Supposedly I have a Convex object $A$ in high dimension(Eculdian space), and I know their low dimensional projection(only look at 2 dimensions at a time) is a zonotope, how can I assert/reject $A$ is a Zonotope?
A concrete example,
$$ A = CH(\{(0, 0, 0), (1, 1, 1), (2, 3, 2), (1, 2, 1)\}) $$, $CH$ is the convex hull operation. it can be shown that if only look at the (0, 1) dimensional projection: $$ CH(\{(0, 0), (1, 1), (2, 3), (1, 2)\}) $$ is a zonotope with generators $((1, 1), (1, 2))$, how can i prove or disprove $A$ is a Zonotope.
More context:
My data has such a generating process, assume I have a sequence of finite random points $$ a_1, \cdots, a_i, \cdots, a_n. a_i \in \mathbb{R}^m_+ $$.
And I'm interested in the Convex hull of the prefix sum of such sequence: $$ B = (b_1, \cdots, b_i, \cdots, b_n) = a_1, a_1+a_2, \cdots, \sum_{j=1}^n a_j $$.
When $m = 2$, i can prove that $CH(B)$ is a union of two Zonotopes.
As $CH(B)$ can be separated into the upper hull and lower hull, where each can be enumerated by a Zonotope, because of the cumulative sum nature.
Some plots showing m=2 condition: 
And I'm looking into extending the property into $m \geq 3$.