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?
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$\begingroup$ This does not answer your question, but Grünbaum's book Convex Polytopes indicates (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'RourkeCommented Mar 28, 2012 at 19:34
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2$\begingroup$ The question is not clear. What do you assume on Q? is it a convex hull of some vertices of P? $\endgroup$– Gil KalaiCommented Dec 30, 2013 at 9:37
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