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Neighborly Cubical Polytopes

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Abstract.

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 \(\partial C_d^n\) 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 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 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.

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Received December 30, 1998. Online publication May 15, 2000.

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Joswig, M., Ziegler, G. Neighborly Cubical Polytopes . Discrete Comput Geom 24, 325–344 (2000). https://doi.org/10.1007/s004540010039

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Keywords

  • Dimensional Cube
  • Neighborly Cubical
  • Cubical Polytopes
  • Polytopal Sphere