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Characterizing Face and Flag Vector Pairs for Polytopes

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Abstract

Grünbaum, Barnette, and Reay in 1974 completed the characterization of the pairs \((f_i,f_j)\) of face numbers of 4-dimensional polytopes. Here we obtain a complete characterization of the pairs of flag numbers \((f_0,f_{03})\) for 4-polytopes. Furthermore, we describe the pairs of face numbers \((f_0,f_{d-1})\) for d-polytopes; this description is complete for even \(d\ge 6\) except for finitely many exceptional pairs that are “small” in a well-defined sense, while for odd d we show that there are also “large” exceptional pairs. Our proofs rely on the insight that “small” pairs need to be defined and to be treated separately; in the 4-dimensional case, these may be characterized with the help of the characterizations of the 4-polytopes with at most eight vertices by Altshuler and Steinberg (1984).

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Authors and Affiliations

  1. Institut für Mathematik, FU Berlin, Arnimallee 2, 14195, Berlin, Germany

    Hannah Sjöberg & Günter M. Ziegler

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  1. Hannah Sjöberg
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  2. Günter M. Ziegler
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Correspondence to Günter M. Ziegler.

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Editor in Charge: János Pach

In memory of Branko Grünbaum (1929–2018)

HS: Supported by DFG via the Berlin Mathematical School.

GMZ: Research supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”

Appendix

Appendix

Table 3 Polytopes \(P_i\) with 7 and 8 vertices
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Table 3 lists all polytopes \(P_i\) with seven and eight vertices from Table 1 used in the construction of all possible pairs \((f_0,f_{03})\). The polytopes are given by their facet list. See Fukuda et al. [14] for a complete list of all 31 polytopes with seven vertices and all 1294 polytopes with eight vertices. Entry 7.x in the last column means that the polytope can be found as the xth polytope listed in the classification of 4-polytopes with seven vertices.

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Sjöberg, H., Ziegler, G.M. Characterizing Face and Flag Vector Pairs for Polytopes. Discrete Comput Geom 64, 174–199 (2020). https://doi.org/10.1007/s00454-018-0044-7

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