Neighborly Cubical Polytopes

Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical convex d-polytope C d whose r-skeleton is combinatorially equivalent to that of the n-dimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂C d of a neighborly cubical polytope C n d maximizes the f -vector among all cubical (d− 1)-spheres with 2 vertices. While we show that this is true for polytopal spheres if n ≤ d+1, we also give a counter-example for d = 4 and n = 6. Further, the existence of neighborly cubical polytopes shows that the graph of the n-dimensional cube, where n ≥ 5, 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.