Neighborly Cubical Polytopes

Neighborly cubical polytopes exist: for any n≥ d≥ 2r+2 , there is a cubical convex d -polytope C _d ⁿ 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 of a neighborly cubical polytope C _d ⁿ maximizes the f -vector among all cubical (d-1) -spheres with 2 ⁿ 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 nge5 , 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 f ₃ =(f ₀ /4) log ₂ (f ₀ /4) vertices, so the facet—vertex ratio f ₃ /f ₀ is not bounded; this solves a problem of Kalai, Perles, and Stanley studied by Jockusch. Received December 30, 1998. Online publication May 15, 2000.