Let $P$ be a centrally symmetric $k$-neighborly $d$-polytope. Let $ P^* $ is polar (also dual) of $P$. Consider a polytope $Q^*$ , $Q^* \subset P^* $(not necessarily a face of $P^*$). By duality, $P\subset Q $. How neighbourly is this $Q$? Is it $k$, or less than $k$, or more than $k$-neighborly?

Convex Polytopesindicates (p.129b) that it is an open problem whether there exists any polytope other than a simplex that is 2-neighborly and its dual is 2-neighborly. $\endgroup$ – Joseph O'Rourke Mar 28 '12 at 19:34