Discrete & Computational Geometry

, Volume 24, Issue 2, pp 325–344

# Neighborly Cubical Polytopes

• M. Joswig
• G. M. Ziegler

DOI: 10.1007/s004540010039

Cite this article as:
Joswig, M. & Ziegler, G. Discrete Comput Geom (2000) 24: 325. doi:10.1007/s004540010039

## Abstract.

Neighborly cubical polytopes exist: for any n≥ d≥ 2r+2 , there is a cubical convex d -polytope Cdn 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 Cdn maximizes the f -vector among all cubical (d-1) -spheres with 2n 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 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.

## Copyright information

© 2000 Springer-Verlag New York Inc.

## 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