second theorem of Minkowski proof

Question

I am wondering if anyone have a proof of second theorem of Minkowski. He says that if $vol(K) = 2^{n} det(L)$ and $K$ is compact and symmetric and convex then $K$ contains a non zero lattice point. I would appreciate your help.

Thanks!

Answer 1

Let me take the slightly stronger condition $\mathrm{vol}(K) > 2^n \det(L)$, rather than $\mathrm{vol}(K) = 2^n \det(L)$. (You can then use the fact that $K$ is compact to argue that equality actually suffices.) By applying a linear transformation, let's also assume that $L$ is the integer lattice $\mathbb{Z}^n$, so that we simply have $\mathrm{vol}(K) > 2^n$. (This step isn't necessary at all, but I think it makes the proof easier to understand.)

For vectors $x,x' \in \mathbb{R}^n$, let's write $x \equiv x' \bmod \mathbb{Z}^n$ if $x - x' \in \mathbb{Z}^n$. (I.e. $x$ and $x'$ have coordinates with the same fractional part.) Suppose that there exists distinct $x, x' \in K/2$ such that $x \equiv x' \bmod \mathbb{Z}^n$. Then, $x - x' \in \mathbb{Z}^n$ by definition, and $x - x' \neq 0$ because $x$ and $x'$ are distinct. However, by symmetry, $-x' \in K/2$, and by convexity, $(x-x')/2 \in K/2$ so that $x - x'$ is a non-zero lattice vector in $K$.

It therefore remains to find such an $x, x' \in K/2$. To that end, we simply notice that for every $x \in \mathbb{R}^n$, there is a unique representative $f(x) \equiv x \bmod \mathbb{Z}^n$ with $f(x) \in [0,1)^n$. Notice that the map $x$ is a piecewise combination of volume-preserving maps. So, if $f$ is an injective mapping when restricted to $K/2$, then we must have $$\mathrm{vol}(f(K/2)) = \mathrm{vol}(K/2) = 2^{-n}\mathrm{vol}(K) > 1 ; . $$ But, this can't be true, since the entire image of $f$ is $[0,1)^n$, which has volume $1$. Therefore, $f$ is not injective over $K/2$. I.e., there exist distinct $x,x' \in K/2$ such that $f(x) = f(x')$, as needed.

Edit: For what it's worth, this is typically referred to as Minkowski's first theorem. His second theorem finds a set of linearly independent vectors satisfying a certain constraint.