Infimum of the norms of all non-zero points in a lattice $\Gamma.$ Let $K$ be a non empty-interior compact convex symmetric set such that $0\in \mathring{A}$. We have a norm the Minkowski functionnal denoted $\Vert \cdot\Vert_K$ so that we can speak about length.
Given a path $\gamma:[a,b]\to \Bbb{R}^n$ we have $$lg_K(\gamma)=\int_{a}^b\Vert \gamma'\Vert_Kdt.$$
Now given a lattice $\Gamma$ the torus can be defined to be $(\Bbb{R^n},\Vert \cdot\Vert_K)/\Gamma:=\Bbb{T}^n_{K}.$
Now the systole is $$sys(\Bbb{T}^n_{K})=\inf\{lg_{K}(\gamma) / [\gamma]\ne 0 \in \pi_1(\Bbb{T}^n_{K})\}$$
Now I am reading a document that say: "The systole of $\Bbb{T}^n_{K}$ is the infimum of the norms of all non-zero points in $\Gamma$."
I am not sure how can I prove that.
I know that $p:\Bbb{R}^n\to \Bbb{T}^n_{K}$ is a covering map so that I can lift any path $\gamma:[0,1]\to \Bbb{T}^n_{K}$ to $\tilde\gamma: [0,1]\to \Bbb{R}^n$ such that $\gamma=p\circ\tilde\gamma.$ 
But what next ?
EDIT: I know that the geodesics of $\Bbb{R}^n$ are line segments. I juste need to make the "link" between a non contractible path in $\Bbb{T}^n_{K}$ and line segments.
 A: The non-contractible paths in $\mathbb{T}_K^n$ are exactly those that correspond to paths in $\mathbb{R}^n$ that connect one lattice point to a different lattice point. It is then immediately obvious that the shortest such path is the one that connects one lattice point (WLOG, 0) to the nearest lattice point via a straight line. That is, the systole is the length of the shortest path from 0 to any other element of the lattice. In other words, it is the infimum of the norms of all non-zero points in $\Gamma$.
A: This is a very general fact from covering space theory.  If $X$ is a space and $p:Y\to X$ is its universal cover, then a loop $\gamma:[0,1]\to X$ is contractible iff when you lift $\gamma$ to $\tilde{\gamma}:[0,1]\to Y$, $\tilde{\gamma}$ is still a loop (that is, $\tilde{\gamma}(0)=\tilde{\gamma}(1)$).  As a sketch of a proof, if $\tilde{\gamma}$ is a loop, then it is contractible since $Y$ is simply connected, and composing the contraction with $p$ you get a contraction of $\gamma$.  Conversely, if there existed a contraction of $\gamma$, you could lift it to a contraction of $\tilde{\gamma}$ which fixes the endpoints, which is only possible if $\tilde{\gamma}$ is a loop.
So in your case, you are looking at the infimum of lengths of all non-contractible loops, which are just loops which lift to paths in $\mathbb{R}^n$ between distinct points.  The endpoints of such a path must differ by an element of $\Gamma$ (since they have the same image in the quotient $\mathbb{R}^n/\Gamma=\mathbb{T}^n_K$).  Translating the path, we may assume it starts at $0$ and ends at some nonzero element of $\Gamma$.
So we get that the systole is the infimum of all lengths of paths from $0$ to nonzero elements of $\Gamma$.  The shortest path from $0$ to any fixed element $g\in\Gamma$ is just the straight line, and the length of the straight line path is just $\|g\|_K$.  So the systole is the infimum of $\|g\|_K$ over all nonzero $g\in \Gamma$.
