Intersection of Zonotopes

Question

A zonotope is specified by a set of generators $G=\{g_1,\dots,g_n\}\subseteq\mathbb{R}^d$, and is defined as $Z(G)=\{z:z=\sum_{i=1}^n x_i g_i, 0\leq x_i\leq 1 \forall i\}$.

Let $v\in Z(G)$, and consider the (non-empty) set $I=Z(G)\cap (Z(G)-v)$. That is, $I$ i s the intersection of $Z(G)$ and a shifted version of itself.

Is $I$ itself guaranteed to be a zonotope?

This is certainly true if $n=d$ and $G$ is linearly independent, in which case a zonotope is just a parallelotope. This is also certainly true if $d=2$, in which case a zonotope is just a centrally-symmetric polygon. But what about the general case?

Answer 1

No.

This intersection of a rhombic dodecahedron with a translate of itself shows a pentagonal face.

Openscad code:

intersection(){
scale(20) show_solid(solid_1);
    color("blue",1.0)
    translate([3,6,9])  
    scale(20) show_solid(solid_1);
}

from https://kitwallace.co.uk/3d/solid-index.xq?mode=solid&id=RhombicDodecahedron