Fat 4polytopes and fatter 3spheres
Abstract
We introduce the fatness parameter of a 4dimensional polytope P, defined as \phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in 4dimensional combinatorial geometry: Is the fatness of convex 4polytopes bounded? We describe and analyze a hyperbolic geometry construction that produces 4polytopes with fatness \phi(P)>5.048, as well as the first infinite family of 2simple, 2simplicial 4polytopes. Moreover, using a construction via finite covering spaces of surfaces, we show that fatness is not bounded for the more general class of strongly regular CW decompositions of the 3sphere.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2002
 DOI:
 10.48550/arXiv.math/0204007
 arXiv:
 arXiv:math/0204007
 Bibcode:
 2002math......4007E
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Metric Geometry
 EPrint:
 12 pages, 12 figures. This version has minor changes proposed by the second referee