Let $B_2^n$ denote the Euclidean unit ball in $\mathbb{R}^n$. Then the Minkowski-sum $B_2^{n-1}+B_2^n$ is a zonoid whose polar is also a zonoid for $n\leq 4$, but not for $n\geq 6$. ([here][1]).

  


P.S -- I never did get to check the case $n=5$.


  [1]: https://arxiv.org/abs/math/9707210