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A Positive Answer to Bárány’s Question on Face Numbers of Polytopes

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Abstract

Despite a full characterization of the face vectors of simple and simplicial polytopes, the face numbers of general polytopes are poorly understood. Around 1997, Bárány asked whether for all convex d-polytopes P and all \(0 \le k \le d-1\), \(f_k(P) \ge \min \{f_0(P), f_{d-1}(P)\}\). We answer Bárány’s question in the affirmative and prove a stronger statement: for all convex d-polytopes P and all \(0 \le k \le d-1\),

$$\begin{aligned} \frac{f_k(P)}{f_0(P)} \ge \frac{1}{2}\biggl [{\lceil \frac{d}{2} \rceil \atopwithdelims ()k} + {\lfloor \frac{d}{2} \rfloor \atopwithdelims ()k}\biggr ], \qquad \frac{f_k(P)}{f_{d-1}(P)} \ge \frac{1}{2}\biggl [{\lceil \frac{d}{2} \rceil \atopwithdelims ()d-k-1} + {\lfloor \frac{d}{2} \rfloor \atopwithdelims ()d-k-1}\biggr ]. \end{aligned}$$

In the former, equality holds precisely when \(k=0\) or when \(k=1\) and P is simple. In the latter, equality holds precisely when \(k=d-1\) or when \(k=d-2\) and P is simplicial.

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Acknowledgements

The author would like to thank Isabella Novik for encouraging him to write this paper, as well as her incredible support and guidance throughout the writing process. The author would also like to thank Rowan Rowlands, Louis Billera, and Maria-Romina Ivan for their generous help in editing this paper. Finally, the author is grateful to the referees for their careful reading of this paper and their gracious feedback.

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Hinman, J. A Positive Answer to Bárány’s Question on Face Numbers of Polytopes. Combinatorica 43, 953–962 (2023). https://doi.org/10.1007/s00493-023-00042-7

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