§ Proof of minkowski convex body theorem

We can derive a proof of the minkowski convex body theorem starting from Blichfeldt’s theorem.

§ 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:
  1. A lattice L(B){Bx:xZn}L(B) \equiv \{ Bx : x \in \mathbb Z^n \} for some basis BRnB \in \mathbb R^n. The lattice LL is spanned by integer linear combinations of rows of BB.
  2. A body SRnS \subseteq R^n which need not be convex! , which has volume greater than det(B)\det(B). Recall that for a lattice L(B)L(B), the volume of a fundamental unit / fundamental parallelopiped is det(B)det(B).
Blichfeldt's theorem tells us that there exists two points x1,x2Sx_1, x_2 \in S such that x1x2Lx_1 - x_2 \in L.

§ Proof

The idea is to:
  1. Chop up sections of SS across all translates of the fundamental parallelopiped that have non-empty intersections with SS back to the origin. This makes all of them overlap with the fundamental parallelopiped with the origin.
  2. Since SS has volume great that det(B)\det(B), but the fundamental paralellopiped only has volume det(B)\det(B), points from two different parallelograms must overlap.
  3. "Undo" the translation to find two points which are of the form x1=l1+δx_1 = l_1 + \delta, x2=l2+δx_2 = l_2 + \delta. they must have the same δ\delta since they overlapped when they were laid on the fundamental paralellopiped. Also notice that l1l2l_1 \neq l_2since they came from two different parallograms on the plane!
  4. Notice that x1x2=l1l2L0x_1 - x_2 = l_1 - l_2\in L \neq 0, since we already argued that l1l2l_1 \neq l_2. This gives us what we want.

§ Minkowskis' Convex body Theorem from Blichfeldt's theorem

Consider a convex set SRnS \subseteq \mathbb R^n that is symmetric about the origin with volume greater than 2ndet(B)2^n det(B). Create a new set TT which is S0.5S * 0.5. Formally:
TS/2={(x1/2,x2,,xn/2):(x1,x2,,xn)S}T \equiv S/2 = \{ (x_1/2, x_2, \dots, x_n/2) : (x_1, x_2, \dots, x_n) \in S \}
We now see that Vol(T)>det(B)Vol(T) > det(B) to invoke Blichfeldt's theorem. Formally:
Vol(T)=1/2nVol(S)>1/2n(2ndet(B))=det(B)Vol(T) = 1/2^n Vol(S) > 1/2^n (2^n det(B)) = det(B)
We can apply Blichfeldt's theorem to get our hands on two points x1,x2Tx_1, x_2 \in T such that x1x2Lx_1 - x_2 \in L.
x1T2x1S (S=2T)x2T2x2S (S=2T)2x2S2x2S (S is symmetric about origin)12(2x1)+12(2x2)S (S is convex)x1x2S (Simplification)nonzero lattice point S \begin{aligned} &x_1 \in T \Rightarrow 2x_1 \in S ~(S = 2T) \\ &x_2 \in T \Rightarrow 2x_2 \in S ~(S = 2T) \\ &2x_2 \in S \Rightarrow -2x_2 \in S~\text{($S$ is symmetric about origin)} \\ &\frac{1}{2}(2x_1) + \frac{1}{2} (-2x_2) \in S~\text{($S$ is convex)}\\ &x_1 - x_2 \in S~\text{(Simplification)}\\ &\text{nonzero lattice point}~\in S \\ \end{aligned}

§ References