Does there exist a polytope in dimensional d consisting of $k>d+1$ faces satisfying that every d faces intersect? I tried 3 dimensional cases, and it seems negative. But is it all negative for any dimension $d\geq 2$?
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4$\begingroup$ This is indeed not possible, since the polar of such a polytope would be $d$-neighbourly: en.wikipedia.org/wiki/Neighborly_polytope $\endgroup$– Tobias FritzCommented Oct 14, 2015 at 17:11
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$\begingroup$ Thanks for the answer! Is it because the dual of such polytope can only be a simplex($d$-neighbourly), so the polytope itself is also a simplex? $\endgroup$– xzlCommented Oct 15, 2015 at 4:54
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$\begingroup$ yes, that's exactly it! $\endgroup$– Tobias FritzCommented Oct 15, 2015 at 10:44
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$\begingroup$ What if the original polytope is not convex? Does the argument still go through? Or the polytope must be convex given that every d faces intersect? $\endgroup$– xzlCommented Oct 15, 2015 at 11:17
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