Fat 4-polytopes and fatter 3-spheres

David Eppstein and Greg Kuperberg and Günter M. Ziegler— 2003

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.

TitelFat 4-polytopes and fatter 3-spheres
VerfasserDavid Eppstein and Greg Kuperberg and Günter M. Ziegler
VerlagMarcel Dekker Inc.
OrtNew York
Erschienen inDiscrete Geometry: In honor of W. Kuperberg's 60th birthday, Pure and Applied Mathematics, volume 253, pages 239--265