Flag vector pairs, fatness, and their bounds for 4-polytopes
DOI:
https://doi.org/10.11575/cdm.v16i3.71007Abstract
Recently Sjoberg and Ziegler showed a remarkable result that completely characterizes the flag vector pair $(f_0, f_{03} )$ of any $4$-dimensional polytopes. Motivated by their results and techniques, in this paper we show some necessary conditions for other remaining flag vector pairs such as $(f_0 , f_{02})$, $(f_{02}, f_{03})$, $(f_{1}, f_{02})$, and $(f_1 , f_{03})$ to be flag vector pairs of $4$-dimensional convex polytopes. Results of this paper give some partial answers to the questions posed by Sj\" oberg and Ziegler. As an application of the bounds for flag vector pairs $(f_1 , f_{03})$, in this paper we also provide some bounds of fatness function for certain $4$-polytopes as well as $3$-polytopes.
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Copyright (c) 2021 Jin Hong Kim, Park Nari
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