Theorem. Minkowski’s theorem: Let \(S\) be a bounded convex centrally-symmetric body in \(\mathbb{R}^n\) with volume larger than \(2^{n}\).
Then \(S\) contains a lattice point differnt from the origin.
Proof. Make a lattice of cubes of side length \(2\) and consider \(S\)’s intersection with these cubes. Translate all the cubes into the origin’s cube. There must be two points \(x,y\) that collide. Then \(x-y \in 2\mathbb{Z}\) because the translations of all the cubes are even integers. \(-y\in S\) becuase \(S\) is centrally-symmetric. But then by convexity \(\frac{x-y}{2}\) is a point in \(S\) and of course it isn’t \(0\).
You can also do this for a parallelpiped spanned by some vectors.
This turns out to be really great for number theory. You define an appropriate lattice and an appropriate bounded convex centrally-symmetric body and then you find a nice non-trivial point in your lattice. sorry I’m tired so no example tonight.
maybe next time.