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

## 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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## References

1. Bárány, I., Larman, D.G.: The convex hull of the integer points in a large ball. Math. Ann. 312(1), 167–181 (1998)

2. Bayer, M.: The extended $$f$$-vectors of $$4$$-polytopes. J. Combin. Theory Ser. A 44(1), 141–151 (1987)

3. Billera, L.J., Björner, A.: Face numbers of polytopes and complexes. In: Goodman, J.E., et al. (eds.) Handbook of discrete and computational geometry, pp. 291–310. CRC Press, Boca Raton (1997)

4. Billera, L.J., Lee, C.W.: Sufficiency of McMullen’s conditions for $$f$$-vectors of simplicial polytopes. Bull. Amer. Math. Soc. (N.S.) 2(1), 11111181–185 (1980)

5. Billera, Louis J., Lee, Carl W.: A proof of the sufficiency of McMullen’s conditions for $$f$$-vectors of simplicial convex polytopes. J. Combin. Theory Ser. A 31(3), 237–255 (1981)

6. Björner, A.: Partial unimodality for $$f$$-vectors of simplicial polytopes and spheres. Contemp. Math. 178, 45–54 (1994)

7. de Loera, J., Rambau, J., Santos, F.: Triangulations: Structures for algorithms and applications. Springer, Berlin (2010)

8. Grünbaum, B.: Convex polytopes graduate texts in mathematics. Springer, Berlin (2003)

9. Kalai, G.: Rigidity and the lower bound theorem. I. Invent. Math. 88(1), 125–151 (1987)

10. McMullen, P.: The maximum numbers of faces of a convex polytope. Mathematika 17, 179–184 (1970)

11. McMullen, P.: The numbers of faces of simplicial polytopes. Israel J. Math. 9, 559–570 (1971)

12. Perles, Micha A., Shephard, Geoffrey C.: Angle sums of convex polytopes. Math. Scand. 21(2), 199–218 (1967)

13. Shephard, Geoffrey C.: Angle deficiencies of convex polytopes. J. London Math. Soc. 43(1), 325–336 (1968)

14. Stanley, Richard P.: The number of faces of a simplicial convex polytope. Adv. Math. 35(3), 236–238 (1980)

15. Xue, L.: A proof of Grünbaum’s lower bound conjecture for general polytopes. Israel J. Math. 245(2), 991–1000 (2021)

16. Ziegler, Günter. M.: Lectures on polytopes graduate texts in mathematics. Springer, Berlin (1995)

## 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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Correspondence to Joshua Hinman.