§ Proof of minkowski convex body theorem
We can derive a proof of the minkowski convex body theorem starting from
§ Blichfeldt's theorem
This theorem allows us to prove that a set
of large-enough-size in any lattice will have two points such that their
difference lies in the lattice. Formally, we have:
Blichfeldt's theorem tells us that there exists two points
such that .
- A lattice for some basis . The lattice is spanned by integer linear combinations of rows of .
- A body which need not be convex! , which has volume greater than . Recall that for a lattice , the volume of a fundamental unit / fundamental parallelopiped is .
The idea is to:
- Chop up sections of across all translates of the fundamental parallelopiped that have non-empty intersections with back to the origin. This makes all of them overlap with the fundamental parallelopiped with the origin.
- Since has volume great that , but the fundamental paralellopiped only has volume , points from two different parallelograms must overlap.
- "Undo" the translation to find two points which are of the form , . they must have the same since they overlapped when they were laid on the fundamental paralellopiped. Also notice that since they came from two different parallograms on the plane!
- Notice that , since we already argued that . This gives us what we want.
§ Minkowskis' Convex body Theorem from Blichfeldt's theorem
Consider a convex set
that is symmetric about the origin with volume greater than .
Create a new set which is . Formally:
We now see that to invoke Blichfeldt's theorem.
We can apply Blichfeldt's theorem to get our hands on two points
such that .