Face numbers of 4-Polytopes and 3-Spheres

By Günter M. Ziegler. 2002

46 citations recorded by the source corpus.

In this paper, we discuss f- and flag-vectors of 4-dimensional convex polytopes and cellular 3-spheres. We put forward two crucial parameters of fatness and complexity: Fatness F(P) := (f_1+f_2-20)/(f_0+f_3-10) is large if there are many more edges and 2-faces than there are vertices and facets, while complexity C(P) := (f_{03}-20)/(f_0+f_3-10) 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 flag-vectors of -- connected Eulerian lattices of length 5 (combinatorial objects), -- strongly regular CW 3-spheres (topological objects), -- convex 4-polytopes (discrete geometric objects), and -- rational convex 4-polytopes (whose study involves arithmetic aspects). Further progress will depend on the derivation of tighter f-vector inequalities for convex 4-polytopes. 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'' 4-polytopes. In this direction, I will report about constructions (from joint work with Michael Joswig, David Eppstein and Greg Kuperberg) that yield -- strongly regular CW 3-spheres of arbitrarily large fatness, -- convex 4-polytopes of fatness larger than 5.048, and -- rational convex 4-polytopes of fatness larger than 5-epsilon.

Subfields: Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/965d7d703a328b1082441a312933eb3ff5bb8086

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In Computational Theory and Mathematics

  1. Francisco Santos
  2. Edward D. Kim
  3. Alexander Postnikov
  4. M. Develin
  5. Federico Ardila
  6. Bernd Sturmfels

In Computer Vision and Pattern Recognition

  1. Francisco Santos
  2. Edward D. Kim

In Discrete Mathematics and Combinatorics

  1. Francisco Santos
  2. Edward D. Kim
  3. M. Develin
  4. Alexander Postnikov
  5. Federico Ardila
  6. Bernd Sturmfels

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