
3. From the above figure we can see
, thus the area covered
by the set s is .
4. In fact, all objects cover more (go over a bit). All objects are
contained in a square of side , where is the maximum distance from
the center to any point of .
5. Now we want to show the inequality
to be true. CLAIM:
Since
implies the leading coefficient is positive for large , if we
choose large then we're done! Conclusion, the blobs will overlap for some pair of
blobs, but by translating and another blob cross too! (All is being repeated).
6. Let be in the intersection of
. Then
![$\displaystyle (a'',b'')=(a',b')+(p,q),$](img34.png) where
Note that
as well.
Consider the midpoint of and ,
then
.
QED
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