Fat 4-polytopes and fatter 3-spheres
Combinatorics
2019-09-16 v3 Metric Geometry
Abstract
We introduce the fatness parameter of a 4-dimensional polytope P, defined as \phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in 4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes bounded? We describe and analyze a hyperbolic geometry construction that produces 4-polytopes with fatness \phi(P)>5.048, as well as the first infinite family of 2-simple, 2-simplicial 4-polytopes. 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 3-sphere.
Cite
@article{arxiv.math/0204007,
title = {Fat 4-polytopes and fatter 3-spheres},
author = {David Eppstein and Greg Kuperberg and Günter M. Ziegler},
journal= {arXiv preprint arXiv:math/0204007},
year = {2019}
}
Comments
12 pages, 12 figures. This version has minor changes proposed by the second referee