


Minkowski's Theorem
Theorem 1
Let be a convex set in
, with centrally
symmetric and
. Then must contain a lattice point
besides .
Definition 2
A set is convex whenever
implies
.
Definition 3
An object is centrally symmetric if whenever
and
implies
and
.
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 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
where
Note that
as well.
QED
 
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