The Mahler conjecture and non-zonoidal 3-polytopes (4-polytopes)

Question

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.

Answer 1

It is correct that the Mahler conjecture was proven for zonoids. However, your definition of zonoid is not quite right.

A zonotope is the Minkowski sum of finitely many line segments. And a zonoid is a compact convex body that is the Hausdorff limit of a sequence of zonotopes. So every zonotope is a zonoid but since zonoids don't have to be polytopes the other implication is not true in general.

For further details, may I suggest the diploma thesis of Matthias Henze? It can be found here.

Of course, one can still wonder if cyclic polytopes are zonoids. I don't know the answer to this question.

But in any case, cyclic polytopes can never be centrally symmetric. Yet, isn't this needed for the Mahler conjecture? In this context it might be worth mentioning that Barvinok and collaborators tried (are still trying?) to find a centrally symmetric equivalent to cyclic polytopes.