**Fat 4-polytopes and fatter 3-spheres**.

D. Eppstein, G. Kuperberg, and G. Ziegler.

arXiv:math.CO/0204007.

*Discrete Geometry: In honor of W. Kuperberg's 60th birthday*, Pure and Appl. Math. 253, Marcel Dekker, pp. 239–265, 2003.We introduce the fatness parameter of a 4-dimensional polytope P, (f

_{1}+f_{2})/(f_{0}+f_{3}). It is open whether all 4-polytopes have bounded fatness. We describe a hyperbolic geometry construction that produces 4-polytopes with fatness > 5.048, as well as the first infinite family of 2-simple, 2-simplicial 4-polytopes and an improved lower bound on the average kissing number of finite sphere packings in R^{3}. We show that fatness is not bounded for the more general class of strongly regular CW decompositions of the 3-sphere.

Co-authors – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine

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