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Discrete & Computational Geometry

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

Neighborly Cubical Polytopes

  • M. Joswig
  • G. M. Ziegler

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.

Keywords

Dimensional Cube Neighborly Cubical Cubical Polytopes Polytopal Sphere 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 joswig@math.tu-berlin.de, ziegler@math.tu-berlin.deDE

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