I am currently learning about zonotopes but I am having troubles understanding the concept of it.
I know that a zonotope is defined as
$$\left\lbrace x : x=c +\sum_{i=1}^k \xi_ig_i, \xi_i \in \lbrack -1,1\rbrack, \forall i=1, \dots, k \right\rbrace$$
where $c$ is called the center and $g_i$ are the generators. As an example I tried to draw the zonotope with center $0$ and generators $\begin{pmatrix} 1 \\ 0\end{pmatrix}, \begin{pmatrix} 2 \\ 1\end{pmatrix}, \begin{pmatrix} 1 \\ 1\end{pmatrix}$ and obtained the two-dimensional zonotp with vertices
$$\begin{pmatrix} 2 \\ 2\end{pmatrix}, \begin{pmatrix} 4 \\ 2\end{pmatrix}, \begin{pmatrix} 2 \\ 0\end{pmatrix}, \begin{pmatrix} -2 \\ -2\end{pmatrix}, \begin{pmatrix} -4 \\ -2\end{pmatrix}, \begin{pmatrix} -2 \\ 0\end{pmatrix}$$
Is this correct? And if so, is there a smarter way to construct it than by testing all possible linear combinations of the generators with coefficients $\pm 1$?
A proof in my home work assignment states that "at the boundary of this zonotope we have at most one fractional $\xi_i$". Unfortunately, I don't see why this is true.
