§ 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 linearcombinations 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 parallelopipedthat have non-empty intersections with back to the origin. This makesall of them overlap with the fundamental parallelopiped with the origin.
- Since has volume great that , but the fundamental paralellopipedonly has volume , points from two different parallelograms mustoverlap.
- "Undo" the translation to find two points which are of the form ,. they must have the same since they overlappedwhen 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 arguedthat . 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 .