Skip to main content
SpringerLink
Log in

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

Combinatorica Aims and scope Submit manuscript

Cite this article

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195-4350, USA

    Joshua Hinman

Authors
  1. Joshua Hinman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joshua Hinman.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-023-00042-7

Keywords

Mathematics Subject Classification

search

Navigation