background

# 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

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

where

Note that as well.
Consider the midpoint of and , then .

QED

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