Discrete & Computational Geometry

, Volume 24, Issue 2–3, pp 325–344 | Cite as

Neighborly Cubical Polytopes

  • M. Joswig
  • G. M. Ziegler


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.


Dimensional Cube Neighborly Cubical Cubical Polytopes Polytopal Sphere 
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Copyright information

© Springer-Verlag New York Inc. 2000

Authors and Affiliations

  • M. Joswig
    • 1
  • G. M. Ziegler
    • 1
  1. 1.Fachbereich Mathematik, MA 7-1, Technische Universität Berlin, 10623 Berlin, Germany, ziegler@math.tu-berlin.deDE

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