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Let $P$ is 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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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. –  Joseph O'Rourke Mar 28 '12 at 19:34
    
Thanks for the comment and reference. –  mmaann Mar 29 '12 at 18:08
    
The question is not clear. What do you assume on Q? is it a convex hull of some vertices of P? –  Gil Kalai Dec 30 '13 at 9:37
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