Neighborly cubical polytopes exist: for any n≥ d≥ 2r+2 , there is a cubical convex d -polytope Cdn whose r -skeleton is combinatorially equivalent to that of the n -dimensional cube. This solves a problem of Babson, Billera, and Chan.
Kalai conjectured that the boundary \(\partial C_d^n\) of a neighborly cubical polytope Cdn maximizes the f -vector among all cubical (d-1) -spheres with 2n vertices. While we show that this is true for polytopal spheres if n≤ d+1 , we also give a counterexample for d=4 and n=6 .
Further, the existence of neighborly cubical polytopes shows that the graph of the n -dimensional cube, where n\ge5 , is ``dimensionally ambiguous'' in the sense of Grünbaum. We also show that the graph of the 5 -cube is ``strongly 4 -ambiguous.''
In the special case d=4 , neighborly cubical polytopes have f3=(f0/4) log2 (f0/4) vertices, so the facet—vertex ratio f3/f0 is not bounded; this solves a problem of Kalai, Perles, and Stanley studied by Jockusch.
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