Face numbers of 4Polytopes and 3Spheres
Abstract
In this paper, we discuss f and flagvectors of 4dimensional convex polytopes and cellular 3spheres. We put forward two crucial parameters of fatness and complexity: Fatness F(P) := (f_1+f_220)/(f_0+f_310) is large if there are many more edges and 2faces than there are vertices and facets, while complexity C(P) := (f_{03}20)/(f_0+f_310) is large if every facet has many vertices, and every vertex is in many facets. Recent results suggest that these parameters might allow one to differentiate between the cones of f or flagvectors of  connected Eulerian lattices of length 5 (combinatorial objects),  strongly regular CW 3spheres (topological objects),  convex 4polytopes (discrete geometric objects), and  rational convex 4polytopes (whose study involves arithmetic aspects). Further progress will depend on the derivation of tighter fvector inequalities for convex 4polytopes. On the other hand, we will need new construction methods that produce interesting polytopes which are far from being simplicial or simple  for example, very ``fat'' or ``complex'' 4polytopes. In this direction, I will report about constructions (from joint work with Michael Joswig, David Eppstein and Greg Kuperberg) that yield  strongly regular CW 3spheres of arbitrarily large fatness,  convex 4polytopes of fatness larger than 5.048, and  rational convex 4polytopes of fatness larger than 5epsilon.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2002
 DOI:
 10.48550/arXiv.math/0208073
 arXiv:
 arXiv:math/0208073
 Bibcode:
 2002math......8073Z
 Keywords:

 Metric Geometry;
 Combinatorics;
 52B11;
 52B10;
 51M20
 EPrint:
 Proceedings of the ICM, Beijing 2002, vol. 3, 625636