Skip to main content
SpringerLink
Log in

A Proof of Grünbaum’s Lower Bound Conjecture for general polytopes

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In 1967, Grünbaum conjectured that any d-dimensional polytope with d + s ≤ 2d vertices has at least

$${\phi _k}(d + s,d) = \left( {\matrix{{d + 1} \cr {k + 1} \cr } } \right) + \left( {\matrix{d \cr {k + 1} \cr } } \right) - \left( {\matrix{ {d + 1 - s} \cr {k + 1} \cr } } \right)$$

k-faces. We prove this conjecture and also characterize the cases in which equality holds.

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

Similar content being viewed by others

References

  1. D. W. Barnette, The minimum number of vertices of a simple polytope, Israel Journal of Mathematics 10 (1971) 121–125.

    Article  MathSciNet  Google Scholar 

  2. D. Barnette, Graph theorems for manifolds, Israel Journal of Mathematics 16 (1973) 62–72.

    Article  MathSciNet  Google Scholar 

  3. D. Barnette, A proof of the lower bound conjecture for convex polytopes, Pacific Journal of Mathematics 46 (1973) 349–354.

    Article  MathSciNet  Google Scholar 

  4. L. J. Billera and C. W. Lee, Sufficiency of McMullen’s conditions for f-vectors of simplicial polytopes, Bulletin of the American Mathematical Society 2 (1980) 181–185.

    Article  MathSciNet  Google Scholar 

  5. B. ‘Grünbaum, Convex Polytopes, Graduate Texts in Mathematics, Vol. 221, Springer, New York, 2003.

    Book  Google Scholar 

  6. B. Grünbaum and G. C. Shephard, Convex polytopes, Bulletin of the London Mathematical Society 1 (1969) 257–300.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. G. Pineda-Villavicencio, J. Ugon and D. Yost, Lower bound theorems for general polytopes, European Journal of Combinatorics 79 (2019) 27–45.

    Article  MathSciNet  Google Scholar 

  9. R. P. Stanley, The number of faces of a simplicial convex polytope, Advances in Mathematics 35 (1980) 236–238.

    Article  MathSciNet  Google Scholar 

  10. G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, Vol. 152, Springer, New York, 1995.

    Book  Google Scholar 

Download references

Acknowledgments

The author would like to thank Isabella Novik for encouraging her to explore this problem and having taken many hours to discuss and to help revise the first few drafts of this paper. The author is also grateful to Steve Klee for numerous comments and suggestions on the draft, and to Günter Ziegler and Guillermo Pineda-Villavicencio for taking the time to look at the previous version of the paper and to provide feedback. We also thank the referee for providing helpful suggestions, which led to a simpler proof of Theorem 4.3.

Author information

Authors and Affiliations

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

    Lei Xue

Authors
  1. Lei Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Xue.

Additional information

This research was partially supported by a graduate fellowship from NSF grant DMS-1664865.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xue, L. A Proof of Grünbaum’s Lower Bound Conjecture for general polytopes. Isr. J. Math. 245, 991–1000 (2021). https://doi.org/10.1007/s11856-021-2234-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2234-x

Search

Navigation