I have been working on the Mahler conjecture for over a year now and have made some progress for certain classes of convex polytopes and I'm now attempting to write up my results specified to $\mathbb{R}^3$ and $\mathbb{R}^4$ for ease of explanation before attempting to generalize to $d \geq 5$. I understand that the Mahler conjecture has been proved for zonoids (finite minkowski sums of line segments), and I just wanted to check that my results are not already subsumed inside this class of convex bodies.

Question: Is every cyclic 3-polytope a zonoid? (Also, is every cyclic 4-polytope a zonoid?)

I have been working on the Mahler conjecture for over a year now and have made some progress for certain classes of convex polytopes and I'm now attempting to write up my results specified to $\mathbb{R}^3$ and $\mathbb{R}^4$ for ease of explanation before attempting to generalize to $d \geq 5$. I understand that the Mahler conjecture has been proved for zonoids (finite minkowski sums of line segments), and I just wanted to check that my results are not already subsumed inside this class of convex bodies.

Question: Is every cyclic 3-polytope a zonoid? (Also, is every cyclic 4-polytope a zonoid?)

Essentially, what is a class of convex 3-polytopes (4-polytopes) that are not zonoids? Because I want to prove the Mahler conjecture for that class with the techniques I have been developing.