## § 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 $L(B) \equiv \{ Bx : x \in \mathbb Z^n \}$ for some basis $B \in \mathbb R^n$. The lattice $L$ is spanned by integer linearcombinations of rows of $B$.
2. A body $S \subseteq R^n$ which need not be convex!, which has volume greater than $\det(B)$. Recall that for a lattice $L(B)$,the volume of a fundamental unit / fundamental parallelopiped is $det(B)$.
Blichfeldt's theorem tells us that there exists two points $x_1, x_2 \in S$ such that $x_1 - x_2 \in L$.

#### § Proof

The idea is to:
1. Chop up sections of $S$ across all translates of the fundamental parallelopipedthat have non-empty intersections with $S$ back to the origin. This makesall of them overlap with the fundamental parallelopiped with the origin.
2. Since $S$ has volume great that $\det(B)$, but the fundamental paralellopipedonly has volume $\det(B)$, points from two different parallelograms mustoverlap.
3. "Undo" the translation to find two points which are of the form $x_1 = l_1 + \delta$,$x_2 = l_2 + \delta$. they must have the same $\delta$ since they overlappedwhen they were laid on the fundamental paralellopiped. Also notice that $l_1 \neq l_2$since they came from two different parallograms on the plane!
4. Notice that $x_1 - x_2 = l_1 - l_2\in L \neq 0$, since we already arguedthat $l_1 \neq l_2$. This gives us what we want.

#### § Minkowskis' Convex body Theorem from Blichfeldt's theorem

Consider a convex set $S \subseteq \mathbb R^n$ that is symmetric about the origin with volume greater than $2^n det(B)$. Create a new set $T$ which is $S * 0.5$. Formally:
$T \equiv S/2 = \{ (x_1/2, x_2, \dots, x_n/2) : (x_1, x_2, \dots, x_n) \in S \}$
We now see that $Vol(T) > det(B)$ to invoke Blichfeldt's theorem. Formally:
$Vol(T) = 1/2^n Vol(S) > 1/2^n (2^n det(B)) = det(B)$
We can apply Blichfeldt's theorem to get our hands on two points $x_1, x_2 \in T$ such that $x_1 - x_2 \in L$.
\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}